David Poole
A
M 0 'D E R N
I N T R 0 D U, C T I 0 N
4th edition
Fourth edition
David Poole
Trent University
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Linear Algebra
A Modern Introduction, 4th Edition
David Poole
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Printed in the United States of America
1 2 3 4 5 6 7 17 16 15 14 13
Dedicated to the memory of
Peter Hilton, who was an
exemplary mathematician,
educator, and citizen-a unit
vector in every sense.
Contents
vii
Preface
To the Instructor
xvii
To the Student
xxiii
Chapter 1
Vectors
1 .0
1.1
1 .2
1 .3
1 .4
Chapter 2
Introduction: The Racetrack Game
The Geometry and Algebra of Vectors
Length and Angle: The Dot Product
Exploration: Vectors and Geometry
Lines and Planes
34
2.3
2.4
2.5
3
18
32
Exploration: The Cross Product
48
Writing Project: The Origins of the Dot Product and Cross Product
50
Applications
Force Vectors
50
55
Chapter Review
Systems of Linear Equations
2.0
2.1
2.2
iv
1
57
Introduction: Triviality
57
Introduction to Systems o f Linear Equations
Direct Methods for Solving Linear Systems
58
64
Writing Project: A History of Gaussian Elimination
Explorations: Lies My Computer Told Me
83
84
Partial Pivoting
Counting Operations: An Introduction to the
Analysis of Algorithms
85
88
Spanning Sets and Linear Independence
99
Applications
Allocation of Resources
99
101
Balancing Chemical Equations
Network Analysis
1 02
1 04
Electrical Networks
Linear Economic Models
1 07
Finite Linear Games
1 09
Vignette: The Global Positioning System
121
Iterative Methods for Solving Linear Systems
134
Chapter Review
1 24
82
49
Chapter 3
Chapter 4
Contents
V
Writing Project: Which Came First: The Matrix or the Determinant?
Vignette: Lewis Carroll's Condensation Method
284
Exploration: Geometric Applications of Determinants
286
Eigenvalues and Eigenvectors of n X n Matrices
292
Writing Project: The History of Eigenvalues
30 1
283
Matrices
136
3.0
3. 1
3.2
3.3
3.4
3.5
3.6
136
Introduction: Matrices in Action
138
Matrix Operations
1 54
Matrix Algebra
1 63
The Inverse of a Matrix
The LU Factorization
1 80
Subspaces, Basis, Dimension, and Rank
Introduction to Linear Transformations
3.7
Applications
230
Markov Chains
230
Linear Economic Models
235
Population Growth
239
Graphs and Digraphs
24 1
251
Chapter Review
Vignette: Robotics
226
Eigenvalues and Eigenvectors
4.0
4. 1
4.2
4.3
4.4
4.5
4.6
191
211
253
Introduction: A Dynamical System on Graphs
Introduction to Eigenvalues and Eigenvectors
263
Determinants
253
254
30 1
Similarity and Diagonalization
31 1
Iterative Methods for Computing Eigenvalues
Applications and the Perron-Frobenius Theorem
325
325
Markov Chains
Population Growth
330
The Perron-Frobenius Theorem
332
Linear Recurrence Relations
335
Systems of Linear Differential Equations
340
Discrete Linear Dynamical Systems
348
356
Vignette: Ranking Sports Teams and Searching the Internet
Chapter Review
Chapter 5
Orthogonality
5.0
5.1
5.2
5.3
5.4
5.5
364
366
Introduction: Shadows on a Wall
366
Orthogonality in IR "
368
Orthogonal Complements and Orthogonal Projections
The Gram-Schmidt Process and the QR Factorization
Explorations: The Modified QR Factorization
396
Approximating Eigenvalues with the QR Algorithm
Orthogonal Diagonalization of Symmetric Matrices
408
Applications
Quadratic Forms
408
415
Graphing Quadratic Equations
425
Chapter Review
378
388
398
400
Vi
Contents
Chapter 6
Vector Spaces
6.0
6. 1
Introduction: Fibonacci in (Vector) Space
Vector Spaces and Subspaces
429
6.2
Linear Independence, Basis, and Dimension
6.3
6.4
6.5
6.6
6.7
Chapter 7
Chapter 8
427
427
Writing Project: The Rise of Vector Spaces
Exploration: Magic Squares
443
460
443
Change of Basis
463
472
Linear Transformations
The Kernel and Range of a Linear Transformation
The Matrix of a Linear Transformation
497
48 1
Exploration: Tilings, Lattices, and the Crystallographic Restriction
518
Applications
Homogeneous Linear Differential Equations
Chapter Review
527
Distance and Approximation
529
7.0
7. 1
529
Introduction: Taxicab Geometry
531
Inner Product Spaces
518
Explorations: Vectors and Matrices with Complex Entries
Geometric Inequalities and Optimization Problems
7.2
7.3
7.4
Norms and Distance Functions
552
Least Squares Approximation
568
The Singular Value Decomposition
590
7.5
Applications
610
Approximation o f Functions
618
Chapter Review
Vignette: Digital Image Compression
607
610
Online only
Codes
8. 1
Code Vectors
620
8.2
8.3
8.4
8.5
627
Error-Correcting Codes
632
Dual Codes
639
Linear Codes
The Minimum Distance of a Code
Vignette: The Codabar System
APPENDIX A
APPENDIXB
APPENDIXC
APPENDIXD
APPENDIX£
543
547
620
626
644
Mathematical Notation and Methods of Proof Al
Mathematical Induction
B1
Complex Numbers
Cl
Polynomials
D1
Online only
Technology Bytes
Answers to Selected Odd-Numbered Exercises
II
Index
ANS I
515
Preface
1he last thing one knows when writing a
book is what to put first.
1670
-Blaise Pascal
Pensees,
For more on the recommendations
of the Linear Algebra Curriculum
Study Group, see 1he College
41-46.
Mathematics Journal 24
(1993),
The fourth edition of Linear Algebra: A Modern Introduction preserves the approach
and features that users found to be strengths of the previous editions. However, I have
streamlined the text somewhat, added numerous clarifications, and freshened up the
exercises.
I want students to see linear algebra as an exciting subject and to appreciate its
tremendous usefulness. At the same time, I want to help them master the basic con cepts and techniques of linear algebra that they will need in other courses, both in
mathematics and in other disciplines. I also want students to appreciate the interplay
of theoretical, applied, and numerical mathematics that pervades the subject.
This book is designed for use in an introductory one- or two-semester course
sequence in linear algebra. First and foremost, it is intended for students, and I have
tried my best to write the book so that students not only will find it readable but also
will want to read it. As in the first three editions, I have taken into account the reality
that students taking introductory linear algebra are likely to come from a variety of
disciplines. In addition to mathematics majors, there are apt to be majors from
engineering, physics, chemistry, computer science, biology, environmental science,
geography, economics, psychology, business, and education, as well as other students
taking the course as an elective or to fulfill degree requirements. Accordingly, the book
balances theory and applications, is written in a conversational style yet is fully rigorous,
and combines a traditional presentation with concern for student-centered learning.
There is no such thing as a universally best learning style. In any class, there will be
some students who work well independently and others who work best in groups;
some who prefer lecture-based learning and others who thrive in a workshop setting,
doing explorations; some who enjoy algebraic manipulations, some who are adept at
numerical calculations (with and without a computer), and some who exhibit strong
geometric intuition. In this edition, I continue to present material in a variety of
ways-algebraically, geometrically, numerically, and verbally-so that all types oflearn­
ers can find a path to follow. I have also attempted to present the theoretical, computa­
tional, and applied topics in a flexible yet integrated way. In doing so, it is my hope that
all students will be exposed to the many sides of linear algebra.
This book is compatible with the recommendations ofthe Linear Algebra Curriculum
Study Group. From a pedagogical point of view, there is no doubt that for most students
Vii
Viii
Preface
concrete examples should precede abstraction. I have taken this approach here. I also
believe strongly that linear algebra is essentially about vectors and that students need to
see vectors first (in a concrete setting) in order to gain some geometric insight. Moreover,
introducing vectors early allows students to see how systems of linear equations arise
naturally from geometric problems. Matrices then arise equally naturally as coefficient
matrices oflinear systems and as agents of change (linear transformations). This sets the
stage for eigenvectors and orthogonal projections, both of which are best understood
geometrically. The dart that appears on the cover of this book symbolizes a vector and
reflects my conviction that geometric understanding should precede computational
techniques.
I have tried to limit the number of theorems in the text. For the most part, results
labeled as theorems either will be used later in the text or summarize preceding work.
Interesting results that are not central to the book have been included as exercises or
explorations. For example, the cross product of vectors is discussed only in explo­
rations (in Chapters 1 and 4). Unlike most linear algebra textbooks, this book has no
chapter on determinants. The essential results are all in Section 4.2, with other inter­
esting material contained in an exploration. The book is, however, comprehensive for
an introductory text. Wherever possible, I have included elementary and accessible
proofs of theorems in order to avoid having to say, "The proof of this result is beyond
the scope of this text:' The result is, I hope, a work that is self-contained.
I have not been stingy with the applications: There are many more in the book than
can be covered in a single course. However, it is important that students see the impressive
range of problems to which linear algebra can be applied. I have included some modern
material on finite linear algebra and coding theory that is not normally found in an intro­
ductory linear algebra text. There are also several impressive real-world applications of
linear algebra and one item of historical, if not practical, interest; these applications are
presented as self-contained "vignettes:'
I hope that instructors will enjoy teaching from this book. More important, I hope
that students using the book will come away with an appreciation of the beauty, power,
and tremendous utility of linear algebra and that they will have fun along the way.
W h a t's New i n t h e Fo u rt h Edition
Th e overall structure and style o f Linear Algebra: A Modern Introduction remain the
same in the fourth edition.
Here is a summary of what is new:
The applications to coding theory have been moved to the new online Chapter 8.
To further engage students, five writing projects have been added to the exer­
cise sets. These projects give students a chance to research and write about aspects of
the history and development oflinear algebra. The explorations, vignettes, and many
of the applications provide additional material for student projects.
•
There are over 200 new or revised exercises. In response to reviewers' com­
ments, there is now a full proof of the Cauchy-Schwarz Inequality in Chapter 1 in the
form of a guided exercise.
•
I have made numerous small changes in wording to improve the clarity or
accuracy of the exposition. Also, several definitions have been made more explicit by
giving them their own definition boxes and a few results have been highlighted by
labeling them as theorems.
•
All existing ancillaries have been updated.
•
•
See pages 49, 82, 283, 301, 443
Preface
ix
Features
Clear Writi ng Stvle
The text is written is a simple, direct, conversational style. As much as possible, I have
used "mathematical English" rather than relying excessively on mathematical nota­
tion. However, all proofs that are given are fully rigorous, and Appendix A contains
an introduction to mathematical notation for those who wish to streamline their own
writing. Concrete examples almost always precede theorems, which are then followed
by further examples and applications. This flow-from specific to general and back
again-is consistent throughout the book.
Kev concepts Introduced Early
Many students encounter difficulty in linear algebra when the course moves from the
computational (solving systems of linear equations, manipulating vectors and matri­
ces) to the theoretical (spanning sets, linear independence, subspaces, basis, and
dimension) . This book introduces all of the key concepts of linear algebra early, in a
concrete setting, before revisiting them in full generality. Vector concepts such as dot
product, length, orthogonality, and projection are first discussed in Chapter 1 in the
concrete setting of IR 2 and IR 3 before the more general notions of inner product, norm,
and orthogonal projection appear in Chapters 5 and 7. Similarly, spanning sets and
linear independence are given a concrete treatment in Chapter 2 prior to their gener­
alization to vector spaces in Chapter 6. The fundamental concepts of subspace, basis,
and dimension appear first in Chapter 3 when the row, column, and null spaces of a
matrix are introduced; it is not until Chapter 6 that these ideas are given a general
treatment. In Chapter 4, eigenvalues and eigenvectors are introduced and explored
for 2 X 2 matrices before their n X n counterparts appear. By the beginning of Chap­
ter 4, all of the key concepts of linear algebra have been introduced, with concrete,
computational examples to support them. When these ideas appear in full generality
later in the book, students have had time to get used to them and, hence, are not so
intimidated by them.
Emphasis on Vectors and Geometry
In keeping with the philosophy that linear algebra is primarily about vectors, this
book stresses geometric intuition. Accordingly, the first chapter is about vectors, and
it develops many concepts that will appear repeatedly throughout the text. Concepts
such as orthogonality, projection, and linear combination are all found in Chapter 1 ,
as is a comprehensive treatment o f lines and planes in IR 3 that provides essential
insight into the solution of systems of linear equations. This emphasis on vectors,
geometry, and visualization is found throughout the text. Linear transformations are
introduced as matrix transformations in Chapter 3, with many geometric examples,
before general linear transformations are covered in Chapter 6. In Chapter 4, eigen­
values are introduced with "eigenpictures" as a visual aid. The proof of Perron's
Theorem is given first heuristically and then formally, in both cases using a geometric
argument. The geometry of linear dynamical systems reinforces and summarizes the
material on eigenvalues and eigenvectors. In Chapter 5, orthogonal projections, or­
thogonal complements of subspaces, and the Gram-Schmidt Process are all presented
in the concrete setting of IR 3 before being generalized to IR " and, in Chapter 7, to inner
X
Preface
product spaces. The nature of the singular value decomposition is also explained in­
formally in Chapter 7 via a geometric argument. Of the more than 300 figures in the
text, over 200 are devoted to fostering a geometric understanding of linear algebra.
Exploralions
See pages 1, 136, 427, 529
See pages 32, 286, 460, 515, 543, 547
See pages 83, 84, 85, 396, 398
The introduction to each chapter is a guided exploration (Section O) in which stu­
dents are invited to discover, individually or in groups, some aspect of the upcoming
chapter. For example, "The Racetrack Game" introduces vectors, "Matrices in Action"
introduces matrix multiplication and linear transformations, "Fibonacci in (Vector)
Space" touches on vector space concepts, and "Taxicab Geometry" sets up general­
ized norms and distance functions. Additional explorations found throughout the
book include applications of vectors and determinants to geometry, an investigation
of 3 X 3 magic squares, a study of symmetry via the tilings of M. C. Escher, an intro­
duction to complex linear algebra, and optimization problems using geometric
inequalities. There are also explorations that introduce important numerical consid­
erations and the analysis of algorithms. Having students do some of these explo­
rations is one way of encouraging them to become active learners and to give them
"ownership" over a small part of the course.
APPlicalions
See pages 623, 641
See pages 121, 226, 356, 607, 626
The book contains an abundant selection of applications chosen from a broad range
of disciplines, including mathematics, computer science, physics, chemistry, engi­
neering, biology, business, economics, psychology, geography, and sociology. Note­
worthy among these is a strong treatment of coding theory, from error-detecting
codes (such as International Standard Book Numbers) to sophisticated error­
correcting codes (such as the Reed-Muller code that was used to transmit satellite
photos from space). Additionally, there are five "vignettes" that briefly showcase some
very modern applications of linear algebra: the Global Positioning System (GPS), ro­
botics, Internet search engines, digital image compression, and the Codabar System.
Examples and Exercises
See pages 248, 359, 526, 588
There are over 400 examples in this book, most worked in greater detail than is cus­
tomary in an introductory linear algebra textbook. This level of detail is in keeping
with the philosophy that students should want (and be able) to read a textbook.
Accordingly, it is not intended that all of these examples be covered in class; many can
be assigned for individual or group study, possibly as part of a project. Most examples
have at least one counterpart exercise so that students can try out the skills covered in
the example before exploring generalizations.
There are over 2000 exercises, more than in most textbooks at a similar level.
Answers to most of the computational odd-numbered exercises can be found in the
back of the book. Instructors will find an abundance of exercises from which to select
homework assignments. The exercises in each section are graduated, progressing from
the routine to the challenging. Exercises range from those intended for hand computa­
tion to those requiring the use of a calculator or computer algebra system, and from
theoretical and numerical exercises to conceptual exercises. Many of the examples and
exercises use actual data compiled from real-world situations. For example, there are
problems on modeling the growth of caribou and seal populations, radiocarbon dating
Preface
Xi
of the Stonehenge monument, and predicting major league baseball players' salaries.
Working such problems reinforces the fact that linear algebra is a valuable tool for mod­
eling real-life problems.
Additional exercises appear in the form of a review after each chapter. In each set,
there are 10 true/false questions designed to test conceptual understanding, followed
by 1 9 computational and theoretical exercises that summarize the main concepts and
techniques of that chapter.
Biographical Sketches and Etvmological Notes
See page 34
It is important that students learn something about the history of mathematics and
come to see it as a social and cultural endeavor as well as a scientific one. Accord­
ingly, the text contains short biographical sketches about many of the mathemati­
cians who contributed to the development of linear algebra. I hope that these will
help to put a human face on the subject and give students another way of relating to
the material.
I have found that many students feel alienated from mathematics because the
terminology makes no sense to them-it is simply a collection of words to be learned.
To help overcome this problem, I have included short etymological notes that give
the origins of many of the terms used in linear algebra. (For example, why do we
use the word normal to refer to a vector that is perpendicular to a plane?)
Margin Icons
The margins of the book contain several icons whose purpose is to alert the reader in
various ways. Calculus is not a prerequisite for this book, but linear algebra has many
interesting and important applications to calculus. The � icon denotes an example or
exercise that requires calculus. (This material can be omitted if not everyone in the
class has had at least one semester of calculus. Alternatively, this material can be as­
signed as projects.) The � icon denotes an example or exercise involving complex
numbers. (For students unfamiliar with complex numbers, Appendix C contains all
the background material that is needed.) The cAs icon indicates that a computer algebra
system (such as Maple, Mathematica, or MATLAB) or a calculator with matrix capa­
bilities (such as almost any graphing calculator) is required-or at least very useful­
for solving the example or exercise.
In an effort to help students learn how to read and use this textbook most ef­
fectively, I have noted various places where the reader is advised to pause. These
may be places where a calculation is needed, part of a proof must be supplied, a
claim should be verified, or some extra thought is required. The _.... icon appears
in the margin at such places; the message is "Slow down. Get out your pencil.
Think about this:'
Technology
This book can be used successfully whether or not students have access to technol­
ogy. However, calculators with matrix capabilities and computer algebra systems
are now commonplace and, properly used, can enrich the learning experience as
well as help with tedious calculations. In this text, I take the point of view that stu­
dents need to master all of the basic techniques of linear algebra by solving by hand
examples that are not too computationally difficult. Technology may then be used
Xii
Preface
(in whole or in part) to solve subsequent examples and applications and to apply
techniques that rely on earlier ones. For example, when systems of linear equations
are first introduced, detailed solutions are provided; later, solutions are simply
given, and the reader is expected to verify them. This is a good place to use some
form of technology. Likewise, when applications use data that make hand calcula­
tion impractical, use technology. All of the numerical methods that are discussed
depend on the use of technology.
With the aid of technology, students can explore linear algebra in some exciting
ways and discover much for themselves. For example, if one of the coefficients of a
linear system is replaced by a parameter, how much variability is there in the solu­
tions? How does changing a single entry of a matrix affect its eigenvalues? This book
is not a tutorial on technology, and in places where technology can be used, I have not
specified a particular type of technology. The student companion website that
accompanies this book offers an online appendix called Technology Bytes that gives
instructions for solving a selection of examples from each chapter using Maple, Math­
ematica, and MATLAB. By imitating these examples, students can do further calcula­
tions and explorations using whichever CAS they have and exploit the power of these
systems to help with the exercises throughout the book, particularly those marked
with the cAs icon. The website also contains data sets and computer code in Maple,
Mathematica, and MATLAB formats keyed to many exercises and examples in the
text. Students and instructors can import these directly into their CAS to save typing
and eliminate errors.
Finite and Numerical linear Algebra
See pages 83, 84, 124, 180, 311, 392,
555, 561, 568, 590
See pages 319, 563, 600
The text covers two aspects of linear algebra that are scarcely ever mentioned to­
gether: finite linear algebra and numerical linear algebra. By introducing modular
arithmetic early, I have been able to make finite linear algebra (more properly, "linear
algebra over finite fields;' although I do not use that phrase) a recurring theme
throughout the book. This approach provides access to the material on coding theory
in Chapter 8 (online) . There is also an application to finite linear games in Section 2.4
that students really enjoy. In addition to being exposed to the applications of finite
linear algebra, mathematics majors will benefit from seeing the material on finite
fields, because they are likely to encounter it in such courses as discrete mathematics,
abstract algebra, and number theory.
All students should be aware that in practice, it is impossible to arrive at exact
solutions of large-scale problems in linear algebra. Exposure to some of the tech­
niques of numerical linear algebra will provide an indication of how to obtain
highly accurate approximate solutions. Some of the numerical topics included in
the book are roundoff error and partial pivoting, iterative methods for solving
linear systems and computing eigenvalues, the LU and QR factorizations, matrix
norms and condition numbers, least squares approximation, and the singular
value decomposition. The inclusion of numerical linear algebra also brings up
some interesting and important issues that are completely absent from the theory
of linear algebra, such as pivoting strategies, the condition of a linear system, and
the convergence of iterative methods. This book not only raises these questions
but also shows how one might approach them. Gerschgorin disks, matrix norms,
and the singular values of a matrix, discussed in Chapters 4 and 7, are useful in
this regard.
Preface
Xiii
Appendices
Appendix A contains an overview of mathematical notation and methods of proof,
and Appendix B discusses mathematical induction. All students will benefit from
these sections, but those with a mathematically oriented major may wish to pay
particular attention to them. Some of the examples in these appendices are uncom­
mon (for instance, Example B.6 in Appendix B) and underscore the power of the
methods. Appendix C is an introduction to complex numbers. For students familiar
with these results, this appendix can serve as a useful reference; for others, this sec­
tion contains everything they need to know for those parts of the text that use com­
plex numbers. Appendix D is about polynomials. I have found that many students
require a refresher about these facts. Most students will be unfamiliar with Descartes's
Rule of Signs; it is used in Chapter 4 to explain the behavior of the eigenvalues of
Leslie matrices. Exercises to accompany the four appendices can be found on the
book's website.
Short answers to most of the odd-numbered computational exercises are given at
the end of the book. Exercise sets to accompany Appendixes A, B, C, and D are avail­
able on the companion website, along with their odd-numbered answers.
Ancillaries
For 1ns1ruc1ors
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Complete Solutions Manual
The Complete Solutions Manual provides detailed solutions to all exercises in the
text, including Exploration and Chapter Review exercises. The Complete Solutions
Manual is available online.
Instructor's Guide
This online guide enhances the text with valuable teaching resources such as group
work projects, teaching tips, interesting exam questions, examples and extra
xiv
Preface
material for lectures, and other items designed to reduce the instructor's prepara­
tion time and make linear algebra class an exciting and interactive experience. For
each section of the text, the Instructor's Guide includes suggested time and empha­
sis, points to stress, questions for discussion, lecture materials and examples, tech­
nology tips, student projects, group work with solutions, sample assignments, and
suggested test questions.
Solution Builder
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Solution Builder allows instructors to create customized, secure PDF printouts of
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*Access Cognero and additional instructor resources online at login.cengage.com.
For Students
Student Solutions Manual (ISBN- 13: 978 - 1 -285-84 195-3)
The Student Solutions Manual and Study Guide includes detailed solutions to all odd­
numbered exercises and selected even-numbered exercises; section and chapter
summaries of symbols, definitions, and theorems; and study tips and hints. Complex
exercises are explored through a question-and-answer format designed to deepen
understanding. Challenging and entertaining problems that further explore selected
exercises are also included.
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Acknowledgments
The reviewers of the previous edition of this text contributed valuable and often in­
sightful comments about the book. I am grateful for the time each of them took to do
this. Their judgement and helpful suggestions have contributed greatly to the devel­
opment and success of this book, and I would like to thank them personally:
Jamey Bass, City College of San Francisco; Olga Brezhneva, Miami University; Karen
Clark, The College of New Jersey; Marek Elzanowski, Portland State University;
Christopher Francisco, Oklahoma State University; Brian Jue, California State
University, Stanislaus; Alexander Kheyfits, Bronx Community College/CUNY;
Henry Krieger, Harvey Mudd College; Rosanna Pearlstein, Michigan State
To the Instructor
See page 284
See page 286
See pages 325, 330
See page 348
xix
course" in determinants contains all the essential material students need, including
an optional but elementary proof of the Laplace Expansion Theorem. The vignette
"Lewis Carroll's Condensation Method" presents a historically interesting, alternative
method of calculating determinants that students may find appealing. The explo­
ration "Geometric Applications of Determinants" makes a nice project that contains
several interesting and useful results. (Alternatively, instructors who wish to give
more detailed coverage to determinants may choose to cover some of this exploration
in class.) The basic theory of eigenvalues and eigenvectors is found in Section 4.3, and
Section 4.4 deals with the important topic of diagonalization. Example 4.29 on powers
of matrices is worth covering in class. The power method and its variants, discussed
in Section 4.5, are optional, but all students should be aware of the method, and an
applied course should cover it in detail. Gerschgorin's Disk Theorem can be covered
independently of the rest of Section 4.5. Markov chains and the Leslie model of pop­
ulation growth reappear in Section 4.6. Although the proof of Perron's Theorem is
optional, the theorem itself (like the stronger Perron-Frobenius Theorem) should at
least be mentioned because it explains why we should expect a unique positive eigen­
value with a corresponding positive eigenvector in these applications. The applica­
tions on recurrence relations and differential equations connect linear algebra to dis­
crete mathematics and calculus, respectively. The matrix exponential can be covered
if your class has a good calculus background. The final topic of discrete linear dynamical systems revisits and summarizes many of the ideas in Chapter 4, looking at them
in a new, geometric light. Students will enjoy reading how eigenvectors can be used to
help rank sports teams and websites. This vignette can easily be extended to a project
or enrichment activity.
Chapter 5: onhouonalilv
See page 366
See pages 396, 398
See pages 408, 415
The introductory exploration, "Shadows on a Wall;' is mathematics at its best: it takes
a known concept (projection of a vector onto another vector) and generalizes it in a
useful way (projection of a vector onto a subspace-a plane), while uncovering some
previously unobserved properties. Section 5 . 1 contains the basic results about or­
thogonal and orthonormal sets of vectors that will be used repeatedly from here on.
In particular, orthogonal matrices should be stressed. In Section 5.2, two concepts
from Chapter 1 are generalized: the orthogonal complement of a subspace and the
orthogonal projection of a vector onto a subspace. The Orthogonal Decomposition
Theorem is important here and helps to set up the Gram-Schmidt Process. Also note
the quick proof of the Rank Theorem. The Gram-Schmidt Process is detailed in
Section 5.3, along with the extremely important QR factorization. The two explo­
rations that follow outline how the QR factorization is computed in practice and how
it can be used to approximate eigenvalues. Section 5.4 on orthogonal diagonalization
of (real) symmetric matrices is needed for the applications that follow. It also contains
the Spectral Theorem, one of the highlights of the theory of linear algebra. The appli­
cations in Section 5.5 are quadratic forms and graphing quadratic equations. I always
include at least the second of these in my course because it extends what students al­
ready know about conic sections.
Chapter 6: vector Spaces
See page 427
The Fibonacci sequence reappears in Section 6.0, although it is not important that
students have seen it before (Section 4.6). The purpose of this exploration is to show
Preface
XV
University; William Sullivan, Portland State University; Matthias Weber, Indiana
University.
I am indebted to a great many people who have, over the years, influenced my
views about linear algebra and the teaching of mathematics in general. First, I would
like to thank collectively the participants in the education and special linear algebra
sessions at meetings of the Mathematical Association of America and the Canadian
Mathematical Society. I have also learned much from participation in the Canadian
Mathematics Education Study Group and the Canadian Mathematics Education
Forum.
I especially want to thank Ed Barbeau, Bill Higginson, Richard Hoshino, John
Grant McLaughlin, Eric Muller, Morris Orzech, Bill Ralph, Pat Rogers, Peter Taylor,
and Walter Whiteley, whose advice and inspiration contributed greatly to the
philosophy and style of this book. My gratitude as well to Robert Rogers, who devel­
oped the student and instructor solutions, as well as the excellent study guide content.
Special thanks go to Jim Stewart for his ongoing support and advice. Joe Rotman and
his lovely book A First Course in Abstract Algebra inspired the etymological notes in
this book, and I relied heavily on Steven Schwartzman's The Words of Mathematics
when compiling these notes. I thank Art Benjamin for introducing me to the Codabar
system and Joe Grear for clarifying aspects of the history of Gaussian elimination. My
colleagues Marcus Pivato and Reem Yassawi provided useful information about dy­
namical systems. As always, I am grateful to my students for asking good questions
and providing me with the feedback necessary to becoming a better teacher.
I sincerely thank all of the people who have been involved in the production of
this book. Jitendra Kumar and the team at MPS Limited did an amazing job produc­
ing the fourth edition. I thank Christine Sabooni for doing a thorough copyedit. Most
of all, it has been a delight to work with the entire editorial, marketing, and produc­
tion teams at Cengage Learning: Richard Stratton, Molly Taylor, Laura Wheel, Cynthia
Ashton, Danielle Hallock, Andrew Coppola, Alison Eigel Zade, and Janay Pryor. They
offered sound advice about changes and additions, provided assistance when I needed
it, but let me write the book I wanted to write. I am fortunate to have worked with
them, as well as the staffs on the first through third editions.
As always, I thank my family for their love, support, and understanding. Without
them, this book would not have been possible.
David Poole
dpoole@trentu.ca
To the
Instructor
"Would you tell me, please,
which way I ought to go from here?"
"That depends a good deal on where
you want to get to," said the Cat.
-Lewis Carroll
1865
Alice's Adventures in
Wonderland,
This text was written with flexibility in mind. It is intended for use in a one- or
two-semester course with 36 lectures per semester. The range of topics and applica­
tions makes it suitable for a variety of audiences and types of courses. However,
there is more material in the book than can be covered in class, even in a two­
semester course. After the following overview of the text are some brief suggestions
for ways to use the book.
An overview of t h e Text
Chanler 1: vec1ors
See page 1
See page 32
See page 48
The racetrack game in Section 1 .0 serves to introduce vectors in an informal way. (It's also
quite a lot of fun to play!) Vectors are then formally introduced from both algebraic and
geometric points of view. The operations of addition and scalar multiplication and their
properties are first developed in the concrete settings of !R 2 and IR3 before being general­
ized to !R n . Modular arithmetic and finite linear algebra are also introduced. Section 1 .2
defines the dot product of vectors and the related notions of length, angle, and orthogo­
nality. The very important concept of (orthogonal) projection is developed here; it will
reappear in Chapters 5 and 7. The exploration "Vectors and Geometry" shows how vec­
tor methods can be used to prove certain results in Euclidean geometry. Section 1 .3 is a
basic but thorough introduction to lines and planes in IR 2 and IR3. This section is crucial
for understanding the geometric significance of the solution of linear systems in Chap­
ter 2. Note that the cross product of vectors in IR3 is left as an exploration. The chapter
concludes with an application to force vectors.
Chapter 2: svstems of linear Equations
See page 57
The introduction to this chapter serves to illustrate that there is more than one way to
think of the solution to a system of linear equations. Sections 2 . 1 and 2.2 develop the
xv i i
xv i i i
To the Instructor
See pages 72, 205, 386, 486
See page 121
See pages 83, 84, 85
main computational tool for solving linear systems: row reduction of matrices (Gaus­
sian and Gauss-Jordan elimination) . Nearly all subsequent computational methods in
the book depend on this. The Rank Theorem appears here for the first time; it shows
up again, in more generality, in Chapters 3, 5, and 6. Section 2.3 is very important; it
introduces the fundamental notions of spanning sets and linear independence of vec­
tors. Do not rush through this material. Section 2.4 contains six applications from
which instructors can choose depending on the time available and the interests of the
class. The vignette on the Global Positioning System provides another application
that students will enjoy. The iterative methods in Section 2.5 will be optional for many
courses but are essential for a course with an applied/numerical focus. The three ex­
plorations in this chapter are related in that they all deal with aspects of the use of
computers to solve linear systems. All students should at least be made aware of these
issues.
Chanler 3: Malrices
See page 136
See pages 172, 206, 296, 512, 605
See page 226
See pages 230, 239
This chapter contains some of the most important ideas in the book. It is a long
chapter, but the early material can be covered fairly quickly, with extra time allowed
for the crucial material in Section 3.5. Section 3.0 is an exploration that introduces
the notion of a linear transformation: the idea that matrices are not just static objects
but rather a type of function, transforming vectors into other vectors. All of the basic
facts about matrices, matrix operations, and their properties are found in the first two
sections. The material on partitioned matrices and the multiple representations of the
matrix product is worth stressing, because it is used repeatedly in subsequent sections.
The Fundamental Theorem of Invertible Matrices in Section 3.3 is very important
and will appear several more times as new characterizations of invertibility are pre­
sented. Section 3.4 discusses the very important LU factorization of a matrix. If this
topic is not covered in class, it is worth assigning as a project or discussing in a work­
shop. The point of Section 3.5 is to present many of the key concepts of linear algebra
(subspace, basis, dimension, and rank) in the concrete setting of matrices before stu dents see them in full generality. Although the examples in this section are all famil­
iar, it is important that students get used to the new terminology and, in particular,
understand what the notion of a basis means. The geometric treatment of linear
transformations in Section 3.6 is intended to smooth the transition to general linear
transformations in Chapter 6. The example of a projection is particularly important
because it will reappear in Chapter 5. The vignette on robotic arms is a concrete
demonstration of composition of linear (and affine) transformations. There are four
applications from which to choose in Section 3.7. Either Markov chains or the Leslie
model of population growth should be covered so that they can be used again in
Chapter 4, where their behavior will be explained.
Chanler 4: Eigenvalues and Eigenveclors
See page 253
The introduction Section 4.0 presents an interesting dynamical system involving
graphs. This exploration introduces the notion of an eigenvector and foreshadows the
power method in Section 4.5. In keeping with the geometric emphasis of the book,
Section 4. 1 contains the novel feature of "eigenpictures" as a way of visualizing the
eigenvectors of 2 X 2 matrices. Determinants appear in Section 4.2, motivated by
their use in finding the characteristic polynomials of small matrices. This "crash
XX
To the Instructor
See page 515
that familiar vector space concepts (Section 3.5) can be used fruitfully in a new
setting. Because all of the main ideas of vector spaces have already been introduced in
Chapters 1 -3, students should find Sections 6. 1 and 6.2 fairly familiar. The emphasis
here should be on using the vector space axioms to prove properties rather than rely­
ing on computational techniques. When discussing change of basis in Section 6.3, it
is helpful to show students how to use the notation to remember how the construc­
tion works. Ultimately, the Gauss-Jordan method is the most efficient here. Sec­
tions 6.4 and 6.5 on linear transformations are important. The examples are related to
previous results on matrices (and matrix transformations) . In particular, it is impor­
tant to stress that the kernel and range of a linear transformation generalize the null
space and column space of a matrix. Section 6.6 puts forth the notion that (almost) all
linear transformations are essentially matrix transformations. This builds on the
information in Section 3.6, so students should not find it terribly surprising. However,
the examples should be worked carefully. The connection between change of basis
and similarity of matrices is noteworthy. The exploration "Tilings, Lattices, and the
Crystallographic Restriction" is an impressive application of change of basis. The con­
nection with the artwork of M. C. Escher makes it all the more interesting. The appli­
cations in Section 6.7 build on previous ones and can be included as time and interest
permit.
Chapter 1: Distance and Approximation
See page 529
See page 543
See page 547
See page 607
Section 7 .0 opens with the entertaining "Taxicab Geometry" exploration. Its
purpose is to set up the material on generalized norms and distance functions
(metrics) that follows. Inner product spaces are discussed in Section 7 . 1 ; the em­
phasis here should be on the examples and using the axioms. The exploration "Vec­
tors and Matrices with Complex Entries" shows how the concepts of dot product,
symmetric matrix, orthogonal matrix, and orthogonal diagonalization can be ex­
tended from real to complex vector spaces. The following exploration, "Geometric
Inequalities and Optimization Problems:' is one that students typically enjoy. (They
will have fun seeing how many "calculus" problems can be solved without using
calculus at all!) Section 7.2 covers generalized vector and matrix norms and shows
how the condition number of a matrix is related to the notion of ill-conditioned
linear systems explored in Chapter 2. Least squares approximation (Section 7.3) is
an important application of linear algebra in many other disciplines. The Best Ap­
proximation Theorem and the Least Squares Theorem are important, but their
proofs are intuitively clear. Spend time here on the examples-a few should suffice.
Section 7.4 presents the singular value decomposition, one of the most impressive
applications of linear algebra. If your course gets this far, you will be amply re­
warded. Not only does the SVD tie together many notions discussed previously; it
also affords some new (and quite powerful) applications. If a CAS is available, the
vignette on digital image compression is worth presenting; it is a visually impres­
sive display of the power of linear algebra and a fitting culmination to the course.
The further applications in Section 7.5 can be chosen according to the time avail­
able and the interests of the class.
Chapter 8: Codes
This online chapter contains applications of linear algebra to the theory of codes.
Section 8 . 1 begins with a discussion of how vectors can be used to design
To the Instructor
See page 626
XXi
error- detecting codes such as the familiar Universal Product Code (UPC) and
International Standard Book Number (ISBN). This topic only requires knowl­
edge of Chapter 1 . The vignette on the Codabar system used in credit and bank
cards is an excellent classroom presentation that can even be used to introduce
Section 8 . 1 . Once students are familiar with matrix operations, Section 8.2 de­
scribes how codes can be designed to correct as well as detect errors. The
Hamming codes introduced here are perhaps the most famous examples of such
error- correcting codes. Dual codes, discussed in Section 8.3, are an important
way of constructing new codes from old ones. The notion of orthogonal comple­
ment, introduced in Chapter 5, is the prerequisite concept here. The most
important, and most widely used, class of codes is the class of linear codes that is
defined in Section 8.4. The notions of subspace, basis, and dimension are key
here. The powerful Reed-Muller codes used by NASA spacecraft are important
examples of linear codes. Our discussion of codes concludes in Section 8.5 with
the definition of the minimum distance of a code and the role it plays in deter­
mining the error-correcting capability of the code.
H o w to u s e t h e B o o k
Students find the book easy to read, s o I usually have them read a section before I
cover the material in class. That way, I can spend class time highlighting the most
important concepts, dealing with topics students find difficult, working examples,
and discussing applications. I do not attempt to cover all of the material from the
assigned reading in class. This approach enables me to keep the pace of the course
fairly brisk, slowing down for those sections that students typically find
challenging.
In a two-semester course, it is possible to cover the entire book, including a rea­
sonable selection of applications. For extra flexibility, you might omit some of the
topics (for example, give only a brief treatment of numerical linear algebra), thereby
freeing up time for more in-depth coverage of the remaining topics, more applica­
tions, or some of the explorations. In an honors mathematics course that emphasizes
proofs, much of the material in Chapters 1 -3 can be covered quickly. Chapter 6 can
then be covered in conjunction with Sections 3.5 and 3.6, and Chapter 7 can be in­
tegrated into Chapter 5. I would be sure to assign the explorations in Chapters 1, 4,
6, and 7 for such a class.
For a one-semester course, the nature of the course and the audience will deter­
mine which topics to include. Three possible courses are described below and on the
following page. The basic course, described first, has fewer than 36 hours suggested,
allowing time for extra topics, in-class review, and tests. The other two courses build
on the basic course but are still quite flexible.
A Basic Course
A course designed for mathematics majors and students from other disciplines is
outlined on the next page. This course does not mention general vector spaces at all
(all concepts are treated in a concrete setting) and is very light on proofs. Still, it is a
thorough introduction to linear algebra.
XXii
To the Instructor
Section
1.1
1.2
1.3
2.1
2.2
2.3
3.1
3.2
3.3
3.5
Number of Lectures
1 - 1 .5
1 - 1 .5
0.5- 1
1 -2
1 -2
1-2
2
2
Section
Number of Lectures
3.6
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
7.3
1 -2
1
2
1
1 -2
1 - 1 .5
1 - 1 .5
0.5
1
2
Total: 23-30 lectures
Because the students in a course such as this one represent a wide variety of dis­
ciplines, I would suggest using much of the remaining lecture time for applications.
In my course, I do code vectors in Section 8. 1 , which students really seem to like, and
at least one application from each of Chapters 2-5. Other applications can be as­
signed as projects, along with as many of the explorations as desired. There is also
sufficient lecture time available to cover some of the theory in detail.
A Course with a Comoulalional Emphasis
For a course with a computational emphasis, the basic course outlined on the previous
page can be supplemented with the sections of the text dealing with numerical linear
algebra. In such a course, I would cover part or all of Sections 2.5, 3.4, 4.5, 5.3, 7.2, and
7.4, ending with the singular value decomposition. The explorations in Chapters 2
and 5 are particularly well suited to such a course, as are almost any of the applications.
A course tor Sludenls Who Have Already
SIUdied Some linear Algebra
Some courses will be aimed at students who have already encountered the basic prin­
ciples of linear algebra in other courses. For example, a college algebra course will
often include an introduction to systems of linear equations, matrices, and deter­
minants; a multivariable calculus course will almost certainly contain material on
vectors, lines, and planes. For students who have seen such topics already, much early
material can be omitted and replaced with a quick review. Depending on the back­
ground of the class, it may be possible to skim over the material in the basic course up
to Section 3.3 in about six lectures. If the class has a significant number of mathemat­
ics majors (and especially if this is the only linear algebra course they will take),
I would be sure to cover Sections 6. 1 - 6.5, 7. 1 , and 7.4 and as many applications as time
permits. If the course has science majors (but not mathematics majors), I would cover
Sections 6. 1 and 7. 1 and a broader selection of applications, being sure to include the
material on differential equations and approximation of functions. If computer sci­
ence students or engineers are prominently represented, I would try to do as much of
the material on codes and numerical linear algebra as I could.
There are many other types of courses that can successfully use this text. I hope
that you find it useful for your course and that you enjoy using it.
To the
Student
"Where shall I begin, please your
Majesty?" he asked.
"Begin at the beginning," the King
said, gravely, "and go on till you come
to the end: then stop."
-Lewis Carroll
1865
Alice's Adventures in
Wonderland,
Linear algebra is an exciting subject. It is full of interesting results, applications to
other disciplines, and connections to other areas of mathematics. The Student Solu­
tions Manual and Study Guide contains detailed advice on how best to use this book;
following are some general suggestions.
Linear algebra has several sides: There are computational techniques, concepts, and
applications. One of the goals of this book is to help you master all of these facets of
the subject and to see the interplay among them. Consequently, it is important that
you read and understand each section of the text before you attempt the exercises in
that section. If you read only examples that are related to exercises that have been
assigned as homework, you will miss much. Make sure you understand the defini­
tions of terms and the meaning of theorems. Don't worry if you have to read some­
thing more than once before you understand it. Have a pencil and calculator with you
as you read. Stop to work out examples for yourself or to fill in missing calculations.
The � icon in the margin indicates a place where you should pause and think over
what you have read so far.
Answers to most odd-numbered computational exercises are in the back of the
book. Resist the temptation to look up an answer before you have completed a ques­
tion. And remember that even if your answer differs from the one in the back, you
may still be right; there is more than one correct way to express some of the solutions.
For example, a value of l / v2 can also be expressed as v2/2 and the set of all scalar
multiples of the vector
[� ]
[� ] .
is the same as the set of all scalar multiples of
1 2
As you encounter new concepts, try to relate them to examples that you know.
Write out proofs and solutions to exercises in a logical, connected way, using com plete sentences. Read back what you have written to see whether it makes sense.
Better yet, if you can, have a friend in the class read what you have written. If it doesn't
make sense to another person, chances are that it doesn't make sense, period.
You will find that a calculator with matrix capabilities or a computer algebra sys­
tem is useful. These tools can help you to check your own hand calculations and are
indispensable for some problems involving tedious computations. Technology also
xx i i i
XXiV
To the Student
enables you to explore aspects of linear algebra on your own. You can play "what if?"
games: What if I change one of the entries in this vector? What if this matrix is of a
different size? Can I force the solution to be what I would like it to be by changing
something? To signal places in the text or exercises where the use of technology is
recommended, I have placed the icon cAs in the margin. The companion website that
accompanies this book contains computer code working out selected exercises from
the book using Maple, Mathematica, and MATLAB, as well as Technology Bytes, an
appendix providing much additional advice about the use of technology in linear
algebra.
You are about to embark on a journey through linear algebra. Think of this book
as your travel guide. Are you ready? Let's go!
Vectors
Here they come pouring out of the blue.
Little arrowsfor me andfor you.
-Albert Hammond and
Mike Hazelwood
1968
Little Arrows
Dutchess Music/BM!,
1.0
I n t ro d u ctio n : T h e R a cetrack G a m e
Many measurable quantities, such as length, area, volume, mass, and temperature,
can be completely described by specifying their magnitude. Other quantities, such
as velocity, force, and acceleration, require both a magnitude and a direction for
their description. These quantities are vectors. For example, wind velocity is a vector
consisting of wind speed and direction, such as 1 0 km/h southwest. Geometrically,
vectors are often represented as arrows or directed line segments.
Although the idea of a vector was introduced in the 1 9th century, its usefulness
in applications, particularly those in the physical sciences, was not realized until the
20th century. More recently, vectors have found applications in computer science,
statistics, economics, and the life and social sciences. We will consider some of these
many applications throughout this book.
This chapter introduces vectors and begins to consider some of their geometric
and algebraic properties. We begin, though, with a simple game that introduces some
of the key ideas. [You may even wish to play it with a friend during those (very rare!)
dull moments in linear algebra class.]
The game is played on graph paper. A track, with a starting line and a finish line,
is drawn on the paper. The track can be of any length and shape, so long as it is wide
enough to accommodate all of the players. For this example, we will have two players
(let's call them Ann and Bert) who use different colored pens to represent their cars
or bicycles or whatever they are going to race around the track. (Let's think of Ann
and Bert as cyclists.)
Ann and Bert each begin by drawing a dot on the starting line at a grid point on
the graph paper. They take turns moving to a new grid point, subject to the following
rules:
1. Each new grid point and the line segment connecting it to the previous grid point
must lie entirely within the track.
2. No two players may occupy the same grid point on the same turn. (This is the
"no collisions" rule.)
3. Each new move is related to the previous move as follows: If a player moves
a units horizontally and b units vertically on one move, then on the next move
he or she must move between a 1 and a + 1 units horizontally and between
-
2
Chapter
1 Vectors
b - 1 and b + 1 units vertically. In other words, if the second move is c units
horizontally and d units vertically, then l a - c l :::::: 1 and l b - d i :::::: 1 . (This is the
"acceleration/deceleration" rule.) Note that this rule forces the first move to be
1 unit vertically and/or 1 unit horizontally.
:r:
r§
(1805-1865)
A player who collides with another player or leaves the track is eliminated. The
winner is the first player to cross the finish line. If more than one player crosses
the finish line on the same turn, the one who goes farthest past the finish line is the
winner.
In the sample game shown in Figure 1 . 1 , Ann was the winner. Bert accelerated too
quickly and had difficulty negotiating the turn at the top of the track.
To understand rule 3, consider Ann's third and fourth moves. On her third move,
she went 1 unit horizontally and 3 units vertically. On her fourth move, her options
were to move 0 to 2 units horizontally and 2 to 4 units vertically. (Notice that some
of these combinations would have placed her outside the track.) She chose to move
2 units in each direction.
The Irish mathematician William
Rowan Hamilton
used vector concepts in his study
of complex numbers and their
generalization, the quaternions.
I
A
I
B
Figure 1 . 1
r
r
l
l
A sample game of racetrack
Problem 2
Problem 1
of moves?
Play a few games of racetrack.
Is it possible for Bert to win this race by choosing a different sequence
Problem 3 Use the notation [a, b] to denote a move that is a units horizontally
and b units vertically. (Either a or b or both may be negative.) If move [3, 4] has just
been made, draw on graph paper all the grid points that could possibly be reached
on the next move.
Problem 4 What is the net effect of two successive moves? In other words, if you
move [a, b] and then [c, d] , how far horizontally and vertically will you have moved
altogether?
Section
1.1 The Geometry and Algebra of Vectors
3
Problem 5 Write out Ann's sequence of moves using the [a, b] notation. Suppose
she begins at the origin (O, 0) on the coordinate axes. Explain how you can find the
coordinates of the grid point corresponding to each of her moves without looking at
the graph paper. If the axes were drawn differently, so that Ann's starting point was
not the origin but the point (2, 3), what would the coordinates of her final point be?
Although simple, this game introduces several ideas that will be useful in our
study of vectors. The next three sections consider vectors from geometric and alge­
braic viewpoints, beginning, as in the racetrack game, in the plane.
T h e G e o m etrv a n d A l g e b ra o f vectors
Vectors in the Plane
The Cartesian plane is named
after the French philosopher and
mathematician Rene Descartes
whose introduction
of coordinates allowed geometric
problems to be handled using
algebraic techniques.
(1596-1650),
The word vector comes from the
Latin root meaning "to carrY:' A
vector is formed when a point is
displaced-or "carried off" -a given
distance in a given direction. Viewed
another way, vectors "carry" two
pieces of information: their length
and their direction.
When writing vectors by hand,
it is difficult to indicate boldface.
Some people prefer to write v for
the vector denoted in print by v,
but in most cases it is fine to use an
ordinary lowercase v. It will usu­
ally be clear from the context when
the letter denotes a vector.
We begin by considering the Cartesian plane with the familiar x- and y-axes.
A vector is a directed line segment that corresponds to a displacement from one point
A to another point B; see Figure 1 .2.
The vector from A to B is denoted by AB ; the point A is called its initial point,
or tail, and the point B is called its terminal point, or head. Often, a vector is simply
denoted by a single boldface, lowercase letter such as v.
The set of all points in the plane corresponds to the set of all vectors whose tails
---->
are at the origin 0. To each point A, there corresponds the vector a = OA; to each
vector a with tail at 0, there corresponds its head A. (Vectors of this form are some­
times called position vectors.)
It is natural to represent such vectors using coordinates. For example, in
---->
Figure 1 .3, A = (3, 2) and we write the vector a = OA = [3, 2] using square brackets.
Similarly, the other vectors in Figure 1 .3 are
[ - 1 , 3 ] and c = [ 2, - 1 ]
The individual coordinates (3 and 2 in the case of a) are called the components of the
vector. A vector is sometimes said to be an ordered pair of real numbers. The order is
important since, for example, [3, 2] � [2, 3] . In general, two vectors are equal if and
only if their corresponding components are equal. Thus, [x, y] = [ l , 5] implies that
x = 1 and y = 5.
b =
[ �]
It is frequently convenient to use column vectors instead of (or in addition to)
row vectors. Another representation of [3, 2] is
. (The important point is that the
y
y
The word component is derived
from the Latin words co, meaning
"together with;' and ponere, mean­
ing "to put:' Thus, a vector is "put
together" out of its components.
A �B
Figure 1 . 2
Figure 1 . 3
4
Chapter
1 Vectors
IR2 is pronounced "r two:'
components are ordered.) In later chapters, you will see that column vectors are some­
what better from a computational point of view; for now, try to get used to both
representations.
It may occur to you that we cannot really draw the vector [O, OJ = 00 from the
origin to itself. Nevertheless, it is a perfectly good vector and has a special name: the
zero vector. The zero vector is denoted by 0.
The set of all vectors with two components is denoted by IR 2 (where IR denotes
the set of real numbers from which the components of vectors in IR 2 are chosen).
Thus, [ - 1, 3.5] , [ \/2, 7f ], and rn, 4 ] are all in IR 2 •
Thinking back to the racetrack game, let's try to connect all of these ideas to vec­
tors whose tails are not at the origin. The etymological origin of the word vector in
the verb "to carry" provides a clue. The vector [3, 2] may be interpreted as follows:
Starting at the origin 0, travel 3 units to the right, then 2 units up, finishing at P. The
same displacement may be applied with other init � oin� igure 1 .4 shows two
equivalent displacements, represented by the vectors AB and CD .
y
c
Figure 1 . 4
When vectors are referred to by
their coordinates, they are being
considered analytically.
We define two vectors as equal if they have the same length and the same direc­
tion. Thus, AB = CD in Figure 1 .4. (Even though they have different initial and ter­
minal points, they represent the same displacement.) Geometrically, two vectors are
equal if one can be obtained by sliding (or translating) the other parallel to itself until
the two vectors coincide. In terms of components, in Figure 1 .4 we have A = ( 3, 1)
and B = (6, 3). Notice that the vector [3, 2] that records the displacement is just the
difference of the respective components:
----->
AB = [ 3, 2 ] = [ 6
3, 3
-
l]
CD = r - 1 - ( -4), 1 - ( - 1) J
Similarly,
----->
-
[ 3, 2 ]
---->
and thus AB = CD , as expected.
A vector such as oP with its tail at the origin is said to be in standard position.
The foregoing discussion shows that every vector can be drawn as a vector in stan­
dard position. Conversely, a vector in standard position can be redrawn (by transla­
tion) so that its tail is at any point in the plane.
Exa m p l e 1 . 1
If A = ( - 1 , 2 ) and B = ( 3, 4), find AB and redraw it (a) in standard position and
(b) with its tail at the point C = (2, - 1).
----->
----->
We compute AB = [3 - ( - 1), 4 - 2] = [4, 2] . If AB is then translated
to CD, where C = ( 2, - 1 ), then we must have D = ( 2 + 4, - 1 + 2) = (6, 1). (See
Figure 1 .5. )
Solulion
Section
1.1 The Geometry and Algebra of Vectors
5
y
D(6, 1 )
Figure 1 . 5
New Veclors from Old
As in the racetrack game, we often want to "follow" one vector by another. This leads
to the notion of vector addition, the first basic vector operation.
If we follow u by v, we can visualize the total displacement as a third vector,
denoted by u + v. In Figure 1 .6, u = [ 1, 2] and v = [ 2, 2 ] , so the net effect of follow­
ing u by v is
[ 1 + 2, 2 + 2 ] = [3, 4]
which gives u + v. In general, if u = [u 1 , u 2 ] and v = [ v 1 , v2 ] , then their s u m u + v
is the vector
It is helpful to visualize u + v geometrically. The following rule is the geometric
version of the foregoing discussion.
y
/
Figure 1 . 6
Vector addition
�(/lj2
:
2
/_ )
U
_
I
I
2_ _
4
_ _ _ _ _ _ _ _ _
3
_n
6
Chapter
1 Vectors
The Head-to-Tail R u l e
Given vectors u and v in IR 2 , translate v s o that its tail coincides with the head
of u. The s u m u + v of u and v is the vector from the tail of u to the head of v.
(See Figure 1 .7. )
Figure 1 . 1
The head-to-tail rule
Figure 1 . 8
The parallelogram
determined by u and v
The Parallelogram Rule
By translating u and v parallel to themselves, we obtain a parallelogram, as
shown in Figure 1 .8. This parallelogram is called the parallelogram determined by u
and v. It leads to an equivalent version of the head-to-tail rule for vectors in standard
position.
Given vectors u and v in IR 2 (in standard position), their s u m u + v is the vector
in standard position along the diagonal of the parallelogram determined by u and
v. (See Figure 1 .9. )
y
Figure 1 . 9
The parallelogram rule
Exa m p l e 1 . 2
If u = [3, - 1 ] and v = [ l , 4], compute and draw u + v.
We compute u + v = [3 + 1, - 1 + 4] = [4, 3] . This vector is drawn
using the head-to-tail rule in Figure l . l O(a) and using the parallelogram rule in
Figure l . l O(b).
Solulion
Section
1.1 The Geometry and Algebra of Vectors
y
1
y
u
(b)
(a)
Figure 1 . 1 0
The second basic vector operation is scalar multiplication. Given a vector v and
a real number c, the scalar multiple c v is the vector obtained by multiplying each
component of v by c. For example, 3 [ - 2, 4] = [ - 6, 12] . In general,
Geometrically, cv is a "scaled" version of v.
Exa m p l e 1 . 3
If v = [ - 2, 4], compute and draw 2v, tv, and - 2v.
Solution
We calculate as follows:
2v = [ 2 ( - 2 ) , 2 ( 4 ) ] = [ - 4, 8 J
tv = [t ( - 2 ) , t ( 4 ) ] = [ - 1 , 2 ]
- 2v = [ - 2 ( - 2 ) , - 2 ( 4 ) ] = [ 4, - 8 ]
These vectors are shown in Figure 1 . 1 1 .
y
2v
- 2v
Figure 1 . 1 1
Chapter
8
1 Vectors
/./
2v
u
- 2v
Figure 1 . 1 3
u
+ ( - v)
Vector subtraction
Figure 1 . 1 2
The term scalar comes from the
Latin word scala, meaning "lad­
dd' The equally spaced rungs on
a ladder suggest a scale, and in vec­
tor arithmetic, multiplication by a
constant changes only the scale (or
length) of a vector. Thus, constants
became known as scalars.
Observe that cv has the same direction as v if c > 0 and the opposite direction if
c < 0. We also see that cv is l e I times as long as v. For this reason, in the context of
vectors, constants (i.e., real numbers) are referred to as scalars. As Figure 1 . 1 2 shows,
when translation of vectors is taken into account, two vectors are scalar multiples of
each other if and only if they are parallel.
A special case of a scalar multiple is ( - l )v, which is written as -v and is called
the negative ofv. We can use it to define vector subtraction: The difference of u and
v is the vector u - v defined by
u - v = u + ( - v)
Figure 1 . 1 3 shows that u - v corresponds to the "other" diagonal of the parallelo­
gram determined by u and v.
Exa m p l e 1 . 4
y
A
Figure 1 . 1 4
�
If u = [ 1 , 2] and v = [ - 3, 1 ] , then u - v = [ 1 - ( - 3), 2 - 1 ] = [ 4, l ] .
The definition of subtraction in Example 1 .4 also agrees with the way we cal­
culate a vector such as AB .---If the
points A and B correspond to the vectors a and b
->
=
in standard position, then AB b - a, as shown in Figure 1 . 14. [Observe that the
head-to-tail rule applied to this diagram gives the equation a + (b - a ) = b. If we
had accidentally drawn b - a with its head at A instead of at B, the diagram would
have read b + (b - a) = a, which is clearly wrong! More will be said about algebraic
expressions involving vectors later in this section.]
vec1ors in � 3
Everything we have just done extends easily to three dimensions. The set of all or­
dered triples of real numbers is denoted by IR 3 . Points and vectors are located using
three mutually perpendicular coordinate axes that meet at the origin 0. A point such
as A = ( 1 , 2, 3) can be located as follows: First travel 1 unit along the x-axis, then
move 2 units parallel to the y-axis, and finally move 3 units parallel to the z-axis. The
corresponding vector a = [ l , 2, 3] is then OA, as shown in Figure 1 . 15.
Another way to visualize vector a in IR 3 is to construct a box whose six sides are de­
termined by the three coordinate planes (the xy-, xz-, and yz-planes) and by three planes
through the point ( 1 , 2, 3 ) parallel to the coordinate planes. The vector [ 1 , 2, 3] then corre­
sponds to the diagonal from the origin to the opposite corner of the box (see Figure 1 . 16).
Section
1.1 The Geometry and Algebra of Vectors
z
z
I
A ( l , 2, 3 )
3
•
I
I
I
,.
:3
-
I
I
I
-- 1
2 -
x
Figure 1 . 1 5
9
�
y
x
Figure 1 . 1 6
The "componentwise" definitions of vector addition and scalar multiplication are
extended to IR 3 in an obvious way.
Vectors in � n
In general, we define !R n as the set of all ordered n-tuples of real numbers written as
row or column vectors. Thus, a vector v in !R n is of the form
�
Figure 1 . 1 1
u+v=v+u
The individual entries of v are its components; V; is called the ith component.
We extend the definitions of vector addition and scalar multiplication to !R n in
the obvious way: If u = [u 1 , u 2 , • • • , u n l and v = [ v 1 , v2 , . . . , vn ] , the ith component of
u + v is U ; + V; and the ith component of cv is just C V;.
Since in !R n we can no longer draw pictures of vectors, it is important to be able to
calculate with vectors. We must be careful not to assume that vector arithmetic will be
similar to the arithmetic of real numbers. Often it is, and the algebraic calculations we
do with vectors are similar to those we would do with scalars. But, in later sections,
we will encounter situations where vector algebra is quite unlike our previous experi­
ence with real numbers. So it is important to verify any algebraic properties before
attempting to use them.
One such property is commutativity of addition: u + v = v + u for vectors u and
v. This is certainly true in IR 2 • Geometrically, the head-to-tail rule shows that both
u + v and v + u are the main diagonals of the parallelogram determined by u and v.
(The parallelogram rule also reflects this symmetry; see Figure 1 . 1 7.)
Note that Figure 1 . 1 7 is simply an illustration of the property u + v = v + u. It
is not a proof, since it does not cover every possible case. For example, we must also
include the cases where u = v, u = - v, and u = 0. (What would diagrams for these
cases look like?) For this reason, an algebraic proof is needed. However, it is just as
easy to give a proof that is valid in !R n as to give one that is valid in IR 2 .
The following theorem summarizes the algebraic properties of vector addition
and scalar multiplication in !R n . The proofs follow from the corresponding properties
of real numbers.
Chapter
10
1 Vectors
Theorem 1 . 1
Algebraic Properties o f Vectors i n !R n
Let u, v, and w be vectors in !R n and let c and d be scalars. Then
a. u + v = v + u
b. (u + v) + w = u + (v + w)
c. u + 0 = u
d. u + ( -u) = 0
e. c (u + v) = cu + cv
f. (c + d)u = cu + du
g. c (du) = (cd)u
h. lu = u
Commutativity
Associativity
Distributivity
Distributivity
R e m a rks
The word theorem is derived from
the Greek word theorema, which
in turn comes from a word mean­
ing "to look af' Thus, a theorem
is based on the insights we have
when we look at examples and
extract from them properties that
we try to prove hold in general.
Similarly, when we understand
something in mathematics-the
proof of a theorem, for example­
we often say, "I see:'
•
Properties (c) and (d) together with the commutativity property (a) imply
that 0 + u = u and - u + u = 0 as well.
•
If we read the distributivity properties (e) and (f) from right to left, they say
that we can factor a common scalar or a common vector from a sum.
We prove properties (a) and (b) and leave the proofs of the remain­
ing properties as exercises. Let u = [u 1 , u 2 , . . . , u n ] , v = [ v 1 , v2 , . . . , vn ] , and w =
Proof
[ W1 , Wz , . . . , Wn ] .
(a)
U
+ = [U1, U z , . . . , U n ] + [V1, Vz , . . . , Vn ]
= [U1 + V1 , U z + Vz , . . . , U n + Vn ]
= [ v 1 + u 1 , v2 + u 2 , , v n + u n ]
= [v 1 , Vz , . . . , Vn ] + [u 1 , U z , . . . , U n ]
=v+u
V
•
.
.
The second and fourth equalities are by the definition of vector addition, and the
third equality is by the commutativity of addition of real numbers.
(b) Figure 1 . 1 8 illustrates associativity in IR 2 . Algebraically, we have
[ (u 1 + v 1 ) + w 1 , (u 2 + vz ) + w2 , . . . , (u n + vn ) + wn ]
[ u 1 + (v l + W1 ), U 2 + ( Vz + Wz ), . . . , U n + (vn + Wn ) J
[ u 1 , u 2 , . . . , u n ] + ( [ v 1 , v2 , . . . , vn ] + [ w 1 , w2 , . . . , w n ] )
= u + (v + w)
u
Figure 1 . 1 8
The fourth equality is by the associativity of addition of real numbers. Note the care­
ful use of parentheses.
Section
1.1 The Geometry and Algebra of Vectors
11
By property (b) of Theorem 1 . 1 , we may unambiguously write u + v + w without
parentheses, since we may group the summands in whichever way we please. By (a),
we may also rearrange the summands-for example, as w + u + v-if we choose.
Likewise, sums of four or more vectors can be calculated without regard to order or
grouping. In general, if v1 , v2 , , vk are vectors in !R n , we will write such sums with­
out parentheses:
•
•
.
The next example illustrates the use of Theorem 1 . 1 in performing algebraic
calculations with vectors.
Exa m p l e 1 . 5
Let a, b, and x denote vectors in !R n .
(a) Simplify 3a + (Sb - 2a) + 2(b - a).
(b) If Sx - a = 2(a + 2x), solve for x in terms of a.
We will give both solutions in detail, with reference to all of the properties
in Theorem 1 . 1 that we use. It is good practice to justify all steps the first few times
you do this type of calculation. Once you are comfortable with the vector properties,
though, it is acceptable to leave out some of the intermediate steps to save time and
space.
Solution
(a) We begin by inserting parentheses.
3a + (Sb - 2a) + 2(b - a) = (3a + (Sb - 2a)) + 2(b - a)
(3a + ( - 2a + Sb)) + (2b - 2a)
((3a + ( - 2a)) + Sb) + (2b - 2a)
((3 + ( - 2))a + Sb) + (2b - 2a)
( la + Sb) + (2b - 2a)
((a + Sb) + 2b) - 2a
(a + (Sb + 2b)) - 2a
(a + (S + 2)b) - 2a
(7b + a) - 2a
= 7b + (a - 2a)
= 7b + ( 1 - 2)a
= 7b + ( - l )a
= 7b - a
(a), (e)
(b)
(f)
(b ), (h)
(b)
(f)
(a)
(b)
(f), (h)
You can see why we will agree to omit some of these steps! In practice, it is acceptable to
simplify this sequence of steps as
3a + (Sb - 2a) + 2(b - a) = 3a + Sb - 2a + 2b - 2a
(3a - 2a - 2a) + (Sb + 2b)
= - a + 7b
or even to do most of the calculation mentally.
12
Chapter
1 Vectors
(b) In detail, we have
Sx - a = 2 ( a + 2x)
Sx - a = 2a + 2 ( 2x)
Sx - a = 2a + ( 2 · 2 ) x
Sx - a = 2a + 4x
( Sx - a ) - 4x = ( 2a + 4x) - 4x
( - a + Sx) - 4x = 2a + ( 4x - 4x)
- a + ( Sx - 4x) = 2a + 0
- a + ( 5 - 4 ) x = 2a
- a + ( l ) x = 2a
a + ( - a + x) = a + 2a
( a + ( - a )) + x = ( 1 + 2 ) a
0 + x = 3a
x = 3a
(e)
(g)
(a), (b)
(b), (d)
(f), (c)
(h)
(b), (f)
(d)
(c)
Again, we will usually omit most of these steps.
Linear Combin alions and Coordin a1es
A vector that is a sum of scalar multiples of other vectors is said to be a linear combi­
nation of those vectors. The formal definition follows.
D e f i n i l i O D A vector v is a linear combination of vectors v1 , v2 , . . . , vk if
there are scalars c 1 , c2 , . . . , c k such that v = c 1 v1 + c2v2 +
+ ckvk . The scalars
c 1 , c2 , . . . , ck are called the coefficients of the linear combination.
·
·
·
Exa m p l e 1 . 6
Remark
Determining whether a given vector is a linear combination of other
vectors is a problem we will address in Chapter 2.
In IR 2 , it is possible to depict linear combinations of two (nonparallel) vectors
quite conveniently.
Exa m p l e 1 . 1
Let u =
[�]
and v =
way that e 1 =
[�]
[�] .
We can use u and v to locate a new set of axes (in the same
and e2 =
[�]
locate the standard coordinate axes) . We can use
Section
1.1 The Geometry and Algebra of Vectors
13
y
I
u
I
Figure 1 . 1 9
I I
I-
these new axes to determine a coordinate grid that will let us easily locate linear
combinations of u and v.
As Figure 1 . 1 9 shows, w can be located by starting at the origin and traveling
- u followed by 2v. That is,
w = -u + 2v
We say that the coordinates of w with respect to u and v are - 1 and 2 . (Note that
this is just another way of thinking of the coefficients of the linear combination.)
It follows that
(Observe that - 1 and 3 are the coordinates of w with respect to e 1 and e2 . )
Switching from the standard coordinate axes to alternative ones is a useful idea. It
has applications in chemistry and geology, since molecular and crystalline structures
often do not fall onto a rectangular grid. It is an idea that we will encounter repeatedly
in this book.
Binarv vec1ors and Modular Arilhmelic
We will also encounter a type of vector that has no geometric interpretation-at least
not using Euclidean geometry. Computers represent data in terms of Os and ls (which
can be interpreted as off/on, closed/open, false/true, or no/yes) . Binary vectors are
vectors each of whose components is a 0 or a 1 . As we will see in Chapter 8, such
vectors arise naturally in the study of many types of codes.
In this setting, the usual rules of arithmetic must be modified, since the result of
each calculation involving scalars must be a 0 or a 1 . The modified rules for addition
and multiplication are given below.
The only curiosity here is the rule that 1 + 1 = 0. This is not as strange as it appears;
if we replace 0 with the word "even" and 1 with the word "odd," these tables simply
14
Chapter
1 Vectors
summarize the familiar parity rules fo r the addition and multiplication of even and
odd integers. For example, 1 + 1 = 0 expresses the fact that the sum of two odd inte­
gers is an even integer. With these rules, our set of scalars {O, l } is denoted by 22 and
is called the set of integers modulo 2.
Exa m p l e 1 . 8
We are using the term length dif­
ferently from the way we used it in
!FR". This should not be confusing,
since there is no geometric notion
of length for binary vectors.
In 22, 1 + 1 + 0 + 1 = 1 and 1 + 1 + 1 + 1 = 0. (These calculations illustrate
the parity rules again: The sum of three odds and an even is odd; the sum of four
odds is even.)
.+
With 22 as our set of scalars, we now extend the above rules to vectors. The set of
all n-tuples of Os and l s (with all arithmetic performed modulo 2) is denoted by 2� .
The vectors in 2� are called binary vectors of length n.
Exa m p l e 1 . 9
The vectors in 2� are [O, OJ , [O, l ] , [ l , OJ , and [ l , l J . (How many vectors does 2�
contain, in general?)
Exa m p l e 1 . 1 0
Let u = [l, 1, 0, 1, OJ and v = [O, 1, 1, 1, OJ be two binary vectors oflength 5. Find u + v.
Solulion
The calculation of u + v takes place over 22 , so we have
u + v = [ 1 , i , o, i , o ] + [o, I , I , i , o ]
= [ 1 + o, I + I , o + I , I + I , o + o ]
= [ 1 , 0, 1 , 0, 0 ]
It is possible to generalize what we have just done for binary vectors to vectors whose
components are taken from a finite set {O, 1, 2, . . . , k} for k 2: 2. To do so, we must
first extend the idea of binary arithmetic.
Exa m p l e 1 . 1 1
The integers modulo 3 is the set 2 3 = {O, 1, 2} with addition and multiplication given
by the following tables:
2
0 1 2
0 0 0 0
0 0 1 2
1 2 0
2
1 0
2 0 2 1
2 2 0
Observe that the result of each addition and multiplication belongs to the set
{O, 1, 2}; we say that 2 3 is closed with respect to the operations of addition and multi­
plication. It is perhaps easiest to think of this set in terms of a 3-hour clock with 0, 1 ,
and 2 o n its face, as shown i n Figure 1 .20.
The calculation 1 + 2 = 0 translates as follows: 2 hours after 1 o'clock, it is
0 o'clock. Just as 24:00 and 1 2:00 are the same on a 1 2-hour clock, so 3 and 0 are
equivalent on this 3-hour clock. Likewise, all multiples of 3-positive and negative­
are equivalent to 0 here; 1 is equivalent to any number that is 1 more than a multiple
of 3 (such as - 2, 4, and 7); and 2 is equivalent to any number that is 2 more than a
+ 0
Section
1.1 The Geometry and Algebra of Vectors
15
multiple of 3 (such as - 1 , 5, and 8). We can visualize the number line as wrapping
around a circle, as shown in Figure 1 .2 1 .
0
. . . , - 3 , 0, 3, . . .
2
. . . , 1 , 2, 5 , . . .
Figure 1 . 2 0
Arithmetic modulo
Exa m p l e 1 . 1 2
3
. . . , - 2, 1 , 4, . . .
Figure 1 . 2 1
To what is 3548 equivalent in Z/
Solution This is the same as asking where 3548 lies on our 3-hour clock. The key is
to calculate how far this number is from the nearest (smaller) multiple of 3; that is,
we need to know the remainder when 3548 is divided by 3. By long division, we find that
3548 = 3 · 1 1 82 + 2, so the remainder is 2. Therefore, 3548 is equivalent to 2 in l'.. 3 •
4
In courses in abstract algebra and number theory, which explore this concept in
greater detail, the above equivalence is often written as 3548 = 2 (mod 3) or 3548 = 2
(mod 3), where = is read "is congruent to." We will not use this notation or termi­
nology here.
Exa m p l e 1 . 1 3
In l'.. 3 , calculate 2 + 2 + 1 + 2.
We use the same ideas as in Example 1 . 12. The ordinary sum is 2 + 2 +
1 + 2 = 7, which is 1 more than 6, so division by 3 leaves a remainder of 1 . Thus, 2 +
2 + 1 + 2 = 1 in l'.. 3 .
Solution 1
Solution 2
A better way to perform this calculation is to do it step by step entirely in l'.. 3 •
2 + 2 + 1 + 2 = (2 + 2) + 1 + 2
= 1 + 1 +2
= (1 + 1) + 2
=2+2
= 1
Here we have used parentheses to group the terms we have chosen to combine. We could
speed things up by simultaneously combining the first two and the last two terms:
(2 + 2) + ( 1 + 2) = 1 + 0
= 1
Chapter
16
1 Vectors
Repeated multiplication can be handled similarly. The idea is to use the addition and
multiplication tables to reduce the result of each calculation to 0, 1 , or 2.
Extending these ideas to vectors is straightforward.
Exa m p l e 1 . 1 4
m
m
-2
0
- 1 _.,---.
_
2
[ 2, 2, 0, 1, 2 ] + [ 1, 2, 2, 2, l ]
[ 2 + 1, 2 + 2, 0 + 2, 1 + 2, 2 + l ]
[ 0, 1 , 2, 0, 0 ]
Vectors in Z� are referred to as ternary vectors of length 5.
In general, we have the set Z m = {O, 1, 2, . . . , m - l } of integers modulo m (cor­
responding to an m-hour clock, as shown in Figure 1 .22). A vector of length n whose
entries are in Z m is called an m-ary vector of length n. The set of all m-ary vectors of
length n is denoted by z::i.
Figure 1.22
Arithmetic modulo m
I
In Z�, let u = [2, 2, 0, 1 , 2] and v = [ l , 2, 2, 2, l ] . Then
u + v =
3
..
4
Exercises 1 . 1
1 . Draw the following vectors in standard position
in IR 2 :
(a) a =
[]
3
0
(b) b =
(d) d =
[]
[]
2
3
3
-2
2 . Draw the vectors in Exercise 1 with their tails at the
point (2, - 3).
3. Draw the following vectors in standard position in IR 3 :
(a) a = [O, 2, OJ
(b) b = [ 3 , 2, l ]
(c) c = [ l , - 2, l ]
(d) d = [ - 1, - 1 , - 2]
4. If the vectors in Exercise 3 are translated so that their
heads are at the point (3, 2, 1), find the points that
correspond to their tails.
5. For each of the following pairs of points, draw the
---->
---->
vector AB. Then compute and redraw AB as a vector
in standard position.
(a) A = ( 1, - 1 ), B = (4, 2)
(b) A = (O, - 2), B = ( 2, - 1 )
(c) A = (2, f), B = (t, 3 )
(d) A = (t, t), B = (i, t)
6. A hiker walks 4 km north and then 5 km northeast.
Draw displacement vectors representing the hiker's
trip and draw a vector that represents the hiker's net
displacement from the starting point.
Exercises 7- 1 0 refer to the vectors in Exercise 1. Compute
the indicated vectors and also show how the results can be
obtained geometrically.
8. b - c
7. a + b
10. a + d
9. d - c
Exercises 1 1 and 12 refer to the vectors in Exercise 3.
Compute the indicated vectors.
1 1 . 2a + 3c
12. 3b - 2c + d
13. Find the components of the vectors u, v, u + v, and
u - v, where u and v are as shown in Figure 1 .23.
14. In Figure 1 . 2 4 , A, B, C, D, E, and F are the vertices of a
regular hexagon centered at the origin.
Express each of the following vectors in terms of
---->
---->
a = OA and b = OB :
(a) Ai
(c) Ai5
(b) BC
(d) a
(e) xc
(f) BC + ill + PX
Section
y
21. u =
22. u =
1.1 The Geometry and Algebra of Vectors
11
[ - � ] , [ � ] , [�]
[ - � ] , [ �] , [ � ]
v=
w=
v=
w=
23. Draw diagrams to illustrate properties (d) and (e) of
Theorem 1 . 1 .
24. Give algebraic proofs of properties ( d) through (g) of
Theorem 1 . 1 .
In Exercises 25-28, u and v are binary vectors. Find u + v
in each case.
25. u =
Figure 1 . 2 3
y
c
B
E
F
Figure 1 . 2 4
[ � ] ,v [ � ]
=
27. u = [ 1 , 0, 1 , 1 ] , v = [ 1, 1 , 1, 1 ]
28. u = [ 1 , 1, 0, 1, 0 ] , v = [ O, 1, 1 , 1, 0 ]
29. Write out the addition and multiplication tables for Z 4 .
30. Write out the addition and multiplication tables for Zs.
In Exercises 3 1 -43, perform the indicated calculations.
31. 2 + 2 + 2 in Z3
32. 2 · 2 · 2 in Z3
(
)
33. 2 2 + 1 + 2 in Z3 34. 3 + 1 + 2 + 3 in Z4
35. 2 · 3 · 2 in Z4
36. 3 ( 3 + 3 + 2 ) in Z4
37. 2 + 1 + 2 + 2 + 1 in Z3 , Z4 , and Zs
38. ( 3 + 4 ) ( 3 + 2 + 4 + 2 ) in Zs
39. 8 ( 6 + 4 + 3 ) in Z9 40. 2 100 in Z 1 1
41. [ 2 , 1 , 2 ] + [ 2, 0, 1 ] in Z� 42. 2 [ 2, 2, 1 ] in Z�
43. 2 ( [ 3, l , 1 , 2 ] + [ 3, 3, 2, l ] ) in Z! and Z�
In Exercises 1 5 and 1 6, simplify the given vector expression.
Indicate which properties in Theorem 1 . 1 you use.
15. 2 (a - 3b) + 3 (2b + a)
16. - 3(a - c) + 2(a + 2b) + 3(c - b)
In Exercises 1 7 and 1 8, solve for the vector x in terms of the
vectors a and b.
17. x - a = 2(x - 2a)
18. x + 2a - b = 3 (x + a) - 2 (2a - b)
In Exercises 19 and 20, draw the coordinate axes relative to
u and v and locate w.
[ � ] [ �l w
[ -�l [ �l
19. u = _ , v =
20. u =
v= _
= 2 u + 3v
w = - u - 2v
In Exercises 21 and 22, draw the standard coordinate axes
on the same diagram as the axes relative to u and v. Use
these to find w as a linear combination of u and v.
In Exercises 44-55, solve the given equation or indicate that
there is no solution.
45. x + 5 = 1 in Z6
44. x + 3 = 2 in Zs
46. 2x = 1 in Z3
47. 2x = 1 in Z4
48. 2x = 1 in Zs
49. 3x = 4 in Zs
51. 6x = 5 in Zs
50. 3x = 4 in Z6
52. Bx = 9 in Z 1 1
53. 2x + 3 = 2 in Z5
55. 6x + 3 = 1 in Zs
54. 4x + 5 = 2 in Z6
56. (a) For which values of a does x + a = 0 have a solu­
tion in Zs?
(b) For which values of a and b does x + a = b have a
solution in Z6?
(c) For which values of a, b, and m does x + a = b
have a solution in Z m ?
57. (a) For which values of a does ax = 1 have a solution
in Zs?
(b) For which values of a does ax = 1 have a solution
in Z6?
(c) For which values of a and m does ax = 1 have a
solution in z m ?
18
Chapter
1 Vectors
Length a n d A n g l e : T h e D o t P ro d u ct
It is quite easy to reformulate the familiar geometric concepts of length, distance,
and angle in terms of vectors. Doing so will allow us to use these important and
powerful ideas in settings more general than IR 2 and IR 3 • In subsequent chapters,
these simple geometric tools will be used to solve a wide variety of problems arising
in applications-even when there is no geometry apparent at all!
The Doi Producl
The vector versions of length, distance, and angle can all be described using the
notion of the dot product of two vectors.
Definilion
If
then the dot product u · v of u and v is defined by
In words, u · v is the sum of the products of the corresponding components of u
and v. It is important to note a couple of things about this "product" that we have just
defined: First, u and v must have the same number of components. Second, the dot
product u · v is a number, not another vector. (This is why u · v is sometimes called
the scalar product of u and v.) The dot product of vectors in !R n is a special and im­
portant case of the more general notion of inner product, which we will explore in
Chapter 7.
Exa m p l e 1 . 1 5
Solution
u · v = l · ( - 3) + 2 · 5 + ( - 3) · 2 = 1
Notice that if we had calculated v · u in Example 1 . 15, we would have computed
v · u = ( - 3) · 1 + 5 · 2 + 2 · ( - 3) = 1
That u · v = v · u in general is clear, since the individual products of the components
commute. This commutativity property is one of the properties of the dot product
that we will use repeatedly. The main properties of the dot product are summarized
in Theorem 1 .2.
Section
Theorem 1 . 2
1.2 Length and Angle: The Dot Product
19
Let u, v, and w be vectors in � n and let c be a scalar. Then
Commutativity
a. u · v = v · u
b. u . ( v + w) = u . v + u . w
Distributivity
c. ( cu) . v = c ( u . v)
d. u · u 2: 0 and u · u = 0 if and only if u = 0
We prove (a) and (c) and leave proof of the remaining properties for the
exercises.
Proof
(a) Applying the definition of dot product to u · v and v · u, we obtain
u · v = U 1 V 1 + U z Vz + . . + U n V n
= V 1 U 1 + Vz U z + . . . + Vn U n
= v·u
·
where the middle equality follows from the fact that multiplication of real numbers
is commutative.
(c) Using the definitions of scalar multiplication and dot product, we have
(cu) · v = [ cu 1 , cu 2 , . . . , cu " ] • [ v 1 , v2 , , v" ]
+ CU n Vn
CU 1 V 1 + CU z V2 +
+ U n Vn )
c( U 1 V 1 + U z Vz +
c(u · v)
•
·
·
·
·
·
·
.
.
R e m a rlls
Property (b) can be read from right to left, in which case it says that we can
factor out a common vector u from a sum of dot products. This property also has
a "right-handed" analogue that follows from properties (b) and (a) together:
(v + w) · u = v · u + w · u.
•
Property (c) can be extended to give u · (cv) = c(u · v) (Exercise 58). This
extended version of ( c) essentially says that in taking a scalar multiple of a dot product
of vectors, the scalar can first be combined with whichever vector is more convenient.
For example,
•
(� [ - 1, -3, 2 ] ) · [ 6, - 4, 0 ] = [ - 1 , -3, 2 ] · (� [6, - 4, 0 ] ) = [ - 1, -3, 2 ] · [3, - 2, 0] = 3
With this approach we avoid introducing fractions into the vectors, as the original
grouping would have.
•
The second part of ( d) uses the logical connective if and only if. Appendix A dis­
cusses this phrase in more detail, but for the moment let us just note that the wording
signals a double i mplication- namely,
if u = 0, then u · u = 0
and
if u · u = 0, then u = 0
Theorem 1 .2 shows that aspects of the algebra of vectors resemble the algebra of
numbers. The next example shows that we can sometimes find vector analogues of
familiar identities.
Chapter
20
1 Vectors
Exa m p l e 1 . 1 6
Prove that (u + v) · (u + v) = u · u + 2(u · v) + v · v for all vectors u and v in !R n .
Solulion
(u + v) · (u + v) = ( u + v) · u + ( u + v) · v
= u·u + v·u + u·v + v·v
= u·u + u·v + u·v + v·v
= u · u + 2(u · v) + v · v
(Identify the parts of Theorem 1 .2 that were used at each step.)
y
b
v =
[�]
Length
To see how the dot product plays a role in the calculation oflengths, recall how lengths
are computed in the plane. The Theorem of Pythagoras is all we need.
In IR 2 , the length of the vector v =
a
[�]
is the distance from the origin to the point
(a, b), which, by Pythagoras' Theorem, is given by Va 2 + b 2 , as in Figure 1 .25.
Observe that a 2 + b 2 = v · v. This leads to the following definition.
Figure 1 . 2 5
Definition
The length (or norm) of a vector v =
tive scalar ll v ll defined by
[ �: ]
·
in !R n is the nonnega-
vn
ll v ll = VV:V = Vvi + v� +
·
·
·
+ v�
In words, the length of a vector is the square root of the sum of the squares of its
components. Note that the square root of v · v is always defined, since v · v 2': 0 by
Theorem l .2 (d). Note also that the definition can be rewritten to give ll v ll 2 = v · v,
which will be useful in proving further properties of the dot product and lengths of
vectors.
Exa m p l e 1 . 1 1
11 [2, 3 J 11 = v2 2 + 3 2 =
vT3
Theorem 1 . 3 lists some of the main properties of vector length.
Theorem 1 . 3
Let v be a vector in IR " and let c be a scalar. Then
a. ll v ll = 0 if and only ifv = 0
b. ll cv ll = l c l ll v ll
Proof
Property (a) follows immediately from Theorem l .2(d). To show (b), we have
ll cv ll 2 = (cv) · (cv) = c 2 (v · v) = c 2 ll v ll 2
using Theorem l .2 (c). Taking square roots ofboth sides, using the fact that W = I c l
for any real number c, gives the result.
Section
1.2 Length and Angle: The Dot Product
21
A vector of length 1 is called a unit vector. In IR 2 , the set of all unit vectors can
be identified with the unit circle, the circle of radius 1 centered at the origin (see
Figure 1 .26). Given any nonzero vector v, we can always find a unit vector in the
same direction as v by dividing v by its own length (or, equivalently, multiplying by
1/ ll v ll ) . We can show this algebraically by using property (b) of Theorem 1.3 above:
If u = ( 1 / ll v ll )v, then
ll u ll = 11 ( 1/ ll v ll )v ll = l l / ll v ll I ll v ll = ( 1/ ll v ll ) ll v ll = 1
and u is in the same direction as v, since 1 / I v I is a positive scalar. Finding a unit vec­
tor in the same direction is often referred to as normalizing a vector (see Figure 1 .27).
y
v
>:;:.
� /
rr�1r
Figure 1 . 2 7
Figure 1 . 2 6
Normalizing a vector
Unit vectors in lffi 2
Exa m p l e 1 . 1 8
In IR 2 , let e1 =
[�]
and e2 =
[�] .
Then e1 and e2 are unit vectors, since the sum of the
squares of their components is 1 in each case. Similarly, in IR 3 , we can construct unit
vectors
Observe in Figure 1 .28 that these vectors serve to locate the positive coordinate axes
in IR 2 and IR 3 .
t
.--+
z
y
Figure 1 . 2 8
Standard unit vectors in lffi 2 and !ffi 3
x
y
Chapter
22
1 Vectors
In general, in !R n , we define unit vectors e1, e2 ,
, en , where e; has 1 in its ith
component and zeros elsewhere. These vectors arise repeatedly in linear algebra and
are called the standard unit vectors.
•
Exa m p l e 1 . 1 9
Nmmalfae the vectm v �
Solulion
.
.
[ -n
ll v ll = V2 2 + ( - 1 ) 2 + 3 2 = \/14, so a unit vector in the same direc­
tion as v is given by
{ :J [
u � ( 1/ ll v ll l v � ( 1/ v'14 -
2/ \/14
- 1/ \/14
3/ \/14
]
Since property (b) of Theorem 1 .3 describes how length behaves with respect to
scalar multiplication, natural curiosity suggests that we ask whether length and vec­
tor addition are compatible. It would be nice if we had an identity such as II u + v II =
ll u 1 + ll v ll , but for almost any choice of vectors u and v this turns out to be false. [See
Exercise 52(a).] However, all is not lost, for it turns out that if we replace the = sign by
:s , the resulting inequality is true. The proof of this famous and important result-the
Triangle Inequality-relies on another important inequality-the Cauchy-Schwarz
Inequality-which we will prove and discuss in more detail in Chapter 7.
Theorem 1 . 4
The Cauchy-Schwarz Inequality
For all vectors u and v in !R n ,
l u · v l :s ll n ll ll v ll
-
u
Figure 1 . 2 9
The Triangle Inequality
Theorem 1 . 5
See Exercises 7 1 and 72 for algebraic and geometric approaches to the proof of this
inequality.
In IR 2 or IR 3 , where we can use geometry, it is clear from a diagram such as
Figure 1 .29 that ll u + v ii :s ll n ll + ll v ll for all vectors u and v. We now show that
this is true more generally.
The Triangle Inequality
For all vectors u and v in !R n ,
ll u + v ii :s ll u ll + ll v ll
Section 1 . 2 Length and Angle: The Dot Product
�
23
Proof
Since both sides of the inequality are nonnegative, showing that the square of
the left-hand side is less than or equal to the square of the right-hand side is equiva­
lent to proving the theorem. (Why?) We compute
ll u + v ll 2 = (u + v) - (u + v)
= u · u + 2(u · v) + v · v
By Example 1.9
2
2
l
l
ll
ll
ll
ll
::; u + 2 u · v + v
::; ll n ll 2 + 2 ll n ll ll v ll + ll v ll 2 By Cauchy-Schwarz
= ( ll n ll + ll v ll ) 2
as required.
Distance
The distance between two vectors is the direct analogue of the distance between two
points on the real number line or two points in the Cartesian plane. On the number
line (Figure 1 .30), the distance between the numbers a and b is given by l a - bl. (Tak­
ing the absolute value ensures that we do not need to know which of a or b is larger.)
This distance is also equal to V ( a - b ) 2 , and its two-dimensional generalization is
the familiar formula for the distance d between points (a 1 , a 2 ) and (b 1 , b 2 )-namely,
d = V ( a 1 - b 1 ) 2 + ( a 2 - bi ) 2 •
4
I
I
I
a
+
-2
0
I
I
Figure 1 . 3 0
d = la - bl = l-2 - 3 1 =
In terms of vectors, if a =
[ :: ]
and b =
[ �:],
b
+
3
I
I
�
5
then d is just the length of a - b,
as shown in Figure 1 . 3 1 . This is the basis for the next definition.
d
_ _ _ _ _ _ _ _ _ _ _
a1 - b1
I
I
I
I
: a 2 - b2
I
I
I
I
_f]
Figure 1 . 3 1
d = V( a , - b,)2 + ( a 2 - b )2 = I l a - b ll
2
Definition
The distance d(u, v) between vectors u and v in u;g n is defined by
d ( U, v) = I u - v II
24
Chapter
1 Vectors
Exa m p l e 1 . 2 0
Find the di,tance between u �
Solution
[ 1] [ _ � l
[ �J '"
•nd v �
·
We rnmpute u - v �
d ( u, v) = ll u - v ii = V( \/2) 2 + ( - 1 ) 2 + 1 2 = V4 = 2
Angles
The dot product can also be used to calculate the angle between a pair of vectors.
In IR 2 or IR 3 , the angle between the nonzero vectors u and v will refer to the angle (}
determined by these vectors that satisfies 0 :::::: (} :::::: 1 80° (see Figure 1 .3 2).
�
v
v4
v
u
(}
0
Figure 1 . 3 2
The angle between u and v
u
'D
. /7
u
u
In Figure 1 .33, consider the triangle with sides u, v, and u - v, where (} is the angle
between u and v. Applying the law of cosines to this triangle yields
v
ll u - v ll 2 = ll u ll 2 + ll v ll 2 - 2 ll u ll ll v ll cos (}
Expanding the left-hand side and using ll v ll 2 = v · v several times, we obtain
Figure 1 . 3 3
ll u ll 2 - 2 ( u · v) + ll v ll 2 = ll u ll 2 + ll v ll 2 - 2 ll u ll ll v ll cos (}
which, after simplification, leaves us with u · v = ll u ll ll v ll cos (} . From this we obtain
the following formula for the cosine of the angle (} between nonzero vectors u and v.
We state it as a definition.
Definition
Exa m p l e 1 . 2 1
For nonzero vectors u and v in !R n ,
u·v
cos (} =
u
ll ll ll v ll
Compute the angle between the vectors u = [2, 1 , - 2] and v = [1, 1 , 1 ] .
Section
1.2 Length and Angle: The Dot Product
25
We calculate u·v = 2 · 1 + l · l + ( - 2 ) · 1 = 1 , JJ u JJ = V2 2 + 1 2 + (- 2 ) 2 =
V9 = 3, and JJ v JJ = Vl 2 + 1 2 + 1 2 = v3. Therefore, cos e = 1 / 3 v3, so
1
e = cos - ( 1 /3 v3) = 1 .377 radians, or 78.9°.
Solution
Exa m p l e 1 . 2 2
...._+
Compute the angle between the diagonals on two adjacent faces of a cube.
The dimensions of the cube do not matter, so we will work with a cube
with sides of length 1 . Orient the cube relative to the coordinate axes in IR 3 , as shown
in Figure 1 .34, and take the two side diagonals to be the vectors [ 1 , 0, 1 ] and [O, 1 , 1 ] .
Then angle e between these vectors satisfies
Solution
cos e =
l·O + O·l + l·l
2
\/2 v'2
from which it follows that the required angle is n /3 radians, or 60°.
-----
z
[O, 1 , l ]
y
x
Figure 1 . 3 4
(Actually, we don't need to do any calculations at all to get this answer. If we draw
a third side diagonal joining the vertices at ( 1 , 0, 1 ) and (O, 1 , 1 ) , we get an equilateral
triangle, since all of the side diagonals are of equal length. The angle we want is one of
the angles of this triangle and therefore measures 60°. Sometimes, a little insight can
save a lot of calculation; in this case, it gives a nice check on our work!)
R e m a rks
As this discussion shows, we usually will have to settle for an approximation
to the angle between two vectors. However, when the angle is one of the so-called
special angles (0°, 30°, 45°, 60°, 90°, or an integer multiple of these), we should be able
to recognize its cosine (Table 1 . 1 ) and thus give the corresponding angle exactly. In
all other cases, we will use a calculator or computer to approximate the desired angle
by means of the inverse cosine function.
•
e
Ta b l e 1 . 1
cos e
cosines or Special Angles
30°
V4
-= 1
2
v3
2
45°
v'2
2
60°
1
v'2
Vi
2
90°
2
Vo
-=O
2
26
Chapter
1 Vectors
The derivation of the formula for the cosine of the angle between two vectors
is valid only in IR 2 or IR 3 , since it depends on a geometric fact: the law of cosines. In
IR n , for n > 3, the formula can be taken as a definition instead. This makes sense, since
u·v
u·v
the Cauc�y-Schwarz ! � equality �mplies that
ranges from
:::::: 1 , so I
I
u
u
ll ll v ll
ll ll v ll
- 1 to 1, J USt as the cosme funct10n does.
•
I I
Orthogonal Veclors
The word orthogonal is derived
from the Greek words orthos, mean­
ing "upright;' and gonia, meaning
"angle:' Hence, orthogonal literally
means "right-angled'.' The Latin
equivalent is rectangular.
The concept of perpendicularity is fundamental to geometry. Anyone studying
geometry quickly realizes the importance and usefulness of right angles. We now gen­
eralize the idea of perpendicularity to vectors in !R n , where it is called orthogonality.
In IR 2 or IR 3 , two nonzero vectors u and v are perpendicular if the angle (J between
u·v
= cos 90° = 0,
them is a right angle-that is, if (J = 1T /2 radians, or 90°. Thus, I
ll u ll v ll
and it follows that u · v = 0. This motivates the following definition.
DefiniliOD
Two vectors u and v in !R n are orthogonal to each other if u ·v = 0.
Since 0 · v = 0 for every vector v in !R n , the zero vector is orthogonal to every
vector.
Exa m p l e 1 . 2 3
In IR 3 , u = [ l , 1 , - 2 ] and v = [3, 1 , 2] are orthogonal, since u · v
3 + 1 - 4 = 0.
Using the notion of orthogonality, we get an easy proof of Pythagoras' Theorem,
valid in !R n .
Theorem 1 . 6
Pythagoras' 1heoreITI
For all vectors u and v in lR n , ll u + v ll 2 = ll u ll 2 + ll v ll 2 if and only if u and v are
orthogonal.
From Example 1 . 16, we have ll u + v ll 2 = ll u ll 2 + 2 ( u · v) + ll v ll 2 for all
vectors u and v in !R n . It follows immediately that ll u + v ll 2 = ll u ll 2 + ll v ll 2 if and
only if u · v = 0. See Figure 1 .35.
Proof
Figure 1 . 3 5
The concept of orthogonality is one of the most important and useful in linear
algebra, and it often arises in surprising ways. Chapter 5 contains a detailed treatment
of the topic, but we will encounter it many times before then. One problem in which
it clearly plays a role is finding the distance from a point to a line, where "dropping a
perpendicular" is a familiar step.
Section
1.2 Length and Angle: The Dot Product
21
Proieclions
We now consider the problem of finding the distance from a point to a line in the
context of vectors. As you will see, this technique leads to an important concept: the
projection of a vector onto another vector.
As Figure 1 .36 shows, the problem of finding the distance from a point B to a
line € (in IR 2 or IR 3 ) reduces to the problem of finding the length of the perpendicular
line segment PB or, equivalently, the length of the vector PB. If we choose a point
A
"---->
on €, then, in the----"
right-angled
triangle
fl.APB,
the
other
two
vectors
are
the
leg
AP
and
----->
-----;
the hypotenuse AB. AP is called the projection of AB onto the line €. We will now look
at this situation in terms of vectors.
B
B
e
A
Figure 1 . 3 6
The distance from a point to a line
_..,.,.-figure 1 . 3 1
The projection of v onto u
u
vu= l Puv.l u, u l u )u u v,
=
= v
Consider two nonzero vectors and Let p be the vector obtained by dropping
a perpendicular from the head of onto and let e be the angle between and as
( I / I is the unit vector in
shown in Figure 1 .37. Then clearly p
where
I II cos e , and
the direction of Moreover, elementary trigonometry gives I p I
Thus, after substitution, we obtain
we know that cos e II II
u. u·v
= u l v1 .
p
= l v l C1 uil · �l vl l ) (ilf)u
u·v )u
= (�
= (�)u
u·u
This is the formula we want, and it is the basis of the following definition for vec­
tors in !R n .
v u
u v
u * 0,
u·v
= (-u·u) u
If and are vectors in !R n and
onto is the vector proju(v) defined by
Definition
proj0 ( v )
then the projection of
An alternative way to derive this formula is described in Exercise 73.
28
Chapter
1 Vectors
R e m a rks
The term projection comes from the idea of projecting an image onto a wall
(with a slide projector, for example) . Imagine a beam oflight with rays parallel to each
other and perpendicular to u shining down on v. The projection of v onto u is just the
shadow cast, or projected, by v onto u.
•
It may be helpful to think of proj0(v) as a function with variable v. Then the
variable v occurs only once on the right-hand side of the definition. Also, it is helpful
to remember Figure 1 .38, which reminds us that proj0(v) is a scalar multiple of the
vector u (not v) .
•
Although in our derivation of the definition of proj0(v) we required v as well
as u to be nonzero (why?), it is clear from the geometry that the projection of the
0
zero vector onto u is 0. The definition is in agreement with this, since "
u=
u·u
Ou = 0.
•
If the angle between u and v is obtuse, as in Figure 1 .38, then proj0(v) will be in
the opposite direction from u; that is, proj0(v) will be a negative scalar multiple of u.
•
If u is a unit vector then proj0(v) = ( u · v) u. (Why?)
•
proju(v)
u
Figure 1 . 3 8
( )
.
Exa m p l e 1 . 2 4
Find the projection of v onto u in each case.
(a) v =
(c) v =
[ - �] [ �]
[ � ] [ ��� ]
[ �] [ - �]
and u =
and u =
3
1 / \/2
(a) We compute u · v =
·
Solution
proj u ( v) =
(b) Since e3 is a unit vector,
[ �] [ �]
[] [ ]
= 1 and u · u =
( )
·
2 /5
u·v
1 2
=
=
"
u·u
1 /5
s 1
(c) We see that ll u ll = V� + � + t = 1 . Thus,
proj0 ( v) = ( u · v) u =
�
(�
3(I
= 5, so
�) [ ��� ]
: [ �l
+ 1 +
V2J
1 / \/2
=
3( 1
� \/2)
[ ��� ]
1 / \/2
J
Section
Exercises 1 . 2
5. u = [ 1 , \/2, \/3, o ] , v = [ 4, - \/2, 0, - 5 ]
6. U = [ 1 . 1 2, - 3.25, 2.07, - 1 .83 ] ,
v = [ - 2.29, 1 .72, 4.33, - 1 .54 ]
In Exercises 7-12, find II u II for the given exercise, and give
a unit vector in the direction of u.
7. Exercise 1
GAs 10. Exercise 4
8. Exercise 2
1 1 . Exercise 5
GAs
9. Exercise 3
12. Exercise 6
In Exercises 13-16, find the distance d(u, v) between u and
v in the given exercise.
14. Exercise 2
13. Exercise 1
GAs
15. Exercise 3
16. Exercise 4
1 7. If u, v, and w are vectors in ll�r, n 2: 2, and c is a
scalar, explain why the following expressions make
no sense:
(b) u · v + w
(a) ll u · v ll
(
)
(c) u · v · w
(d) c . ( u + w )
In Exercises 18-23, determine whether the angle between
u and v is acute, obtuse, or a right angle.
20. u = [4, 3, - 1 ] , v = [ l , - 1 , l ]
GAS 2 1 . U = [0.9, 2 . 1 , 1 .2] , V = [ - 4.5, 2.6, - 0.8]
22. u = [ 1 , 2, 3, 4], v = [ - 3, 1 , 2, - 2 ]
23. u = [ 1 , 2, 3, 4], v = [5, 6, 7, 8]
In Exercises 24-29, find the angle between u and v in the
given exercise.
24. Exercise 1 8
29
..
In Exercises 1-6, find u · v.
GAS
1.2 Length and Angle: The Dot Product
25. Exercise 1 9
GAs
27. Exercise 2 1
26. Exercise 20
GAs
GAs 28. Exercise 22
29. Exercise 23
30. Let A = ( - 3, 2), B = ( 1 , 0), and C = (4, 6). Prove that
MBC is a right-angled triangle.
31. Let A = ( 1 , 1, - 1 ), B = ( - 3, 2, - 2), and C = (2, 2, - 4).
Prove that �ABC is a right-angled triangle.
GAs 32. Find the angle between a diagonal of a cube and an ad­
jacent edge.
33. A cube has four diagonals. Show that no two of them
are perpendicular.
34. A parallelogram has diagonals determined by the
vectors
d, �
[H
and d, �
[ - �-
Show that the parallelogram is a rhombus (all sides of
equal length) and determine the side length.
35. The rectangle ABCD has vertices at A = ( 1 , 2, 3),
B = (3, 6, - 2), and C = (O, 5, - 4) . Determine the
coordinates of vertex D.
36. An airplane heading due east has a velocity of
200 miles per hour. A wind is blowing from the north
at 40 miles per hour. What is the resultant velocity of
the airplane?
37. A boat heads north across a river at a rate of 4 miles
per hour. If the current is flowing east at a rate of
3 miles per hour, find the resultant velocity of
the boat.
38. Ann is driving a motorboat across a river that is 2 km
wide. The boat has a speed of 20 km/h in still water, and
the current in the river is flowing at 5 km/h. Ann heads
out from one bank of the river for a dock directly across
from her on the opposite bank. She drives the boat in a
direction perpendicular to the current.
(a) How far downstream from the dock will
Ann land?
(b) How long will it take Ann to cross the river?
39. Bert can swim at a rate of 2 miles per hour in still
water. The current in a river is flowing at a rate of
1 mile per hour. If Bert wants to swim across the river
to a point directly opposite, at what angle to the bank
of the river must he swim?
30
Chapter
1 Vectors
In Exercises 40-45, find the projection of v onto u. Draw a
sketch in Exercises 40 and 41.
In Exercises 48 and 49, find all values of the scalar k for
which the two vectors are orthogonal.
48. u �
[ : Jv [ � � : l
�
50. Describe all vectors v =
[
]
to u =
]
[
1 .34
3.01
CAS 45. U =
4.25
- 0.33 V =
- 1 .66
2.52
Figure 1 .39 suggests two ways in which vectors
may be used to compute the area of a triangle.
The area A of
,
51. Describe all vectors v =
to u =
[�] .
[ -J [ _fl
�
[;]
that are orthogonal
[;]
that are orthogonal
52. Under what conditions are the following true for
vectors u and v in IR 2 or IR 3 ?
(a) ll u + v ii = ll u ll + ll v ll (b) ll u + v ii = ll u ll - ll v ll
53. Prove Theorem 1 .2 (b).
54. Prove Theorem 1 .2 (d) .
In Exercises 55-57, prove the stated property of distance
between vectors.
55. d(u, v) = d(v, u) for all vectors u and v
56. d(u, w) :s d(u, v) + d(v, w) for all vectors u, v, and w
57. d(u, v) = 0 if and only if u = v
58. Prove that u · c v = c( u · v) for all vectors u and v in !R n
and all scalars c.
59. Prove that ll u - v ii 2': ll u ll - ll v ll for all vectors u and
v in !R n . [Hint: Replace u by u - v in the Triangle
(a)
(b)
[�] .
49. u �
u
Figure 1 . 3 9
the triangle in part (a) is given by t ll u ll ll v - proj u ( v ) II ,
and part (b) suggests the trigonometric form of the
area of a triangle: A = t I u I I v II sin e (We can use the
identity sin e = v 1 - cos 2 e to find sin e.)
In Exercises 46 and 47, compute the area of the triangle
with the given vertices using both methods.
46. A = ( 1 , - 1 ), B = (2, 2), C = (4, O)
47. A = (3, - 1 , 4), B = (4, - 2 , 6), C = (5, 0, 2)
Inequality.]
60. Suppose we know that u · v = u · w. Does it follow that
v = w? If it does, give a proof that is valid in !R n ;
otherwise, give a counterexample (i.e., a specific set of
vectors u, v, and w for which u · v = u · w but v -=F w) .
61. Prove that (u + v) · (u - v) = ll u ll 2 - ll v ll 2 for all vec­
tors u and v in !R n .
62. (a) Prove that ll u + v ll 2 + ll u - v ll 2 = 2 ll u ll 2 + 2 ll v ll 2
for all vectors u and v in !R n .
(h) Draw a diagram showing u, v, u + v, and u - v
in IR 2 and use (a) to deduce a result about
parallelograms.
1
1
63. Prove that u · v = - ll u + v ll 2 - - ll u - v ll 2 for all
4
4
vectors u and v in !R n .
Section
64. (a) Prove that ll u + v ii = ll u - v ii if and only if u and
v are orthogonal.
(b) Draw a diagram showing u, v, u + v, and u - v
in IR 2 and use (a) to deduce a result about
parallelograms.
65. ( a) Prove that u + v and u - v are orthogonal in !R n if
and only if ll u ll = ll v ll .
(b) Draw a diagram showing u, v, u + v, and u - v
in IR 2 and use (a) to deduce a result about
parallelograms.
1.2 Length and Angle: The Dot Product
31
72. Figure 1 .40 shows that, in IR 2 or IR 3 ,
ll proju ( v ) I :::::: ll v ll ·
(a) Prove that this inequality is true in general. [Hint:
Prove that proju(v) is orthogonal to v - proju(v)
and use Pythagoras' Theorem.]
(b) Prove that the inequality ll proju ( v ) I :::::: ll v ll is
equivalent to the Cauchy-Schwarz Inequality.
66. If ll u ll = 2, ll v ll = v'3 , and u · v = 1, find ll u + v ii .
67. Show that there are no vectors u and v such that ll u ll = 1,
ll v ll = 2, and u · v = 3.
68. ( a) Prove that if u is orthogonal to both v and w, then
u is orthogonal to v + w.
(b) Prove that if u is orthogonal to both v and w, then
u is orthogonal to sv + tw for all scalars s and t.
69. Prove that u is orthogonal to v - proju(v) for all
vectors u and v in !R n , where u * 0.
70. ( a) Prove that proju(proju(v)) = proju(v) ·
(b) Prove that proju(v - proju(v)) = 0.
proju(v)
u
Figure 1 . 4 0
73. Use the fact that proju(v) = c u for some scalar c, to­
gether with Figure 1 . 4 1 , to find c and thereby derive
the formula for proju(v) .
(c ) Explain (a) and (b) geometrically.
71. The Cauchy-Schwarz Inequality I u · v i :::::: ll u ll ll v ll is
v
equivalent to the inequality we get by squaring both
sides: (u · v) 2 :::::: ll u ll 2 ll v ll 2 .
( a) In IR 2 , with u =
[ �: ]
and v =
[ :J
this becomes
Prove this algebraically. [Hint: Subtract the left-hand
side from the right-hand side and show that the
difference must necessarily be nonnegative.]
(b) Prove the analogue of (a) in IR 3 .
-
cu
Figure 1 . 4 1
cu
u
74. Using mathematical induction, prove the following
generalization of the Triangle Inequality:
ll v1 + Vz + · · · + vn ll ::=::: ll v1 ll + ll vz ll + · · · + ll vn ll
for all n 2: 1.
Exp loration
Ve ctors and G e o m etry
Many results i n plane Euclidean geometry can b e proved using vector techniques.
For example, in Example 1 .24, we used vectors to prove Pythagoras' Theorem. In
this exploration, we will use vectors to develop proofs for some other theorems from
Euclidean geometry.
As an introduction to the notation and the basic approach, consider the following
easy example.
Exa m p l e 1 . 2 5
Give a vector description of the midpoint M of a line segment AB.
We first convert everything to vector notation. If 0 denotes the origin and
P----is-> a point,
let---->
p be the vector GP. In this situation, a = GA, b = GB, m = oM, and
------>
AB = OB - OA = b - a (Figure 1.42).
Now, since M is the midpoint of AB, we have
Solution
A
B
m
so
Figure 1 . 4 2
The midpoint of AB
m
- a = AM = fAB = t ( b - a )
= a + t (b - a) = t ( a + b)
1 . Give a vector description of the point P that is one-third of the way from A to
B on the line segment AB. Generalize.
c
A ----------.... B
Figure 1 . 4 3
32
2. Prove that the line segment joining the midpoints of two sides of a triangle is
parallel to the third side and half as long. (In vector notation, prove that PQ = t AB in
Figure 1 .43.)
3. Prove that the quadrilateral PQRS (Figure 1 .44), whose vertices are the mid­
points of the sides of an arbitrary quadrilateral ABCD, is a parallelogram.
4. A median of a triangle is a line segment from a vertex to the midpoint of
the opposite side (Figure 1 .45). Prove that the three medians of any triangle are con­
current (i.e., they have a common point of intersection) at a point G that is two­
thirds of the distance from each vertex to the midpoint of the opposite side. [ Hint: In
Figure 1 .46, show that the point that is two-thirds of the distance from A to P is given
by t ( a + b + c). Then show that t ( a + b + c ) is two-thirds of the distance from B
to Q and two-thirds of the distance from C to R.] The point G in Figure 1 .46 is called
the centroid of the triangle.
A
A
B
B
Figure 1 . 4 4
c
c
Figure 1 . 4 5
Figure 1 . 4 6
A median
The centroid
5. An altitude of a triangle is a line segment from a vertex that is perpendicu­
lar to the opposite side (Figure 1 .47) . Prove that the three altitudes of a triangle are
concurrent. [Hint: Let H be the point of intersection of the altitudes from A and B in
Figure 1 .48. Prove that cH is orthogonal to AB .] The point H in Figure 1 .48 is called
the orthocenter of the triangle.
6. A perpendicular bisector of a line segment is a line through the midpoint of
the segment, perpendicular to the segment (Figure 1 .49). Prove that the perpendicular
bisectors of the three sides of a triangle are concurrent. [Hint: Let K be the point of in­
tersection of t� erpendicular bisectors of AC and BC in Figure 1 .50. Prove that RK is
orthogonal to AB .] The point K in Figure 1 .50 is called the circumcenter of the triangle.
c
A
B
A --�-*��������--=e B
Figure 1 . 41
Figure 1 . 4 8
Figure 1 . 4 9
An altitude
The orthocenter
A perpendicular bisector
7. Let A and B be the endpoints of a diameter of a circle. If C is any point on the
circle, prove that LACE is a right angle. [Hint: In Figure 1 . 5 1 , let 0 be the center of the
circle. Express everything in terms of a and c and show that AC is orthogonal to BC. J
8. Prove that the line segments joining the midpoints of opposite sides of a
quadrilateral bisect each other (Figure 1 .52).
c
c
0
D
R
Figure 1 . 5 0
The circumcenter
Figure 1 . 5 1
Figure 1 . 5 2
33
34
Chapter
1 Vectors
lines a n d P l a n es
We are all familiar with the equation of a line in the Cartesian plane. We now want
to consider lines in IR 2 from a vector point of view. The insights we obtain from this
approach will allow us to generalize to lines in IR 3 and then to planes in IR 3 . Much of
the linear algebra we will consider in later chapters has its origins in the simple geom­
etry of lines and planes; the ability to visualize these and to think geometrically about
a problem will serve you well.
Lines in � 2 and � 3
In the xy-plane, the general form of the equation of a line is ax + by = c. If b * 0, then
the equation can be rewritten as y = - ( a/b )x + c/b, which has the form y = mx + k.
[This is the slope-intercept form; m is the slope of the line, and the point with coordi­
nates (O, k) is its y-intercept.J To get vectors into the picture, let's consider an example.
Exa m p l e 1 . 2 6
The line C with equation 2x + y = 0 is shown in Figure 1 .53. It is a line with slope - 2
passing through the origin. The left-hand side of the equation is in the form of a dot
product; in fact, if we let n =
[ �]
and x =
[; ]
, then the equation becomes n · x = 0.
The vector n is perpendicular to the line-that is, it is orthogonal to any vector x that
is parallel to the line (Figure 1 .54)-and it is called a normal vector to the line. The
equation n x = 0 is the normal form of the equation of e.
Another way to think about this line is to imagine a particle moving along the
line. Suppose the particle is initially at the origin at time t = 0 and it moves along
the line in such a way that its x-coordinate changes 1 unit per second. Then at t = 1
the particle is at ( 1 , - 2 ), at t = 1 .5 it is at ( 1 .5, - 3 ), and, if we allow negative values of t
(i.e., we consider where the particle was in the past), at t = - 2 it is (or was) at ( - 2, 4).
.
y
y
The Latin word norma refers to a
carpenter's square, used for draw­
ing right angles. Thus, a normal
vector is one that is perpendicular
to something else, usually a plane.
2x + 0
Figure 1 . 5 3
The line
y =
Figure 1 . 5 4
A normal vector n
Section
1. 3 Lines and Planes
35
This movement is illustrated in Figure 1 .55. In general, if x = t, then y = -2t, and we
may write this relationship in vector form as
What is the significance of the vector d =
[ �] ?
_
It is a particular vector parallel
to e, called a direction vector for the line. As shown in Figure 1 .56, we may write the
equation of e as x = td. This is the vector form of the equation of the line.
If the line does not pass through the origin, then we must modify things
slightly.
y
y
e
Figure 1 . 5 6
A direction vector d
Figure 1 . 5 5
Exa m p l e 1 . 2 1
Consider the line C with equation 2x + y = 5 (Figure 1 .57). This is just the line from
Example 1 .26 shifted upward 5 units. It also has slope - 2, but its y-intercept is the
point (O, 5). It is clear that the vectors d and n from Example 1 .26 are, respectively, a
direction vector and a normal vector for this line too.
Thus, n is orthogonal to every vector that is parallel to e. The point P = ( 1, 3)
---+
is on C. If X = ( x, y) represents a general point on C, then the vector PX = x p is
parallel to e and n · (x p ) = 0 (see Figure 1 .58 ) . Simplified, we have n · x = n · p .
As a check, we compute
-
-
n·x =
[ �] [;]
·
= 2x + y
and
n
·p =
[ �] [ �]
·
= 5
Thus, the normal form n · x = n · p is just a different representation of the general
form of the equation of the line. (Note that in Example 1 .26, p was the zero vector, so
n · p = 0 gave the right-hand side of the equation.)
36
Chapter
1 Vectors
y
y
n
-t--j--+--t--t--f--\-t-jl-t-... x
2x + 5
figure 1 . 5 1
The line
y =
(x - p) =
figure 1 . 5 8
n •
0
These results lead to the following definition.
Definition
The normal form of the equation of a line e in IR 2 is
n · ( x - p ) = 0 or n · x = n · p
where p is a specific point on e and n =F 0 is a normal vector for e.
The general form of the equation of e is ax +
= where n =
by
normal vector for e.
C,
[ ab]
i· s a
Continuing with Example 1 .27, let us now find the vector form of the equation
of e. Note that, for each choice of X, x - p must be parallel to-and thus a multiple
of-the direction vector d. That is, x - p = td or x = p + td for some scalar t. In
terms of components, we have
or
The word parameter and the cor­
responding adjective parametric
come from the Greek words para,
meaning "alongside;' and metron,
meaning "measure:' Mathemati­
cally speaking, a parameter is a
variable in terms of which other
variables are expressed-a new
"measure" placed alongside
old ones.
[;] [ �] [ - � ]
y +t
(1)
x= 1 + t
= 3
2t
(2)
Equation ( 1) is the vector form of the equation of €, and the componentwise Equa­
tions ( 2) are called parametric equations of the line. The variable t is called a parameter.
How does all of this generalize to IR 3 ? Observe that the vector and parametric
forms of the equations of a line carry over perfectly. The notion of the slope of a line
in IR 2 -which is difficult to generalize to three dimensions-is replaced by the more
convenient notion of a direction vector, leading to the following definition.
The vector form of the equation of a line e in IR 2 or IR 3 is
x = p + td
where p is a specific point on e and d =F 0 is a direction vector for e.
The equations corresponding to the components of the vector form of the
equation are called parametric equations of e.
Definition
Section
1.3 Lines and Planes
31
We will often abbreviate this terminology slightly, referring simply to the general,
normal, vector, and parametric equations of a line or plane.
Exa m p l e 1 . 2 8
[ -:J
Find vector and parametric equations of the line in IR 3 through the point P = ( 1, 2, - 1 ),
pornllel to the mtm d �
Solution
The vector equation x = p + td is
The parametric form is
x = 1 + St
y= 2 - t
z = - 1 + 3t
R e m a rks
The vector and parametric forms of the equation of a given line e are not
unique-in fact, there are infinitely many, since we may use any point on e to de­
termine p and any direction vector for e. However, all direction vectors are clearly
multiples of each other.
1o
In Example 1 .28, (6, 1 , 2) is another point on the line (take t = 1), and - 2 is
another direction vector. Therefore,
6
•
[ l
gives a different (but equivalent) vector equation for the line. The relationship between
the two parameters s and t can be found by comparing the parametric equations: For
a given point ( x, y, z) on e, we have
x = 1 + St = 6 + 10s
y = 2 - t = 1 - 2s
z = - 1 + 3t = 2 + 6s
implying that
- 10s + St = S
2s - t = -1
- 6s + 3t = 3
Each of these equations reduces to t = 1 + 2s.
38
Chapter
1 Vectors
3
Intuitively, we know that a line is a one-dimensional object. The idea of
"dimension" will be clarified in Chapters and 6, but for the moment observe that
this idea appears to agree with the fact that the vector form of the equation of a line
requires one parameter.
•
Exa m p l e 1 . 2 9
One often hears the expression "two points determine a line:' Find a vector equation
of the line f in IR 3 determined by the points P = ( - 1 , 5, o) and Q = ( 2, 1 , 1 ) .
Solulion
fine) .
3[ ]
We may choose any point on f for p, so we will use P ( Q would also be
A convenient direction vector is d = PQ =
Thus, we obtain
-4
1
(or any scalar multiple of this).
Planes i n lR 3
n is orthogonal to infinitely many
vectors
Figure 1 . 5 9
n
p
n · (x - p ) =
Figure 1 . 6 0
0
The next question we should ask ourselves is, How does the general form of the equa­
tion of a line generalize to IR 3 ? We might reasonably guess that if ax + by = c is the
general form of the equation of a line in IR 2 , then ax + by + cz = d might represent a
line in IR 3 . In normal form, this equation would be n · x = n · p, where n is a normal
vector to the line and p corresponds to a point on the line.
To see if this is a reasonable hypothesis, let's think about the special case of the
equation ax + by + cz � 0. In normal furn, it becomes n • x � 0, wh"e n �
[�l
·
However, the set of all vectors x that satisfy this equation is the set of all vectors or­
thogonal to n. As shown in Figure 1 .59, vectors in infinitely many directions have
this property, determining a family of parallel planes. So our guess was incorrect: It
appears that ax + by + cz = d is the equation of a plane-not a line-in IR 3 .
Let's make this finding more precise. Every plane <!]' in IR 3 can be determined by
specifying a point p on <!I' and a nonzero vector n normal to <!I' (Figure 1 .60) . Thus,
if x represents an arbitrary point on <!I', we have n · (x - p) = 0 or n · x = n · p. If
n
�
[�]
and x
�
[� l
then, in teems of rnmponents, the equation bernmes
ax + by + cz = d (where d = n · p ) .
The normal form of the equation of a plane <!I' in IR 3 is
n · (x - p) = 0 or n · x = n · p
where p is a specific point on <!I' and n * 0 is a normal vector for <!I' .
Definition
The general form of the equation of <!I' i s ax + by + cz = d , where n =
is a normal vector for <!I' .
Section
1.3 Lines and Planes
39
Note that any scalar multiple of a normal vector for a plane is another normal
vector.
Exa m p l e 1 . 3 0
Find the normal and general forms of the equation of the plane that contains the
point P � (6, O, 1 ) and har normal vodor n �
- With p �
[fl
and x �
[� ].
[H
we havr n · p � 1 · 6 + 2 ' 0 � 3 · 1 � 9, 'o
the normal equation n · x = n · p becomes the general equation x + 2y + 3z = 9 .
.+
Geometrically, it is clear that parallel planes have the same normal vector(s) .
Thus, their general equations have left-hand sides that are multiples of each other. So, for
example, 2x + 4y + 6z = 10 is the general equation of a plane that is parallel to the
plane in Example 1 .30, since we may rewrite the equation as x + 2y + 3z = 5-from
which we see that the two planes have the same normal vector n. (Note that the planes
do not coincide, since the right-hand sides of their equations are distinct.)
We may also express the equation of a plane in vector or parametric form. To do
so, we observe that a plane can also be determined by specifying one of its points
P (by the vector p ) and two direction vectors u and v parallel to the plane (but not
parallel to each other). As Figure 1 .6 1 shows, given any point X in the plane (located
,<= I
x
tv
-
p = s u + tv
-x
SU
figure 1 . 6 1
X -
p = SU + t v
by x), we can always find appropriate multiples s u and tv of the direction vectors such
that x - p = su + tv or x = p + s u + tv. If we write this equation componentwise,
we obtain parametric equations for the plane.
The vector form of the equation of a plane <lP in IR 3 is
X = p + SU + tv
where p i s a point o n <lP and u and v are direction vectors fo r <lP ( u and v are non zero and parallel to <lP , but not parallel to each other) .
The equations corresponding to the components of the vector form of the
equation are called parametric equations of <lP .
Definition
Chapter
40
1 Vectors
Exa m p l e 1 . 3 1
Find vector and parametric equations for the plane in Example 1 .30.
We need to find two direction vectors. We have one point P = (6, 0, 1 ) in
the plane; if we can find two other points Q and R in <!J', then the vectors PQ and PR
can serve as direction vectors (unless by bad luck they happen to be parallel!). By
trial and error, we observe that Q = (9, 0, O) and R = (3, 3, O) both satisfy the general
equation x + 2y + 3z = 9 and so lie in the plane. Then we compute
Solulion
Figure 1 . 6 2
\
which, since they are not scalar multiples of each other, will serve as direction vectors.
Therefore, we have the vector equation of <!J',
Two normals determine a line
and the corresponding parametric equations,
<;JP I
Figure 1 . 6 3
The intersection of
two planes is a line
x = 6 + 3s - 3t
�
y = 3t
z= l - s- t
[What would have happened had we chosen R = (O, 0, 3)?]
R e m a rks
A plane is a two-dimensional object, and its equation, in vector or parametric
form, requires two parameters.
•
As Figure 1 .59 shows, given a point P and a nonzero vector n in IR 3 , there are
infinitely many lines through P with n as a normal vector. However, P and two non­
parallel normal vectors n 1 and n2 do serve to locate a line e uniquely, since e must
then be the line through P that is perpendicular to the plane with equation x = p +
sn 1 + tn2 (Figure 1 .62) . Thus, a line in IR 3 can also be specified by a pair of equations
•
a 1 x + b1y + c 1 z = d 1
a 1X + b 2y + C 2 Z = d1
one corresponding to each normal vector. But since these equations correspond to a
pair of nonparallel planes (why nonparallel?), this is just the description of a line as
the intersection of two nonparallel planes (Figure 1 .63). Algebraically, the line con­
sists of all points (x, y, z) that simultaneously satisfy both equations. We will explore
this concept further in Chapter 2 when we discuss the solution of systems of linear
equations.
Tables 1 .2 and 1 . 3 summarize the information presented so far about the equa­
tions of lines and planes.
Observe once again that a single (general) equation describes a line in IR 2 but
a plane in IR 3 . [In higher dimensions, an object (line, plane, etc.) determined by a
single equation of this type is usually called a hyperplane.] The relationship among
Section
Ta b l e 1 . 3
1. 3 Lines and Planes
Ta b l e 1 . 2
Equations or lines in !R 2
Normal Form
General Form
Vector Form
n·x=n·p
ax + by = c
x = p + td
41
{ yx == P 1 ++ tdtd1
Parametric Form
P2
2
lines and Planes in !R 3
Lines
{n1 · x = n1 · p1
n2 • x = n2 • P2
{a 1X + b 1y + C1Z = d1
x = p + td
Planes
n·x = n·p
ax + by + cz = d
X = p + SU + tv
Normal Form
Vector Form
General Form
a zX + b1Y + C2 Z = dz
Parametric Form
{ ;:
{;:
P1 + td 1
P2 + td2
z = p 3 + td3
p 1 + su 1 + tv 1
p2 + su 2 + tv2
z = p 3 + S U 3 + tv3
the dimension of the object, the number of equations required, and the dimension of
the space is given by the "balancing formula'':
(dimension of the object) + (number of general equations) = dimension of the space
The higher the dimension of the object, the fewer equations it needs. For
example, a plane in IR 3 is two-dimensional, requires one general equation, and lives
in a three-dimensional space: 2 + 1 = 3. A line in IR 3 is one-dimensional and so
needs 3 1 = 2 equations. Note that the dimension of the object also agrees with
the number of parameters in its vector or parametric form. Notions of "dimension"
will be clarified in Chapters 3 and 6, but for the time being, these intuitive observa­
tions will serve us well.
We can now find the distance from a point to a line or a plane by combining the
results of Section 1 .2 with the results from this section.
-
Exa m p l e 1 . 3 2
Find the distance from the point B = ( 1 , 0, 2) to the line € through the point
A � (3, ! , I ) with dimhon vedoc d �
[ -u
--+
As we have already determined, we need to calculate the length of ---->
PB,
where P is the point on e at the foot of the perpendicular from B. If we label v = AB,
then AP = projd(v) and PB = v projd ( v) (see Figure 1 .64) . We do the necessary
calculations in several steps.
Solution
-
42
Chapter
1 Vectors
B
v - profa(v)
___...--- e
d
A
= llv - projd ( v ) II
Figure 1 . 6 4
d ( B, € )
Step 2: The projection of v onto d is
projd ( v ) =
=
(:��)
d
(( - 1 ) · ( (--21) ) 1·1 ( - 10) 0· 1 ) [ - �
ol
+
2+
+
+
Step 3: The vector we want is
Step 4:
The distance d(B, f) from B to e is
Using Theorem
1.3(b)
ll v - prnj , ( v ) I
�
[ =! ]
[ =� -
to simplify the calculation, we have
ll v - prnj , ( v ) II � l
= t v9 + 9 + 4
= t v22
Nole
•
In terms of our earlier notation, d(B, f) = d(v, projd(v)).
Section
1.3 Lines and Planes
43
In the case where the line £ is in IR 2 and its equation has the general form
ax + by = c, the distance d(B, €) from B = (x0, y0 ) is given by the formula
d ( B, € ) =
l ax0 + by0 c l
Va z + b 2
-
(3)
You are invited to prove this formula in Exercise 39.
Exa m p l e 1 . 3 3
Find the distance from the point B = ( 1 , 0, 2) to the plane <fP whose general equation
is x + y z = 1 .
-
In this case, we need to calculate the length of PB, where P is the point on
<fP at the foot of the perpendicular from B. As Figure 1 .65 shows, if A is any point on
Solution
'ii' and we 'ituate thr nurnrnl vectu' n �
[_:]
of 'ii' ' ° that it; tail [, at A, then we
need to find the length of the projection of AB onto n. Again we do the necessary
calculations in steps.
n
B
I
I
I
I
I
11 p
Figure 1 . 6 5
d ( B, <!P ) = ll proj n (AB) II
Step 1:
x+y
-
Step 2:
Step 3:
By trial and error, we find any point whose coordinates satisfy the equation
z = 1 . A = ( 1 , 0, O) will do.
Set
The projection of v onto n is
(n n)
( 1 + 1 + ( - 1 ) 2 ) [ �1
proj n ( V ) = � n
·
=
1 ·0 + 1·0 - 1•2
-t: l [=! ]
-1
Chapter
44
1 Vectors
Step 4:
The distance d(B, <JP) from B to <JP is
[J
[J
l l r rnj " ( vl ll � H
�i
In general, the distance d(B, <JP) from the point B = (x0, y0 , z0 ) to the plane whose
general equation is ax + by + cz = d is given by the formula
d ( B, <JP )
l ax0 + by0 + CZ0 - d i
Va 2 + b 2 + c 2
(4)
You will be asked to derive this formula in Exercise 40.
..
I
Exercises 1 . 3
In Exercises 1 and 2, write the equation of the line passing
through P with normal vector n in (a) normal form and
(b) general form.
I . P = ( 0, 0 ) , n =
[�]
2. P = ( 1, 2 ) , n =
[ _! ]
In Exercises 3-6, write the equation of the line passing
through P with direction vector d in (a) vector form and
(b) parametric form.
3. P = ( 1, 0 ) , d =
S. P � (0, 0, 0), d �
[ - �]
[ -: l
[�]
_ m
4. p = ( - 4, 4 ) , d =
•.
p � ( 3 . o. 2 ) . d �
In Exercises 7 and 8, write the equation of the plane passing
through P with normal vector n in (a) normal form and
(b) general form.
7. P � ( 0, 1, 0 ) , n �
[;]
8. P � ( 3, 0, - 2 ) , n �
m
In Exercises 9 and 1 0, write the equation of the plane pass­
ing through P with direction vectors u and v in (a) vector
form and (b) parametric form.
9. p � ( o. o. 0 ) . u �
}
[ [ :l
[ l [ - :i
10. p � ( 6, - 4, - 3), u �
�
�
In Exercises 1 1 and 12, give the vector equation of the line
passing through P and Q.
1 1 . P = ( 1, - 2), Q = (3, O)
12. P = (O, 1, - 1), Q = ( - 2, 1, 3)
In Exercises 1 3 and 1 4, give the vector equation of the plane
passing through P, Q, and R.
13. P = ( 1, 1, 1 ) , Q = (4, 0, 2), R = (O, 1, - 1 )
14. P = ( 1, 1, O), Q = ( 1 , 0, 1 ), R = (O, 1 , 1)
15. Find parametric equations and an equation in vector
form for the lines in IR 2 with the following equations:
(a) y = 3x - 1
(b) 3x + 2y = 5
Section
16. Consider the vector equation x = p + t(q - p ) , where
p and q correspond to distinct points P and Q in IR 2
or IR 3 .
(a) Show that this equation describes the line segment
PQ as t varies from 0 to 1 .
(b) For which value of t is x the midpoint of PQ,
and what is x in this case?
(c) Find the midpoint of PQ when P = (2, - 3 ) and
Q = ( 0, 1 ) .
(d) Find the midpoint o f P Q when P = ( 1 , 0, 1 )
and Q = ( 4, 1 , - 2) .
(e) Find the two points that divide P Q i n part ( c ) into
three equal parts.
(f) Find the two points that divide PQ in part (d) into
three equal parts.
17. Suggest a "vector proof" of the fact that, in IR 2 , two
lines with slopes m 1 and m 2 are perpendicular if and
only if m 1 m 2 = - 1 .
18. The line e passes through the point P = ( 1 , - 1 , 1 ) and
h"' dfr,dion mtoc d �
[ _;l
Foe mh of th,
following planes <if', determine whether e and <if' are
parallel, perpendicular, or neither:
(a) 2x + 3y - z = 1 (h) 4x - y + 5z = 0
(c) x - y - z = 3
(d) 4x + 6y - 2z = 0
19. The plane <if' 1 has the equation 4x - y + 5z = 2. For
each of the planes <if' in Exercise 18, determine whether
<if' 1 and <if' are parallel, perpendicular, or neither.
20. Find the vector form of the equation of the line in IR 2
that passes through P = (2, - 1 ) and is perpendicular
to the line with general equation 2x - 3y = 1 .
21. Find the vector form of the equation of the line in IR 2
that passes through P = (2, - 1 ) and is parallel to the
line with general equation 2x - 3y = 1 .
22. Find the vector form of the equation of the line in IR 3
that passes through P = ( - 1 , 0, 3 ) and is perpendicular
to the plane with general equation x - 3y + 2z = 5.
23. Find the vector form of the equation of the line in IR 3
that passes through P = ( - 1 , 0, 3 ) and is parallel to
the line with parametric equations
passes through P = (O, - 2, 5 ) and is parallel to the
plane with general equation 6x - y + 2z = 3.
45
25. A cube has vertices at the eight points (x, y, z) , where
each of x, y, and z is either 0 or 1 . (See Figure 1 .34.)
(a) Find the general equations of the planes that
determine the six faces (sides) of the cube.
(b) Find the general equation of the plane that con­
tains the diagonal from the origin to ( 1, 1, 1 ) and
is perpendicular to the xy-plane.
(c) Find the general equation of the plane that
contains the side diagonals referred to in
Example 1 .22.
26. Find the equation of the set of all points that are
equidistant from the points P = ( 1 , 0, - 2) and
Q = ( 5, 2, 4 ) .
In Exercises 27 and 28, find the distance from the point Q to
the line e.
27. Q = (2, 2), f with equation
[;] [ - � ] [ _ � ]
�[ ] [ : J { - � ]
28. Q � (O, ! , O), M h 'qoation
+ t
=
�
+
In Exercises 29 and 30, find the distance from the point Q to
the plane <if' .
29. Q = (2, 2, 2), <if' with equation x + y - z = 0
30. Q = (O, 0, O), <if' with equation x - 2y + 2z = 1
Figure 1 . 66 suggests a way to use vectors to locate the point
R on e that is closest to Q.
31. Find the point R on f that is closest to Q in Exercise 27.
32. Find the point R on e that is closest to Q in Exercise 28.
Q
I
x= 1 - t
y = 2 + 3t
z = -2 - t
24. Find the normal form of the equation of the plane that
1. 3 Lines and Planes
Fioure 1 . 6 6
r
----->
= p + PR
46
Chapter
1 Vectors
Figure 1.67 suggests a way to use vectors to locate the point
R on <!J' that is closest to Q.
k --
the angle between <!J' and <!J' z to be either e or 1 80° - e,
whichever is an acute angle. (Figure 1.68)
l
e
0
\
r
= p + PQ + QR
------>
Figure 1 . 6 8
33. Find the point R on <!J' that is closest to Q in Exercise 29.
34. Find the point R on <!J' that is closest to Q in Exercise 30.
In Exercises 35 and 36, find the distance between the
parallel lines.
35.
36.
[;] [ � ] [ � ] [;] [ � ] [ � ]
[�] [ _�] {] [;] [:J {]
+s
-
+
and
+ t
and
-
+
In Exercises 37 and 38, find the distance between the
parallel planes.
37. 2x + y 2z = 0 and 2x + y 2z = 5
38. x + y + z = 1 and x + y + z = 3
39. Prove Equation (3) on page 43.
40. Prove Equation (4) on page 44.
41. Prove that, in !R z , the distance between parallel lines
with equations n · x = c1 and n · x = Cz is given by
l c 1 - Cz l
ll n ll .
42. Prove that the distance between parallel planes with
equations n · x = d 1 and n · x = dz is given by
I d , - dz l
ll n ll
-
�
1 80
-
Figure 1 . 6 1
------>
\lJ> I
-
If two nonparallel planes <!J' 1 and <!J' z have normal vectors n 1
and nz and e is the angle between n1 and Dz, then we define
e
In Exercises 43-44, find the acute angle between the planes
with the given equations.
43. x + y + z = 0 and 2x + y - 2z = 0
44. 3x - y + 2z = 5 and x + 4y - z = 2
In Exercises 45-46, show that the plane and line with the
given equations intersect, and then find the acute angle of
intersection between them.
45. The plane given by x + y + 2z = 0 and the line
given by x = 2 + t
y = 1 - 2t
z= 3 + t
46. The plane given by 4x - y - z 6 and the line
given by x = t
y = 1 + 2t
z = 2 + 3t
Exercises 47-48 explore one approach to the problem of
finding the projection of a vector onto a plane. As Figure 1.69 shows, if <!J' is a plane through the origin in IR 3 with
normal vector n, and v is a vector in IR 3, then p = proj9p(v)
is a vector in <!J' such that v - en = p for some scalar c.
n
Figure 1 . 6 9
Projection onto a plane
Section
47. Using the fact that n is orthogonal to every vector in <!f'
(and hence to p ), solve for c and thereby find an expres­
sion for p in terms of v and n.
48. Use the method of Exercise 43 to find the projection of
1. 3 Lines and Planes
onto the planes with the following equations:
(b) 3x - y + z = 0
(a) x + y + z = 0
(c) x - 2z = 0
(d) 2x - 3y + z = 0
41
Exp loration
T h e C ro s s Pro duct
It would b e convenient i f we could easily convert the vector form x = p + s u + t v of
the equation of a plane to the normal form n · x = n · p. What we need is a process
that, given two nonparallel vectors u and v, produces a third vector n that is orthogo­
nal to both u and v. One approach is to use a construction known as the cross product
of vectors. Only valid in IR 3 , it is defined as follows:
Definition
defined by
U
XV=
[U2U3VV31 -- U3U1VV2]3
U 1 V2 - U 2 V 1
A shortcut that can help you remember how to calculate the cross product of
two vectors is illustrated below. Under each complete vector, write the first two com­
ponents of that vector. Ignoring the two components on the top line, consider each
block of four: Subtract the products of the components connected by dashed lines
from the products of the components connected by solid lines. (It helps to notice that
the first component of u X v has no ls as subscripts, the second has no 2s, and the
third has no 3s.)
UU21 VVz1
U3U1 xx V3V1 UU23VV31 -- U3U1 VV23
Uz x Vz U1V2 - U2V1
The following problems briefly explore the cross product.
1 . Compute u X v.
48
u X v
v
Figure 1 . 1 0
2. Show that e 1 X e = e3 , e X e3 = e 1 , and e3 X e 1 = e .
2
2
2
3. Using the definition of a cross product, prove that u X v (as shown in Figure 1 .70)
is orthogonal to u and v.
4. Use the cross product to help find the normal form of the equation of the plane.
(a) The pfane P"'ing thrnugh P � ( 1 . 0, - 2 ), parn!kl to u �
[:] [ - �]
and v �
(b) The plane passing through P = (0, - 1 , 1 ) , Q = (2, 0, 2), and R = ( 1 , 2, - 1 )
5. Prove the following properties of the cross product:
(a) v X u = - ( u X v) (b) u X 0 = 0
(c) u X u = 0
(d) u X kv = k ( u X v)
(e) u X ku = 0
(f) u X ( v + w) = u X v + u X w
6. Prove the following properties of the cross product:
(a) u · ( v X w) = ( u X v ) · w
(b) u X (v X w) = (u · w)v - (u · v)w
(c) ll u X v ll 2 = ll u ll 2 ll v ll 2 - ( u · v ) 2
7. Redo Problems 2 and 3, this time making use of Problems 5 and 6.
8. Let u and v be vectors in IR 3 and let (} be the angle between u and v.
(a) Prove that ll u X v ii = ll u ll ll v ll sin (}. [Hint: Use Problem 6(c) .]
(b) Prove that the area A of the triangle determined by u and v (as shown in Fig­
ure 1. 7 1 ) is given by
Figure 1 . 1 1
A = H u X v ii
( c) Use the result in part (b) to compute the area of the triangle with vertices
A = ( 1 , 2, 1 ) , B = (2, 1 , O), and C = (5, - 1 , 3).
Writi n g Project
The Origins o f the Dot Product and Cross Product
The notations for dot and cross product that we use today were introduced in the
late 1 9th century by Josiah Willard Gibbs, a professor of mathematical physics at
Yale University. Edwin B. Wilson was a graduate student in Gibbs's class, and he
later wrote up his class notes, expanded upon them, and had them published in
1 90 1 , with Gibbs's blessing, as Vector Analysis: A Text-Book for the Use of Students
of Mathematics and Physics. However, the concepts of dot and cross product arose
earlier and went by various other names and notations.
Write a report on the evolution of the names and notations for the dot product
and cross product.
1 . Florian Cajori, A History of Mathematical Notations (New York: Dover, 1 993).
2. J. Willard Gibbs and Edwin Bidwell Wilson, Vector Analysis: A Text-Book for the
Use of Students of Mathematics and Physics (New York: Charles Scribner's Sons,
1 90 1 ) . Available online at http://archive.org/details/ 1 1 77 1 4283.
3. Ivor Grattan-Guinness, Companion Encyclopedia of the History and Philosophy
of the Mathematical Sciences (London: Routledge, 2013).
49
50
Chapter
1 Vectors
A p p l icati o n s
Force Veclors
Force is defined as the product of
mass and acceleration due to grav­
ity (which, on Earth, is m/s2 ).
Thus, a kg mass exerts a down­
ward force of kg X m/s2 or
kg • m/s2 • This unit of measure­
ment is a newton (N). So the force
exerted by a kg mass is N.
1 1 9.89.8
9.8
1
9.8
We can use vectors to model force. For example, a wind blowing at 30 km/h in a west­
erly direction or the Earth's gravity acting on a 1 kg mass with a force of 9.8 newtons
downward are each best represented by vectors since they each consist of a magnitude
and a direction.
It is often the case that multiple forces act on an object. In such situations, the
net result of all the forces acting together is a single force called the resultant, which
is simply the vector sum of the individual forces (Figure 1 .72). When several forces
act on an object, it is possible that the resultant force is zero. In this case, the object
is clearly not moving in any direction and we say that it is in equilibrium. When an
object is in equilibrium and the force vectors acting on it are arranged head-to-tail,
the result is a closed polygon (Figure 1 .73).
Figure 1 . 1 2
Figure 1 . 1 3
The resultant of two forces
Equilibrium
Exa m p l e 1 . 3 4
Ann and Bert are trying to roll a rock out of the way. Ann pushes with a force of 20 N
in a northerly direction while Bert pushes with a force of 40 N in an easterly direction.
(a) What is the resultant force on the rock?
(b) Carla is trying to prevent Ann and Bert from moving the rock. What force must
Carla apply to keep the rock in equilibrium?
Solution
(a) Figure 1 .74 shows the position of the two forces. Using the paral­
lelogram rule, we add the two forces to get the resultant r as shown. By Pythagoras'
a
Figure 1 . 1 4
a
b
The resultant of two forces
b
Section
1.4 Applications
51
Theorem, we see that ll r ll = V20 2 + 40 2 = V20QO = 44.72 N. For the direc­
tion of r, we calculate the angle e between r and Bert's easterly force. We find that
sine = 20/ ll r ll = 0.447, so e = 26.57°.
(b) If we denote the forces exerted by Ann, Bert, and Carla by a, b, and c, respec­
tively, then we require a + b + c = 0. Therefore c = - ( a + b) = - r, so Carla
needs to exert a force of 44. 72 N in the direction opposite to r.
Often, we are interested in decomposing a force vector into other vectors whose
resultant is the given vector. This process is called resolving a vector into com­
ponents. In two dimensions, we wish to resolve a vector into two components.
However, there are infinitely many ways to do this; the most useful will be to re­
solve the vector into two orthogonal components. (Chapters 5 and 7 explore this
idea more generally.) This is usually done by introducing coordinate axes and by
choosing the components so that one is parallel to the x-axis and the other to the
y-axis. These components are usually referred to as the horizontal and vertical
components, respectively. In Figure 1 .75, f is the given vector and fx and fy are its
horizontal and vertical components.
y
x
Figure 1 . 1 5
Resolving a vector into components
Exa m p l e 1 . 3 5
Ann pulls on the handle of a wagon with a force of 100 N. If the handle makes an
angle of20° with the horizontal, what is the force that tends to pull the wagon forward
and what force tends to lift it off the ground?
Solution
consider.
Figure 1 .76 shows the situation and the vector diagram that we need to
Figure 1 . 1 6
52
Chapter
1 Vectors
We see that
l l fx I I = 11 £ 11 cos 20° and l l fy I = 1 £ 11 sin 20°
Thus, ll fx ll = 1 00 ( 0.9397 ) = 93.97 and ll fy ll = 1 00 ( 0.3420 ) = 34.20. So the
wagon is pulled forward with a force of approximately 93.97 N and it tends to lift off
the ground with a force of approximately 34.20 N.
We solve the next example using two different methods. The first solution considers
a triangle of forces in equilibrium; the second solution uses resolution of forces into
components.
Exa m p l e 1 . 3 6
Figure 1. 77 shows a painting that has been hung from the ceiling by two wires. If the
painting has a mass of 5 kg and if the two wires make angles of 45 and 60 degrees with
the ceiling, determine the tension in each wire.
Figure 1 . 1 1
We assume that the painting is in equilibrium. Then the two wires must
supply enough upward force to balance the downward force of gravity. Gravity
exerts a downward force of 5 X 9.8 = 49 N on the painting, so the two wires must
collectively pull upward with 49 N of force. Let f1 and f2 denote the tensions in the
wires and let r be their resultant (Figure 1 . 78 ). It follows that II r II = 49 since we are
in equilibrium.
Solulion 1
Section
1.4 Applications
53
Using the law of sines, we have
ll f1 ll
sin 45°
so
sin 30°
ll r ll
sin 1 05°
11 f1 11 = sin 1 05°
ll r ll sin 45° = 49 ( 0.7071 ) =
ll r ll sin 30° = 49 ( 0.5 ) =
35.87 and I f2 II =
25.36
sin 1 05°
0.9659
0.9659
Therefore, the tensions in the wires are approximately 35.87 N and 25.36 N.
We resolve f1 and f2 into horizontal and vertical components, say, f1 =
h1 + v1 and f2 = h2 + v2 , and note that, as above, there is a downward force of 49 N
(Figure 1 .79).
Solution 2
30 °
It follows that
Figure 1 . 1 8
ll vz ll = ll f2 ll sin 45° =
Vz
60 °
/
I�
Since the painting is in equilibrium, the horizontal components must balance, as
must the vertical components. Therefore, I h1 I = ll h2 II and ll v1 II + ll v2 II = 49, from
which it follows that
and
45 °
Substituting the first of these equations into the second equation yields
v'3 ll f2 ll
, ;;:;
VL
+ fil - 49,
,v;;:;2
_
or ll f2 I _
49 \/2
- 25.36
1 + ,v;;:;3
_
Thus, 11 £1 11 = v2 ll f2 ll = 1 .4 1 42 ( 25.36 ) = 35.87, so the tensions in the hires are
approximately 35.87 N and 25.36 N, as before.
49 N
Figure 1 . 1 9
I
Exercises 1 . 4
Force Vectors
In Exercises 1 -6, determine the resultant of the given
forces.
1 . f1 acting due north with a magnitude of 1 2 N and
f2 acting due east with a magnitude of 5 N
2. f1 acting due west with a magnitude of 1 5 N and
f2 acting due south with a magnitude of 20 N
3. f1 acting with a magnitude of 8 N and f2 acting at an
angle of 60° to f1 with a magnitude of 8 N
4. f1 acting with a magnitude of 4 N and f2 acting at an
angle of 1 35° to f1 with a magnitude of 6 N
54
Chapter
1 Vectors
5. f1 acting due east with a magnitude of 2 N, f2 acting
due west with a magnitude of 6 N, and f3 acting at an
angle of 60° to f1 with a magnitude of 4 N
6. f1 acting due east with a magnitude of 1 0 N, f2 acting
due north with a magnitude of 1 3 N, f3 acting due west
with a magnitude of 5 N, and f4 acting due south with
a magnitude of 8 N
7. Resolve a force of 1 0 N into two forces perpendicular
to each other so that one component makes an angle
of 60° with the 10 N force.
8. A 10 kg block lies on a ramp that is inclined at an angle
of 30° (Figure 1 .80). Assuming there is no friction, what
force, parallel to the ramp, must be applied to keep the
block from sliding down the ramp?
10. A lawn mower has a mass of 30 kg. It is being pushed
with a force of 1 00 N. If the handle of the lawn mower
makes an angle of 45° with the ground, what is the
horizontal component of the force that is causing the
mower to move forward?
1 1 . A sign hanging outside Joe's Diner has a mass of 50 kg
(Figure 1 .82). If the supporting cable makes an angle
of 60° with the wall of the building, determine the
tension in the cable.
Figure 1 . 8 2
12. A sign hanging in the window of Joe's Diner has a
Figure 1 . 8 0
9. A tow truck is towing a car. The tension in the tow
cable is 1 500 N and the cable makes a 45° with the
horizontal, as shown in Figure 1 . 8 1 . What is the verti­
cal force that tends to lift the car off the ground?
mass of 1 kg. If the supporting strings each make an
angle of 45° with the sign and the supporting hooks
are at the same height (Figure 1 .83), find the tension in
each string.
•
_
_
f = 1500 N
,
,
'
_
,
,
•
, ' ,f2
45 °
OPEN FOR BUSINESS
Figure 1 . 8 3
13. A painting with a mass of 1 5 kg is suspended by two
Figure 1 . 8 1
wires from hooks on a ceiling. If the wires have lengths
of 1 5 cm and 20 cm and the distance between the
hooks is 25 cm, find the tension in each wire.
14. A painting with a mass of 20 kg is suspended by two
wires from a ceiling. If the wires make angles of 30°
and 45° with the ceiling, find the tension in each wire.
Chapter Review
Kev Defi nitions and concepts
head-to-tail rule, 6
integers modulo m (Z m ), 14- 1 6
length (norm) of a vector, 2 0
linear combination of
vectors, 1 2
normal vector, 34, 38
m-ary vector 16
orthogonal vectors, 26
parallel vectors, 8
parallelogram rule, 6
algebraic properties of vectors, 10
angle between vectors, 24
binary vector, 1 3
Cauchy-Schwarz Inequality, 22
cross product, 48
direction vector, 3 5
distance between vectors, 2 3
dot product, 1 8
equation o f a line, 3 6
equation o f a plane, 38-39
Review Questions
1 . Mark each of the following statements true or false:
(a) For vectors U, v, and w in rr;r, if u + w = v + w,
then u = v.
(b) For vectors U, v, and w in rr;r, if u . w = v . w, then
u = v.
(c) For vectors u, v, and w in IR 3 , if u is orthogonal
to v, and v is orthogonal to w, then u is orthogonal
to w.
(d) In IR 3 , if a line C is parallel to a plane <:IP, then a di­
rection vector d for e is parallel to a normal vector
n for <:IP .
(e) I n IR 3 , i f a line e i s perpendicular t o a plane <:IP , then
a direction vector d for e is a parallel to a normal
vector n for <:IP .
(f) I n IR 3 , i f two planes are not parallel, then they must
intersect in a line.
( g) In IR 3 , if two lines are not parallel, then they must
intersect in a point.
(h) If v is a binary vector such that v · v = 0, then
v = 0.
(i) In l:'.5, if ab = 0 then either a = 0 or b = 0.
(j) In l:'.6, if ab = 0 then either a = 0 or b = 0.
2. If u =
[ -�l [�l
[ -�l [�l
v=
and the vector 4u + v is drawn
with its tail at the point ( 1 0 , 1 0 ), find the coordinates
of the point at the head of 4u + v.
3. If u =
for x.
v=
and 2x + u 3 (x - v), solve
=
projection o f a vector onto
a vector, 27
Pythagoras' Theorem, 26
scalar multiplication, 7
standard unit vectors, 22
Triangle Inequality, 22
unit vector, 2 1
vector, 3
vector addition, 5
zero vector, 4
4. Let A, B, C, and D be the vertices of a square centered
---->
at the origin 0, labeled in clockwise order. If a = OA
and b = OB , find BC in terms of a and b.
5. Find the angle between the vectors [ - 1 , 1, 2] and
[2, 1, - 1 ] .
6. Hnd ilie prnjection of v �
[:]
m
onto u �
[ -n
7. Find a unit vector in the xy-plane that is orthogonal
to
8. Find the general equation of the plane through the
point ( 1, 1, 1 ) that is perpendicular to the line with
parametric equations
x= 2- t
y = 3 + 2t
z = -1 + t
9. Find the general equation of the plane through the
-
point (3, 2, 5) that is parallel to the plane whose general
equation is 2 x + 3y z = 0.
10. Find the general equation of the plane through the
points A ( l , 1 , O ) , B ( l , 0, 1 ) , and C ( O, 1 , 2).
11. Find the area of the triangle with vertices A(l, 1 , O),
B ( l , 0, 1 ) , and C(O, 1 , 2).
55
56
Chapter
1 Vectors
12. Find the midpoint of the line segment between
A = (5, 1 , - 2) and B = (3, - 7, 0).
13. Why are there no vectors u and v in !R n such that
ll u ll = 2, ll v ll = 3, and u · v = - 7?
14. Find the distance from the point (3, 2, 5) to the plane
whose general equation is 2x + 3y - z = 0.
15. Find the distance from the point (3, 2, 5) to the line
with parametric equations x = t, y = 1 + t, z = 2 + t.
16. Compute 3 - (2 + 4) 3 (4 + 3) 2 in Z5.
17. If possible, solve 3(x + 2) = 5 in Z7•
18. If possible, solve 3 (x + 2) = 5 in Z9•
19. Compute [2, 1, 3, 3] · [3, 4, 4, 2] in zt.
20. Let u [ l , 1, 1, O] in Zi. How many binary vectors v
=
satisfy u · v = O?
Systems of Lineari
Eq u ations
The world was full of equations . . . .
There must be an answerfor everything,
if only you knew how to setforth
the questions.
-Anne Tyler
1985, p. 235
The Accidental Tourist
Alfred A. Knopf,
2.0
I n t ro d u ctio n : Triv i a l ilV
The word trivial is derived from the Latin root tri ("three") and the Latin word via
("road"). Thus, speaking literally, a triviality is a place where three roads meet. This
common meeting point gives rise to the other, more familiar meaning of trivial­
commonplace, ordinary, or insignificant. In medieval universities, the trivium con­
sisted of the three "common" subjects (grammar, rhetoric, and logic) that were taught
before the quadrivium (arithmetic, geometry, music, and astronomy) . The "three
roads" that made up the trivium were the beginning of the liberal arts.
In this section, we begin to examine systems oflinear equations. The same system
of equations can be viewed in three different, yet equally important, ways-these will
be our three roads, all leading to the same solution. You will need to get used to this
threefold way of viewing systems of linear equations, so that it becomes common­
place (trivial!) for you.
The system of equations we are going to consider is
2x + y = 8
x - 3y = - 3
Problem 1 Draw the two lines represented by these equations. What is their point
of intersection?
Problem 2
Consider the vectors u =
[�]
and v =
[ _ �]
. Draw the coordinate
grid determined by u and v. [Hint: Lightly draw the standard coordinate grid first and
use it as an aid in drawing the new one.]
Problem 3
On the u-v grid, find the coordinates of w =
[ _: ]
.
Problem 4 Another way to state Problem 3 is to ask for the coefficients x and y
for which xu + yv = w. Write out the two equations to which this vector equation is
equivalent (one for each component) . What do you observe?
Problem 5 Return now to the lines you drew for Problem 1 . We will refer to the
line whose equation is 2x + y = 8 as line 1 and the line whose equation is x - 3y = - 3
as line 2. Plot the point (O, O) on your graph from Problem 1 and label it P0 . Draw a
51
58
Chapter
2 Systems of Linear Equations
Ta b l e 2 . 1
Point
Po
P1
P2
P3
P4
Ps
p6
x
0
y
0
horizontal line segment from P0 to line 1 and label this new point P1 . Next draw a
vertical line segment from P1 to line 2 and label this point P2 • Now draw a horizontal
line segment from P2 to line 1, obtaining point P3 • Continue in this fashion, drawing
vertical segments to line 2 followed by horizontal segments to line 1 . What appears to
be happening?
Problem 6 Using a calculator with two-decimal-place accuracy, find the (approxi­
mate) coordinates of the points P 1 , P2 , P3 , . . . , P6• (You will find it helpful to first
solve the first equation for x in terms of y and the second equation for y in terms
of x.) Record your results in Table 2 . 1 , writing the x- and y-coordinates of each point
separately.
The results of these problems show that the task of "solving" a system of linear
equations may be viewed in several ways. Repeat the process described in the prob­
lems with the following systems of equations:
( a ) 4x - 2y = 0 ( b ) 3x + 2y = 9 ( c ) x + y = S ( d ) x + 2y = 4
x + 2y = 5
x + 3y = 1 0
x-y=3
2x - y = 3
Are all of your observations from Problems 1-6 still valid for these examples? Note
any similarities or differences. In this chapter, we will explore these ideas in more detail.
I n t ro d u ct i o n to svste m s o t l i n e a r E q u a t i o n s
Recall that the general equation o f a line in IR 2 is of the form
ax + by = c
and that the general equation of a plane in IR 3 is of the form
ax + by + cz = d
Equations of this form are called linear equations.
D e f i n i t i o n A linear equation in the n variables x 1 , x2 , • • • , xn is an equation
that can be written in the form
where the coefficients a 1 , a 2 , . • . , a n and the constant term b are constants.
Exa m p l e 2 . 1
The following equations are linear:
3x - 4y = - 1
\/2x +
r - ts - lft = 9
:y - (sin :)z =
1
3.2x1 - O.O lx2 = 4 . 6
Observe that the third equation is linear because it can be rewritten in the form x 1 +
5x 2 + x 3 - 2x4 = 3. It is also important to note that, although in these examples (and
in most applications) the coefficients and constant terms are real numbers, in some
examples and applications they will be complex numbers or members of "11.P for some
prime number p.
Section
2. 1 Introduction to Systems of Linear Equations
59
The following equations are not linear:
xy + 2z = 1
xi - x� = 3
x
-+z=2
y
Thus, linear equations do not contain products, reciprocals, or other functions of the
variables; the variables occur only to the first power and are multiplied only by con stants. Pay particular attention to the fourth example in each list: Why is it that the
fourth equation in the first list is linear but the fourth equation in the second list is not?
4
A solution of a linear equation a 1 x 1 + a 2 x 2 +
+ a nxn = b is a vector
, sn l whose components satisfy the equation when we substitute x 1 = s 1 ,
[ s 1 , s2 ,
X2 = S 2> , Xn = Sn ·
•
Exa m p l e 2 . 2
•
•
•
•
·
·
•
(a) [5, 4] is a solution of 3x - 4y = - 1 because, when we substitute x = 5 and y = 4,
the equation is satisfied: 3 (5) - 4(4) = - 1 . [ l , 1] is another solution. In general, the
solutions simply correspond to the points on the line determined by the given equa­
tion. Thus, setting x = t and solving for y, we see that the complete set of solutions
can be written in the parametric form [t, � + �t] . (We could also set y equal to some
parameter-say, s -and solve for x instead; the two parametric solutions would look
different but would be equivalent. Try this.)
(b) The linear equation x 1 - x 2 + 2x 3 = 3 has [3, 0, O ] , [O, 1, 2], and [6, 1 , - 1 ]
as specific solutions. The complete set of solutions corresponds to the set of points
in the plane determined by the given equation. If we set x 2 = s and x 3 = t, then a
parametric solution is given by [3 + s - 2t, s, t] . (Which values of s and t produce the
three specific solutions above?)
A system of linear equations is a finite set of linear equations, each with the same
variables. A solution of a system of linear equations is a vector that is simultaneously
a solution of each equation in the system. The solution set of a system of linear equa­
tions is the set of all solutions of the system. We will refer to the process of finding the
solution set of a system of linear equations as "solving the system:'
Exa m p l e 2 . 3
The system
2x - y = 3
x + 3y 5
has [2, 1 ] as a solution, since it is a solution of both equations. On the other hand,
[ 1 , - 1 ] is not a solution of the system, since it satisfies only the first equation.
=
Exa m p l e 2 . 4
Solve the following systems of linear equations:
(a) x - y = 1
x+y=3
(b) x - y = 2
2x - 2y = 4
(c) x - y = 1
x-y= 3
60
Chapter
2 Systems of Linear Equations
Solution
(a) Adding the two equations together gives 2x = 4, so x = 2, from which we find
that y = 1 . A quick check confirms that [2, 1 ] is indeed a solution of both equations.
That this is the only solution can be seen by observing that this solution corresponds
to the (unique) point of intersection ( 2, 1 ) of the lines with equations x - y = 1 and
x + y = 3, as shown in Figure 2 . l (a). Thus, [2, 1 ] is a unique solution.
(b) The second equation in this system is just twice the first, so the solutions are the
solutions of the first equation alone-namely, the points on the line x - y = 2. These
can be represented parametrically as [2 + t, t] . Thus, this system has infinitely many
solutions [Figure 2 . 1 (b) ] .
(c) Two numbers x and y cannot simultaneously have a difference of 1 and 3 . Hence,
this system has no solutions. (A more algebraic approach might be to subtract the second
equation from the first, yielding the absurd conclusion 0 = - 2.) As Figure 2.l (c) shows,
the lines for the equations are parallel in this case .
.J
y
y
(b)
(c)
4
( a)
Figure 2 . 1
A system oflinear equations is called consistent if it has at least one solution. A sys­
tem with no solutions is called inconsistent. Even though they are small, the three sys­
tems in Example 2.4 illustrate the only three possibilities for the number of solutions of
a system of linear equations with real coefficients. We will prove later that these same
three possibilities hold for any system of linear equations over the real numbers.
A system of linear equations with real coefficients has either
(a) a unique solution (a consistent system) or
(b) infinitely many solutions (a consistent system) or
( c) no solutions (an inconsistent system) .
Solving a svstem or Linear Equations
�
Two linear systems are called equivalent if they have the same solution sets. For
example,
and
x-y= l
x-y= l
x+y=3
y= 1
are equivalent, since they both have the unique solution [2, l ] . (Check this.)
Section
2. 1 Introduction to Systems of Linear Equations
61
Our approach to solving a system of linear equations is to transform the given
system into an equivalent one that is easier to solve. The triangular pattern of the
second example above (in which the second equation has one less variable than the
first) is what we will aim for.
Exa m p l e 2 . 5
Solve the system
x-y- z= 2
y + 3z = 5
5z = 1 0
Starting from the last equation and working backward, we find successively
that z = 2, y = 5 - 3(2) = - 1 , and x = 2 + ( - 1 ) + 2 = 3. So the unique solution is
[3, - 1 , 2] .
Solution
The procedure used to solve Example 2.5 is called back substitution.
We now turn to the general strategy for transforming a given system into an
equivalent one that can be solved easily by back substitution. This process will be
described in greater detail in the next section; for now, we will simply observe it in
action in a single example.
Exa m p l e 2 . 6
Solve the system
x- y- z= 2
3x - 3y + 2z = 1 6
2x - y + z = 9
To transform this system into one that exhibits the triangular structure
of Example 2.5, we first need to eliminate the variable x from Equations 2 and 3.
Observe that subtracting appropriate multiples of equation 1 from Equations 2 and
3 will do the trick. Next, observe that we are operating on the coefficients, not on
the variables, so we can save ourselves some writing if we record the coefficients and
constant terms in the matrix
-1 -1 2
-3 2 16
1 9
-1
Solution
The word matrix is derived from
the Latin word mater, meaning
"mother:' When the suffix - ix
is added, the meaning becomes
"womb:' Just as a womb surrounds
a fetus, the brackets of a matrix
surround its entries, and just as
the womb gives rise to a baby, a
matrix gives rise to certain types of
functions called linear transforma­
tions. A matrix with m rows and
n columns is called an m X n
matrix (pronounced "m by n").
The plural of matrix is matrices,
not "matrixes:'
[:
]
where the first three columns contain the coefficients of the variables in order, the final
column contains the constant terms, and the vertical bar serves to remind us of the
equal signs in the equations. This matrix is called the augmented matrix of the system.
There are various ways to convert the given system into one with the triangular
pattern we are after. The steps we will use here are closest in spirit to the more general
method described in the next section. We will perform the sequence of operations on
the given system and simultaneously on the corresponding augmented matrix. We
begin by eliminating x from Equations 2 and 3.
x- y- z= 2
3x - 3y + 2z = 1 6
2x - y + z = 9
-1
-3
-1
2
2 16
1 9
-1
]
62
Chapter
2 Systems of Linear Equations
Subtract 3 times the first equation
from the second equation:
Subtract 3 times the first row from the
second row:
x-y- z= 2
5z = 1 0
2x - y + z = 9
Subtract 2 times the first equation
from the third equation:
[:
-1
0
-1
Subtract 2 times the first row from the
third row:
x-y- z= 2
5z = 1 0
y + 3z = 5
Interchange Equations 2 and 3:
[:
-1
0
1
Interchange rows 2 and 3:
x-y- z= 2
y + 3z = 5
5z = 1 0
[i
-1
1
0
-1
5
3
-1
3
5
!�:
lt
This is the same system that we solved using back substitution in Example 2.5, where
we found that the solution was [ 3, - 1 , 2] . This is therefore also the solution to the sys­
tem given in this example. Why? The calculations above show that any solution of the
given system is also a solution of thefinal one. But since the steps we just performed are
reversible, we could recover the original system, starting with the final system. (How?)
So any solution of the final system is also a solution of the given one. Thus, the systems
are equivalent (as are all of the ones obtained in the intermediate steps above). More­
over, we might just as well work with matrices instead of equations, since it is a simple
matter to reinsert the variables before proceeding with the back substitution. (Work­
ing with matrices is the subject of the next section.)
Remark
Calculators with matrix capabilities and computer algebra systems can
facilitate solving systems of linear equations, particularly when the systems are large
or have coefficients that are not "nice;' as is often the case in real-life applications. As
always, though, you should do as many examples as you can with pencil and paper
until you are comfortable with the techniques. Even if a calculator or CAS is called
for, think about how you would do the calculations manually before doing anything.
After you have an answer, be sure to think about whether it is reasonable.
Do not be misled into thinking that technology will always give you the answer
faster or more easily than calculating by hand. Sometimes it may not give you the
answer at all! Roundoff errors associated with the floating-point arithmetic used by
calculators and computers can cause serious problems and lead to wildly wrong an­
swers to some problems. See Exploration: Lies My Computer Told Me for a glimpse
of the problem. (You've been warned! )
Section
1
2. 1 Introduction to Systems of Linear Equations
63
Exercises 2 . 1
In Exercises 1-6, determine which equations are linear
equations in the variables x, y, and z. If any equation is not
linear, explain why not.
1 . x - 'TTY + efsz = 0
2. x 2 + y 2 + z 2 = 1
3. x - 1 + 7y + z = sin
(;)
4. 2x - xy - 5z = 0
6. (cos3)x - 4y + z = v3
5. 3 cos x - 4y + z = v3
_
In Exercises 7-1 0, find a linear equation that has the same
solution set as the given equation (possibly with some
restrictions on the variables).
x2 y2
8. x - y
7. 2x + y = 7 - 3y
1 1 4
9. - + - = 10. log 1 0 x - log 1 0 y = 2
x y xy
--­
In Exercises 1 1 -14, find the solution set of each equation.
12. 2x 1 + 3x 2 = 5
1 1 . 3x - 6y = 0
13. x + 2y + 3 z = 4
In Exercises 15-18, draw graphs corresponding to the given
linear systems. Determine geometrically whether each sys­
tem has a unique solution, infinitely many solutions, or no
solution. Then solve each system algebraically to confirm
your answer.
16. x - 2y = 7
15. x + y = 0
2x + y = 3
3x + y = 7
18. O. lOx - 0.05y = 0.20
17. 3x - 6y = 3
-0.06x + 0.03y = -0.12
-x + 2y = l
In Exercises 1 9-24, solve the given system by back
substitution.
19. x - 2y = 1
20. 2u - 3v = 5
2v = 6
y=3
21. x - y + z = 0
22. X 1 + 2X 2 + 3X 3 = 0
- 5x2 + 2x 3 = 0
2y - z = 1
4X 3 = 0
3z = - 1
23. X 1 + Xz - X3 - X4 = 1 24. x - 3y + z = 5
y - 2z = - 1
X2 + X3 + X4 = 0
X 3 - X4 = 0
X4 = 1
The systems in Exercises 25 and 26 exhibit a "lower trian­
gular" pattern that makes them easy to solve by forward
substitution. (We will encounter forward substitution again
in Chapter 3.) Solve these systems.
= -1
25. X
2 26. X1
!
= -3
2x + y
5
- x1 + x2
�X 1 + 2X 2 + X3 = 7
- 3x - 4y + z = - 10
Find the augmented matrices of the linear systems in
Exercises 27-30.
28. 2X1 + 3X2 - X3 = 1
27. x - y = 0
2x + y = 3
+ X3 = 0
X1
- X 1 + 2X2 - 2X 3 = 0
29. x + Sy = - 1
30. a - 2b
+ d=2
-x + y = - 5
- a + b - c - 3d = 1
2x + 4y = 4
In Exercises 31 and 32, find a system of linear equations
that has the given matrix as its augmented matrix.
31.
32.
[ � =: � :J
[i
�]
-1 0 3
2
-1
0 2 3
For Exercises 33-38, solve the linear systems in the given
exercises.
33. Exercise 27
34. Exercise 28
35. Exercise 29
36. Exercise 3 0
38. Exercise 32
37. Exercise 3 1
39. (a) Find a system of two linear equations in the vari­
ables x and y whose solution set is given by the
parametric equations x = t and y = 3 - 2t.
(b) Find another parametric solution to the system in
part (a) in which the parameter is s and y = s.
40. ( a) Find a system of two linear equations in the vari­
ables x 1 , x 2 , and x 3 whose solution set is given by
the parametric equations x 1 = t, x 2 = 1 + t, and
X 3 = 2 - t.
(b) Find another parametric solution to the system in
part (a) in which the parameter is s and x3 = s.
64
Chapter
2 Systems of Linear Equations
In Exercises 4 1 -44, the systems of equations are nonlinear.
Find substitutions (changes of variables) that convert each
system into a linear system and use this linear system to help
solve the given system.
2 3
41. -x + - = 0
y
-3 + -4 = l
x y
42. x 2 + 2y 2 = 6
x2 - y 2 = 3
43. tan x 2 sin y
2
sin y + cos z = 2
tan x
sin y cos z = - 1
44. - 2a + 2 (3 b ) = 1
3(2a) - 4(3 b ) = 1
-- -
D i rect M e t h o d s f o r S o lvi n g l i n e a r Svst e m s
In this section, we will look at a general, systematic procedure fo r solving a system
of linear equations. This procedure is based on the idea of reducing the augmented
matrix of the given system to a form that can then be solved by back substitution.
The method is direct in the sense that it leads directly to the solution (if one exists) in
a finite number of steps. In Section 2.5, we will consider some indirect methods that
work in a completely different way.
Matrices a n d Echelon Form
There are two important matrices associated with a linear system. The coefficient
matrix contains the coefficients of the variables, and the augmented matrix (which
we have already encountered) is the coefficient matrix augmented by an extra column
containing the constant terms.
For the system
2x + y - z = 3
x
+ Sz = 1
-x + 3y - 2z = O
the coefficient matrix is
and the augmented matrix is
[
-1]
- � � -�
2 1
1 -1 3
0
5 1
3 -2 0
]
Note that if a variable is missing (as y is in the second equation), its coefficient 0 is
entered in the appropriate position in the matrix. If we denote the coefficient matrix
of a linear system by A and the column vector of constant terms by b, then the form
of the augmented matrix is [A I b J .
Section
2.2 Direct Methods for Solving Linear Systems
65
In solving a linear system, it will not always be possible to reduce the coefficient
matrix to triangular form, as we did in Example 2.6. However, we can always achieve
a staircase pattern in the nonzero entries of the final matrix.
The word echelon comes from
the Latin word scala, meaning
"ladder" or "stairs:' The French
word for "ladder:' echelle, is also
derived from this Latin base. A
matrix in echelon form exhibits
a staircase pattern.
Definition
properties:
A matrix is in row echelon form if it satisfies the following
1. Any rows consisting entirely of zeros are at the bottom.
2. In each nonzero row, the first nonzero entry (called the leading entry) is
in a column to the left of any leading entries below it.
Note that these properties guarantee that the leading entries form a staircase pat­
tern. In particular, in any column containing a leading entry, all entries below the
leading entry are zero, as the following examples illustrate.
Exa m p l e 2 . 1
The following matrices are in row echelon form:
[: �] [: :J [i
4
-1
0
0
1
0
1 2
0 1
0 0
�- l �
1 -1
2
0 -1 1
4
0
0 0
2
0
0
0 0
0
�-4
If a matrix in row echelon form is actually the augmented matrix of a linear sys­
tem, the system is quite easy to solve by back substitution alone.
Exa m p l e 2 . 8
Assuming that each of the matrices in Example 2.7 is an augmented matrix, write out
the corresponding systems of linear equations and solve them.
We first remind ourselves that the last column in an augmented matrix is
the vector of constant terms. The first matrix then corresponds to the system
Solution
2x1 + 4X2 = 1
-x2 = 2
(Notice that we have dropped the last equation 0 = 0, or Ox 1 + Ox 2 = 0, which is
clearly satisfied for any values of x 1 and x 2 .) Back substitution gives x 2 = - 2 and then
2x1 = 1 - 4( - 2) = 9, so x1 = � . The solution is [ �, - 2 ] .
The second matrix has the corresponding system
x, = 1
X2 = 5
0 =4
The last equation represents Ox 1 + Ox2 = 4, which clearly has no solutions. Therefore,
the system has no solutions. Similarly, the system corresponding to the fourth matrix
has no solutions. For the system corresponding to the third matrix, we have
66
Chapter
2 Systems of Linear Equations
X 1 + X2 + 2X3 = 1
X3 = 3
so x1 = 1 - 2 ( 3 ) - x2 = - 5 x2• There are infinitely many solutions, since we may
assign x2 any value t to get the parametric solution [ 5 t, t, 3 ] .
-
-
-
Elemenlarv Row Operalions
We now describe the procedure by which any matrix can be reduced to a matrix
in row echelon form. The allowable operations, called elementary row operations,
correspond to the operations that can be performed on a system of linear equations
to transform it into an equivalent system.
DefiniliOD
matrix:
The following elementary row operations can be performed on a
1 . Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
Observe that dividing a row by a nonzero constant is implied in the above
definition, since, for example, dividing a row by 2 is the same as multiplying it by ! .
Similarly, subtracting a multiple o f one row from another row i s the same a s adding a
negative multiple of one row to another row.
We will use the following shorthand notation for the three elementary row
operations:
1 . R; � Rj means interchange rows i and j.
2. kR; means multiply row i by k.
3. R; + kRj means add k times row j to row i (and replace row i with the result).
The process of applying elementary row operations to bring a matrix into row
echelon form, called row reduction, is used to reduce a matrix to echelon form.
Exa m p l e 2 . 9
Reduce the following matrix to echelon form:
[ _:
-4 -4
0 0
2
3
1
3
6
1
2
4
�]
We work column by column, from left to right and from top to bottom.
The strategy is to create a leading entry in a column and then use it to create zeros
below it. The entry chosen to become a leading entry is called a pivot, and this phase
of the process is called pivoting. Although not strictly necessary, it is often convenient
to use the second elementary row operation to make each leading entry a 1 .
Solution
Section
2.2 Direct Methods for Solving Linear Systems
5]
55 [
61
-- :5 ]
We begin by introducing zeros into the first column below the leading 1 in the
first row:
2 -4 -4
4 0 0 2
1
3 2
3 6
R, - 2R1
R3 - 2R1
�
R, + R1
l
2 -4 -4
0 0 8 8
0 - 1 10 9
0 3 -1 2
10
The first column i s now a s we want it, s o the next thing t o do i s to create a leading entry
in the second row, aiming for the staircase pattern of echelon form. In this case, we do
this by interchanging rows. (We could also add row 3 or row 4 to row 2.)
]�
2 -4 -4
- 1 10 9
0 8 8 -8
3 - 1 2 10
]
The pivot this time was - 1 . We now create a zero at the bottom o f column 2, using
the leading entry - 1 in row 2:
-�
-5
2 -4 -4
- 1 10 9
0 8 8 -8
0 29 29
Column 2 is now done. Noting that we already have a leading entry in column 3,
we just pivot on the 8 to introduce a zero below it. This is easiest if we first divide
row 3 by 8:
�
� R,
[1
]�
2 -4 -4
- 1 10 9
1 -1
0 1
0 29 29 - 5
]
Now we use the leading 1 in row 3 to create a zero below it:
·�· [ i
-�
2 -4 -4
- 1 10 9
1 -1
0 1
0 0 0 24
With this final step, we have reduced our matrix to echelon form.
R e m arks
The row echelon form of a matrix is not unique. (Find a different row echelon
form for the matrix in Example 2.9. )
•
68
Chapter
2 Systems of Linear Equations
The leading entry in each row is used to create the zeros below it.
The pivots are not necessarily the entries that are originally in the posi­
tions eventually occupied by the leading entries. In Example 2.9, the pivots were
1 , - 1 , 8, and 24. The original matrix had 1 , 4, 2, and 5 in those positions on the
"staircase."
•
Once we have pivoted and introduced zeros below the leading entry in a
column, that column does not change. In other words, the row echelon form emerges
from left to right, top to bottom.
•
•
Elementary row operations are reversible-that is, they can be "undone:' Thus,
if some elementary row operation converts A into B, there is also an elementary row
operation that converts B into A. (See Exercises 1 5 and 16.)
Matrices A and B are row equivalent if there is a sequence of
elementary row operations that converts A into B.
Definition
The matrices in Example 2.9,
2 -4 -4
4 0 0
1
3 2
3 6
�] [�
md
-�]
2 -4 -4
- 1 10 9
1 -1
0 1
0 0 0 24
are row equivalent. In general, though, how can we tell whether two matrices are row
equivalent?
Theorem 2 . 1
Matrices A and B are row equivalent if and only if they can be reduced to the same
row echelon form.
Proof If A and B are row equivalent, then further row operations will reduce B (and
therefore A) to the (same) row echelon form.
Conversely, if A and B have the same row echelon form R, then, via elementary
row operations, we can convert A into R and B into R. Reversing the latter sequence of
operations, we can convert R into B, and therefore the sequence A � R � B achieves
the desired effect.
Remark In practice, Theorem 2 . 1 is easiest to use if R is the reduced row echelon
form of A and B, as defined on page 73. See Exercises 17 and 18.
Gaussian Elimination
When row reduction is applied to the augmented matrix of a system of linear
equations, we create an equivalent system that can be solved by back substitution.
The entire process is known as Gaussian elimination.
Section 2.2 Direct Methods for Solving Linear Systems
G a u s s i a n Elimination
69
1. Write the augmented matrix of the system of linear equations.
2. Use elementary row operations to reduce the augmented matrix to row
echelon form.
3. Using back substitution, solve the equivalent system that corresponds to the
row-reduced matrix.
Remark When performed by hand, step 2 of Gaussian elimination allows quite a
bit of choice. Here are some useful guidelines:
(a) Locate the leftmost column that is not all zeros.
(b) Create a leading entry at the top of this column. (It will usually be easiest if you
make this a leading 1 . See Exercise 22.)
(c) Use the leading entry to create zeros below it.
( d) Cover up the row containing the leading entry, and go back to step (a) to repeat the pro­
cedure on the remaining submatrix. Stop when the entire matrix is in row echelon form.
Exa m p l e 2 . 1 0
Solve the system
2x2 + 3x3 = 8
2x1 + 3x2 + x3 = 5
X1 - Xz - 2X3 = 5
-
Solution
The augmented matrix is
[�
2 3
3
- 1 -2
-
�l
We proceed to reduce this matrix to row echelon form, following the guidelines given
for step 2 of the process. The first nonzero column is column 1 . We begin by creating
Carl Friedrich Gauss (1777-1855) is generally considered to be one of the three greatest
mathematicians of all time, along with Archimedes and Newton. He is often called the "prince of
mathematicians;' a nickname that he richly deserves. A child prodigy, Gauss reportedly could do
arithmetic before he could talk. At the age of he corrected an error in his father's calculations for
the company payroll, and as a young student, he found the formula n(n + 1) /2 for the sum of the
first n natural numbers. When he was 19, he proved that a 17-sided polygon could be constructed
using only a straightedge and a compass, and at the age of21, he proved, in his doctoral
dissertation, that every polynomial of degree n with real or complex coefficients has exactly
n zeros, counting multiple zeros-the Fundamental Theorem of Algebra.
Gauss's 1801 publication Disquisitiones Arithmeticae is generally considered to be the
foundation of modern number theory, but he made contributions to nearly every branch of
mathematics as well as to statistics, physics, astronomy, and surveying. Gauss did not publish
all of his findings, probably because he was too critical of his own work. He also did not like
to teach and was often critical of other mathematicians, perhaps because he discovered-but
did not publish-their results before they did.
The method called Gaussian elimination was known to the Chinese in the third century B . c .
and was well known by Gauss's time. The method bears Gauss's name because of his use of it in a
paper in which he solved a system oflinear equations to describe the orbit of an asteroid.
3,
10
Chapter
2 Systems of Linear Equations
a leading entry at the top of this column; interchanging rows 1 and 3 is the best way
to achieve this.
[�
2
3
-1
SJ [
3
s
-2 -s
[
Ri ++ R,
�
l
2
0
-1
3
2
-2 -S
1 s
3 8
i
We now create a second zero in the first column, using the leading 1 :
R, 2R
�
-
1
l - l -2 -S
0 s s lS
0 2 3 8
i
We now cover up the first row and repeat the procedure. The second column is
the first nonzero column of the submatrix. Multiplying row 2 by � will create a
leading 1 .
[i
-1
s
2
-2
s
3
- 5 1 [ i -! ]
I[ lS
8
tR2
�
-1
-2
2
3
We now need another zero at the bottom of column 2:
�
,, - rn,
0
0
- 1 -2
1
1
0 1
5i
3
2
The augmented matrix is now in row echelon form, and we move to step 3. The cor­
responding system is
X 1 - Xz - 2X3 = - S
Xz + x3 = 3
X3 = 2
and back substitution gives x3 = 2, then x2 = 3 - x3 = 3 - 2 = 1, and
finally x 1 = - S + x2 + 2x3 = - S + 1 + 4 = 0. We write the solution in vector
form as
(We are going to write the vector solutions of linear systems as column vectors from
now on. The reason for this will become clear in Chapter 3.)
Exa m p l e 2 . 1 1
Solve the system
w - x - y + 2z =
2w - 2x - y + 3z = 3
= -3
-w + x - y
Section
Solution
2.2 Direct Methods for Solving Linear Systems
11
The augmented matrix is
u
-1 -1 2
-2 - 1 3
-1 0
which can be row reduced as follows:
u
J [:
[:
-1 -1 2
-2 - 1 3
-1 0
R, - 2R,
R, + R,
-----+
R3 + 2R2
-----+
The associated system is now
w
J
-1 -1 2
0
-1
0 -2 2
-1 -1 2
0 1 -1
0 0 0
J
il
- x - y + 2z = 1
y - z = 1
which has infinitely many solutions. There is more than one way to assign param eters, but we will proceed to use back substitution, writing the variables correspond­
ing to the leading entries (the leading variables) in terms of the other variables (the
free variables) .
In this case, the leading variables are w and y, and the free variables are x and z.
Thus, y = 1 + z, and from this we obtain
w
= 1 + x + y - 2z
= 1 + x + (1 + z) - 2z
=2+x-z
If we assign parameters x = s and z = t, the solution can be written in vector
form as
Example 2. 1 1 highlights a very important property: In a consistent system, the
free variables are just the variables that are not leading variables. Since the number
of leading variables is the number of nonzero rows in the row echelon form of the
coefficient matrix, we can predict the number of free variables (parameters) before
we find the explicit solution using back substitution. In Chapter 3, we will prove that,
although the row echelon form of a matrix is not unique, the number of nonzero rows
is the same in all row echelon forms of a given matrix. Thus, it makes sense to give a
name to this number.
12
Chapter
2 Systems of Linear Equations
Definition
echelon form.
The rank of a matrix is the number of nonzero rows in its row
We will denote the rank of a matrix A by rank(A) . In Example 2 . 1 0, the rank of
the coefficient matrix is 3, and in Example 2. 1 1 , the rank of the coefficient matrix is
2. The observations we have just made justify the following theorem, which we will
prove in more generality in Chapters 3 and 6.
Theorem 2 . 2
The Rank Theorem
-
Let A be the coefficient matrix of a system of linear equations with n variables. If
the system is consistent, then
number of free variables = n
-
rank (A )
Thus, in Example 2 . 1 0, we have 3 - 3 = 0 free variables (in other words, a unique
solution), and in Example 2. 1 1 , we have 4 2 = 2 free variables, as we found.
Exa m p l e 2 . 1 2
Solve the system
X 1 - X2 + 2X3 = 3
X 1 + 2X2 - X3 = - 3
2x2 - 2X3 =
Solulion
When we row reduce the augmented matrix, we have
[i
-1
2
2
2
-1
-2
- : l [i
[i
[I
Ri - R 1
�
!. R2
�
�
� - ,,,
(1842-1899)
Wilhelm Jordan
was
a German professor of geodesy
whose contribution to solving
linear systems was a systematic
method of back substitution
closely related to the method
described here.
0
0
-1
2
3 -3
2
-2
-1
1
2
2
-1
-2
-1
1
2
-1
0
0
-;]
�]
-�]
-
-
leading to the impossible equation 0 = 5. (We could also have performed R3 t R2 as the
second elementary row operation, which would have given us the same contradiction
but a different row echelon form.) Thus, the system has no solutions-it is inconsistent.
Gauss-Jordan Eliminalion
4
A modification of Gaussian elimination greatly simplifies the back substitution phase
and is particularly helpful when calculations are being done by hand on a system with
Section
2.2 Direct Methods for Solving Linear Systems
13
infinitely many solutions. This variant, known as Gauss-Jordan elimination, relies
on reducing the augmented matrix even further.
Definition
properties:
A matrix is in reduced row echelon form ifit satisfies the following
1. It is in row echelon form.
2. The leading entry in each nonzero row is a 1 (called a leading 1 ) .
3. Each column containing a leading 1 has zeros everywhere else.
The following matrix is in reduced row echelon form:
1
0
0
0
0
2
0
0
0
0
1 0
0 0 -3
0 4 -1 0
0
3 -2 0
0 0 0 0 1
0 0 0 0 0
For 2 X 2 matrices, the possible reduced row echelon forms are
For a short proof that the reduced
row echelon form of a matrix is
unique, see the article by Thomas
Yuster, "The Reduced Row Echelon
Form of a Matrix Is Unique: A
Simple Proof;' in the March
issue of Mathematics Magazine
(vol. no. pp.
57, 2, 93-94).
1984
G a u ss-Jordan
Elimination
where * can be any number.
It is clear that after a matrix has been reduced to echelon form, further elementary
row operations will bring it to reduced row echelon form. What is not clear (although
intuition may suggest it) is that, unlike the row echelon form, the reduced row ech­
elon form of a matrix is unique.
In Gauss-Jordan elimination, we proceed as in Gaussian elimination but reduce
the augmented matrix to reduced row echelon form.
1 . Write the augmented matrix of the system of linear equations.
2. Use elementary row operations to reduce the augmented matrix to reduced
row echelon form.
3. If the resulting system is consistent, solve for the leading variables in terms
of any remaining free variables.
Exa m p l e 2 . 1 3
Solve the system in Example 2.1 1 by Gauss-Jordan elimination.
SolUtion
[:
The reduction proceeds as it did in Example 2.1 1 until we reach the echelon form:
ll
-1 -1 2
0 1 -1 1
0 0 0 0
14
Chapter
2 Systems of Linear Equations
We now must create a zero above the leading 1 in the second row, third column. We
do this by adding row 2 to row 1 to obtain
[:
-1 0
0
0 0
The system has now been reduced to
w-x
+z=2
y-z= I
It is now much easier to solve for the leading variables:
w=2+x-z
and
y= I +z
If we assign parameters x = s and z = t as before, the solution can be written in vector
form as
Remark
From a computational point of view, it is more efficient (in the sense
that it requires fewer calculations) to first reduce the matrix to row echelon form
and then, working from right to left, make each leading entry a 1 and create zeros
above these leading 1 s. However, for manual calculation, you will find it easier to
just work from left to right and create the leading ls and the zeros in their columns
as you go.
Let's return to the geometry that brought us to this point. Just as systems of linear
equations in two variables correspond to lines in IR 2 , so linear equations in three vari­
ables correspond to planes in IR 3 . In fact, many questions about lines and planes can
be answered by solving an appropriate linear system.
Exa m p l e 2 . 1 4
Find the line of intersection of the planes x + 2y - z = 3 and 2x + 3y + z = 1 .
Solution
First, observe that there will b e a line of intersection, since the normal
vectors of the two planes- [ I, 2, - I ] and [2, 3, 1 ] -are not parallel. The points that
lie in the intersection of the two planes correspond to the points in the solution set
of the system
x + 2y - z = 3
2x + 3y + z = 1
Gauss-Jordan elimination applied to the augmented matrix yields
Section
2.2 Direct Methods for Solving Linear Systems
2
2
[� 3
1
3
1
-1 3 ]
1- 3 - 7 J
7
-7
-5
0
s
Replacing variables, we have
5
+ Sz =
x
z
15
y
3z = 5
We set the free variable z equal to a parameter t and thus obtain the parametric equa­
tions of the line of intersection of the two planes:
x=
- St
y = 5 + 3t
z=
In vector form, the equation is
x
y
Figure 2 . 2
The intersection of two planes
Exa m p l e 2 . 1 5
See Figure 2.2.
Lct p �
[ _ H m' [:J.
q �
u�
md v �
[J
Dcte;m;ne whcthec the lilles
x = p + tu and x = q + tv intersect and, if so, find their point of intersection.
We need to be careful here. Although t has been used as the parameter
in the equations of both lines, the lines are independent and therefore so are their
parameters. Let's use a different parameter-say, s -for the first line, so its equation
Solution
bewmes x � p + su. If the lines illtmed, then we wmt to find on x �
[;]
thot
satisfies both equations simultaneously. That is, we want x = p + su = q + tv or
SU
- tv = q
-
p.
Substituting the given p, q, u, and v, we obtain the equations
s - 3t =
s+ t=
s+ t=
-1
2
2
whose solution is easily found to be s = �, t = � . The point of intersection is therefore
Chapter
16
2 Systems of Linear Equations
See Figure 2.3. (Check that substituting t = i in the other equation gives the same
point.)
z
x
In IR 3 , it is possible for two lines to intersect in a point, to be parallel, or
to do neither. Nonparallel lines that do not intersect are called skew lines.
Remark
y
Figure 2 . 3
Two intersecting lines
Homogeneous svste ms
We have seen that every system of linear equations has either no solution, a unique
solution, or infinitely many solutions. However, there is one type of system that
always has at least one solution.
A system of linear equations is called homogeneous ifthe constant
term in each equation is zero.
Definition
In other words, a homogeneous system has an augmented matrix of the form
[A I O] . The following system is homogeneous:
2x + 3y - z = 0
-x + Sy + 2z = 0
Since a homogeneous system cannot have no solution (forgive the double negative!),
it will have either a unique solution (namely, the zero, or trivial, solution) or infinitely
many solutions. The next theorem says that the latter case must occur if the number
of variables is greater than the number of equations.
Theorem 2 . 3
If [A I O] is a homogeneous system of m linear equations with n variables, where
m < n, then the system has infinitely many solutions.
Since the system has at least the zero solution, it is consistent. Also,
rank(A) :::::: m (why?) . By the Rank Theorem, we have
Proof
�
number of free variables = n - rank (A ) 2: n - m > 0
So there is at least one free variable and, hence, there are infinitely many solutions.
Note Theorem 2.3 says nothing about the case where m 2: n. Exercise 44 asks
you to give examples to show that, in this case, there can be either a unique solution
or infinitely many solutions.
Section
IR and ZP are examples offields.
The set of rational numbers Q and
the set of complex numbers C are
other examples. Fields are covered
in detail in courses in abstract
algebra.
2.2 Direct Methods for Solving Linear Systems
linear svs1ems over Z p
Thus far, all o f the linear systems we have encountered have involved real numbers,
and the solutions have accordingly been vectors in some !R n . We have seen how other
number systems arise-notably, "ll_P ' When p is a prime number, "ll_P behaves in many
respects like IR; in particular, we can add, subtract, multiply, and divide (by nonzero
numbers). Thus, we can also solve systems of linear equations when the variables and
coefficients belong to some "ll_P ' In such instances, we refer to solving a system over "ll_P '
For example, the linear equation x1 + x2 + x3 = 1 , when viewed as an equation
over Z 2 , has exactly four solutions:
(where the last solution arises because 1 + 1 + 1 = 1 in Z 2 ) .
wh,n wni,w th"q"'Hon X ; + x, + X; � 1 ov" 2,, thno!utiom
�
Exa m p l e 2 . 1 6
11
[ :: l
m
(Check these.)
But we need not use trial-and-error methods; row reduction of augmented matri­
ces works just as well over "ll_P as over IR.
Solve the following system of linear equations over Z 3 :
x3 0
+ X3 = 2
x2 + 2x3 1
X 1 + 2x2 +
X1
=
=
The first thing to note in examples like this one is that subtraction and
division are not needed; we can accomplish the same effects using addition and mul­
tiplication. (This, however, requires that we be working over "ll_P , where p is a prime;
see Exercise 60 at the end of this section and Exercise 57 in Section 1 . 1 .)
We row reduce the augmented matrix of the system, using calculations modulo 3.
Solution
[i ;] [i ;]
[i :J
2
0
1 2
�
R2 + 2R1
R, + R2
�
R3 + 2R2
2 1
1 0
1 2
0
1 0
0 2
18
Chapter
2 Systems of Linear Equations
R, + R,
�
[
1 Q 0 1
0 1 02
0 0 1 1
l
Thus, the solution is Xi = 1, x2 = 2, x3 = 1 .
Exa m p l e 2 . 1 1
Solve the following system of linear equations over 2 2 :
X i + Xz + X 3 + X4 = 1
= 1
X i + X2
= 0
X2 + X3
X3 + X4 = 0
+ X4 = 1
Xi
Solution
The row reduction proceeds as follows:
1 1
1
0 0 1
0 1
0 0
1 0
0 0
0 0
R2 + R1
R5 + R,
�
R2 <--> R3
R1 + R2
�
R5 + R2
R2 + R3
�
R4 + R3
1
1
0 0 1 1 0
0 1 1 0 0
0 0 1
0
0
0 0
1
0
0
0
0
0 0 1 1
0 0
0
0
0 1 1 0
0 0 0 0
1
0
0
0
0
0 0
1
0
0
0 1 1 0
0 0 0 0
0 0 0 0
Therefore, we have
X1
Xz
Setting the free variable x4 = t yields
+ X4 = 1
+ X4 = 0
X3 + X4 = Q
Section
2.2 Direct Methods for Solving Linear Systems
0 1,
Since t can take on the two values and
�
1
..
In Exercises 1 -8, determine whether the given matrix is in
row echelon form. If it is, state whether it is also in reduced
row echelon form.
!.
3.
5.
3
3
2.
4.
6.
5
7
.
01 -11 ]
[� 0 0 �
00
[ : 0 :J
01
[: 0 :J
001 01
[ � 0 0 -� ]
3
8
.
there are exactly two solutions:
Remark For linear systems over "11..P , there can never be infinitely many
solutions. (Why not?) Rather, when there is more than one solution, the number
of solutions is finite and is a function of the number of free variables and p. (See
Exercise 59.)
Exercises 2 . 2
0 ]
0
[i �
[ � 01 0 � ]
00 0 - 40
[i 0 :J
20
[ j 01 �:
19
In Exercises 9-14, use elementary row operations to reduce
the given matrix to (a) row echelon form and (b) reduced
row echelon form.
0
•
[: : J
[: -!]
13. [ : -- 21 --- 111 i
9.
II
10.
[� � ]
1 2. [� - 41 - 62 : J
.
-[ 2 - 4 1 : ]
2
14. - � -6
-3
-3
15. Reverse the elementary row operations used in
Example 2.9 to show that we can convert
2- 1 -104 - 4
1
1
0
1
[� 0 0 0 -�]24
9
into
Chapter
80
2 Systems of Linear Equations
2 -4 -4
4 0 0
3 2
3 6
�]
16. In general, what is the elementary row operation that
"undoes" each of the three elementary row operations
R; � �' kR;, and R; + kR/
In Exercises 1 7 and 1 8, show that the given matrices are row
equivalent and find a sequence of elementary row operations
that will convert A into B.
1
3 -1
'B =
17. A =
3 4
1
0
[ 2] [
18.
]
A= [ � � i [ � - � ]
0 -1
-1
1
,B =
3
5
2
19. What is wrong with the following "proof" that every
matrix with at least two rows is row equivalent to a
matrix with a zero row?
Perform R 2 + R 1 and R 1 + R2 • Now rows 1 and 2
are identical. Now perform R 2 - R 1 to obtain a
row of zeros in the second row.
20. What is the net effect of performing the following
sequence of elementary row operations on a matrix
(with at least two rows)?
R2 + R , , R , - R2 , R2 + R , , - R ,
21. Students frequently perform the following type of cal­
culation to introduce a zero into a matrix:
However, 3R 2 - 2R 1 is not an elementary row opera­
tion. Why not? Show how to achieve the same result
using elementary row operations.
22. Consider the matrix A =
[ � !]
. Show that any of
the three types of elementary row operations can be
used to create a leading 1 at the top of the first column.
Which do you prefer and why?
23. What is the rank of each of the matrices in Exercises 1 -8?
24. What are the possible reduced row echelon forms of
3 X 3 matrices?
In Exercises 25-34, solve the given system of equations using
either Gaussian or Gauss-Jordan elimination.
26. x - y + z = 0
25. X 1 + 2X2 - 3X3 = 9
2X 1 - X2 + X3 = 0
-x + 3y + z = 5
4x 1 - X2 + x3 = 4
3x + y + 7z = 2
27. X 1 - 3X2 - 2X3 = 0 28. 2w + 3x - y + 4z = 1
3w - x
-X1 + 2X2 + X3 = 0
+ z=1
3w - 4x + y - z = 2
2x1 + 4x2 + 6x3 = 0
29. 2r + s = 3
4r + s = 7
2r + Ss = - 1
30. -X1 + 3X2 - 2X3 + 4X4 = 0
2X 1 - 6X2 + X3 - 2X4 = - 3
X 1 - 3X2 + 4X3 - 8x4 = 2
2
31. � X 1 + X2 - X3 - 6X4
ix , + � Xz
- 3X4 + X5 = - 1
- 4X5 = 8
32. Vlx + y + 2z =
1
Vly - 3z = - V2
-y + Vlz =
33. w + x + 2y + z = 1
w-x- y+z= 0
x+ y
-1
+z= 2
w+x
34. a + b + c + d = 4
a + 2b + 3c + 4d = 1 0
a + 3b + 6c + lOd = 20
a + 4b + lOc + 20d = 35
In Exercises 35-38, determine by inspection (i.e., without
performing any calculations) whether a linear system with
the given augmented matrix has a unique solution, infinitely
many solutions, or no solution. Justify your answers.
0
-2 0
35.
2 -3
36.
1
0
4 -6
2 3 4 O
2 3 4
6 7 8 0
5 4 3
37.
38.
10 1 1 12 0
7 7 7
39. Show that if ad - be -=!= 0, then the system
ax + by = r
ex + dy = s
[:
[i
[�
J [�
has a unique solution.
Section
In Exercises 40-43,for what value(s) of k, if any, will the
systems have (a) no solution, (b) a unique solution, and
(c) infinitely many solutions?
kx + 2y = 3
x + ky = 1
kx + y = 1
2x - 4y =
x - 2y + 3 z = 2
x + y + kz =
x + ky + z =
x+ y+ z=k
2
kx + y + z = - 2
2x - y + 4z = k
Give examples of homogeneous systems of m linear
equations in n variables with m = n and with m > n
40.
42.
44.
445..
467.
-6
41.
43.
that have (a) infinitely many solutions and (b) a
unique solution.
2.2 Direct Methods for Solving Linear Systems
6
(c ) Give an example of three planes, exactly two of
which are parallel (Figure 2. ) .
figure 2 . 6
(d) Give an example of three planes that intersect in a
single point (Figure 2.7).
In Exercises 45 and 46, find the line of intersection of the
given planes.
3x + 2y + z = - 1 and 2x - y + 4z = 5
4x + y + z = 0 and 2x - y + 3z = 2
(a) Give an example of three planes that have a com­
mon line of intersection (Figure 2.4).
p
q
48.p � [ - Hq � [H u � [ _+ � [ - i l
49.p � [�J q � [: J . u � [ + � m
50. p � m. u � [ J. � m
p
51. u
uu
� [ :: l [ :: l
figure 2 . 1
In Exercises 48 and 49, determine whether the lines
x = + su and x = + tv intersect and, if they do, find
their point of intersection.
Figure 2 . 4
(b) Give an example of three planes that intersect in
pairs but have no common point of intersection
(Figure 2.5 ).
let
ond v
Desaibe
all points Q = ( a, b, c) such that the line through
Q with direction vector v intersects the line with
equation x = + su.
Recall that the cross product of vectors and v is a
vector X v that is orthogonal to both and v. (See
Exploration: The Cross Product in Chapter 1 .) If
Figure 2 . 5
81
U
and
F
82
Chapter
2 Systems of Linear Equations
show that there are infinitely many vectors
that simultaneously satisfy u · x = 0 and v · x = 0
and that all are multiples of
U
XV=
[
U 2 V3 - U 3 V2
U 3 V 1 - U 1 V3
U 1 V2 - U 2 V 1
]
Show that the lines x = p + su and x = q + tv are
skew lines. Find vector equations of a pair of parallel
planes, one containing each line.
In Exercises 53-58, solve the systems of linear equations
over the indicated ZP.
53. x + 2y = 1 over Z 3
x+ y=2
54. x + y
= 1 over Z 2
y+z=O
x
+z= 1
Writi n g Proiect
55. x + y
= 1 over Z 3
y+z=O
x
+z= 1
56. 3 x + 2y = 1 over Z s
x + 4y = 1
57. 3 x + 2y = 1 over Z 7
x + 4y = 1
58. x 1
+ 4x4 = 1 over Z s
=3
X 1 + 2X2 + 4x3
2X 1 + 2X2
+ X4 = 1
=2
X1
+ 3X3
59. Prove the following corollary to the Rank Theorem:
Let A be an m X n matrix with entries in Zp . Any
consistent system of linear equations with coefficient
matrix A has exactly p n - rank(A) solutions over ZP '
60. When p is not prime, extra care is needed in solving
a linear system (or, indeed, any equation) over ZP '
Using Gaussian elimination, solve the following system
over Z6• What complications arise?
2x + 3y = 4
4x + 3y = 2
A History of Gaussian Elimination
As noted in the biographical sketch of Gauss in this section, Gauss did not actually
"invent" the method known as Gaussian elimination. It was known in some form as
early as the third century B . C . and appears in the mathematical writings of cultures
throughout Europe and Asia.
Write a report on the history of elimination methods for solving systems of
linear equations. What role did Gauss actually play in this history, and why is his
name attached to the method?
1. S. Athloen and R. McLaughlin, Gauss-Jordan reduction: A brief history,
American Mathematical Monthly 94 ( 1 987), pp. 1 30- 142.
2. Joseph F. Grear, Mathematicians of Gaussian Elimination, Notices of the AMS,
Vol. 58, No. 6 (20 1 1 ) , pp. 782-792. (Available online at http://www.ams.org/
notices/20 1 1 06/index.html)
3. Roger Hart, The Chinese Roots of Linear Algebra (Baltimore: Johns Hopkins
University Press, 20 1 1 ) .
4. Victor J. Katz, A History of Mathematics: An Introduction (Third Edition)
(Reading, MA: Addison Wesley Longman, 2008).
cAs
L i e s My C o mp uter Told M e
Computers and calculators store real numbers in floating-point form. For example,
2001 is stored as 0.200 1 X 10 4 , and - 0.00063 is stored as - 0.63 X 10 - 3 . In general,
the floating-point form of a number is :±:.M X lO k , where k is an integer and the
mantissa M is a (decimal) real number that satisfies 0. 1 ::::: M < 1 .
The maximum number of decimal places that can be stored in the mantissa depends
on the computer, calculator, or computer algebra system. If the maximum number of
decimal places that can be stored is d, we say that there are d significant digits. Many
calculators store 8 or 12 significant digits; computers can store more but still are subject
to a limit. Any digits that are not stored are either omitted (in which case we say that the
number has been truncated) or used to round the number to d significant digits.
For example,-rr = 3.141592654, and its floating-point form is 0.3 141592654 X 10 1 .
In a computer that truncates to five significant digits, 1T would be stored as 0.3 1415 X
10 1 (and displayed as 3.1415); a computer that rounds to five significant digits would
store 1T as 0.3 1416 X 10 1 (and display 3.1416). When the dropped digit is a solitary
5, the last remaining digit is rounded so that it becomes even. Thus, rounded to two
significant digits, 0.735 becomes 0.74 while 0.725 becomes 0.72.
Whenever truncation or rounding occurs, a roundoff error is introduced, which
can have a dramatic effect on the calculations. The more operations that are per­
formed, the more the error accumulates. Sometimes, unfortunately, there is nothing
we can do about this. This exploration illustrates this phenomenon with very simple
systems of linear equations.
1 . Solve the following system of linear equations exactly (that is, work with
rational numbers throughout the calculations).
x+ y=O
01 = 1
X + 8sooY
2. As a decimal, ��b = 1 .00 125, so, rounded to five significant digits, the system
becomes
x+
y=O
x + 1 .00 1 2y = 1
Using your calculator or CAS, solve this system, rounding the result of every calcula­
tion to five significant digits.
83
3. Solve the system two more times, rounding first to four significant digits and
then to three significant digits. What happens?
4. Clearly, a very small roundoff error (less than or equal to 0.00 125) can re­
sult in very large errors in the solution. Explain why geometrically. (Think about the
graphs of the various linear systems you solved in Problems 1 - 3 .)
Systems such as the one you just worked with are called ill-conditioned. They
are extremely sensitive to roundoff errors, and there is not much we can do about it.
We will encounter ill-conditioned systems again in Chapters 3 and 7. Here is another
example to experiment with:
4.552x + 7.083y = 1 .93 1
1 .73 1x + 2.693y = 2.00 1
Play around with various numbers of significant digits to see what happens, starting
with eight significant digits (if you can).
Partial P ivoting
In Exploration: Lies My Computer Told Me, we saw that ill-conditioned linear sys­
tems can cause trouble when roundoff error occurs. In this exploration, you will dis­
cover another way in which linear systems are sensitive to roundoff error and see that
very small changes in the coefficients can lead to huge inaccuracies in the solution.
Fortunately, there is something that can be done to minimize or even eliminate this
problem (unlike the problem with ill-conditioned systems).
84
1 . (a) Solve the single linear equation 0.0002 lx = 1 for x.
(b) Suppose your calculator can carry only four decimal places. The equa­
tion will be rounded to 0.0002x = 1 . Solve this equation.
The difference between the answers in parts (a) and (b) can be thought of as the
effect of an error of 0.0000 1 on the solution of the given equation.
2. Now extend this idea to a system of linear equations.
(a) With Gaussian elimination, solve the linear system
0.400x + 99.6y = 1 00
75.3x - 45. 3y = 30.0
using three significant digits. Begin by pivoting on 0.400 and take each
calculation to three significant digits. You should obtain the "solution" x =
- 1 .00, y = 1 .0 1 . Check that the actual solution is x = 1 .00, y = 1 .00. This is a
huge error-200% in the x value! Can you discover what caused it?
(b) Solve the system in part (a) again, this time interchanging the two equa­
tions (or, equivalently, the two rows of its augmented matrix) and pivoting
on 75.3. Again, take each calculation to three significant digits. What is the
solution this time?
The moral of the story is that, when using Gaussian or Gauss-Jordan elimination
to obtain a numerical solution to a system of linear equations (i.e., a decimal approxi­
mation), you should choose the pivots with care. Specifically, at each pivoting step,
choose from among all possible pivots in a column the entry with the largest absolute
value. Use row interchanges to bring this element into the correct position and use it to
create zeros where needed in the column. This strategy is known as partial pivoting.
3.
Solve the following systems by Gaussian elimination, first without and then
with partial pivoting. Take each calculation to three significant digits. (The exact
solutions are given.)
(a)
O.- O10.O l2xx 0.1.99050yy - 51.0.000 -l Ox3x - 2.09y7y 6z 73.91
5x - y [5zx l 6[ 0.00
]
.
0
0
5
0
0
y
1.
[;] [ 1.00 ]
z - 1.00
+
+
(b)
=
=
Exact solution:
+
+
+
Exact solution:
=
=
=
=
C o unting O p e r at i o n s : An I n t r o d u c t i o n
to t h e An alys i s of Algorith m s
Gaussian and Gauss-Jordan elimination are examples of algorithms: systematic pro­
cedures designed to implement a particular task-in this case, the row reduction of
the augmented matrix of a system of linear equations. Algorithms are particularly well
suited to computer implementation, but not all algorithms are created equal. Apart
from the speed, memory, and other attributes of the computer system on which they
are running, some algorithms are faster than others. One measure of the so-called
of an algorithm (a measure of its efficiency, or ability to perform its task in a
reasonable number of steps) is the number of basic operations it performs as a func­
tion of the number of variables that are input.
Let's examine this proposition in the case of the two algorithms we have for
solving a linear system: Gaussian and Gauss-Jordan elimination. For our pur­
poses, the basic operations are multiplication and division; we will assume that
all other operations are performed much more rapidly and can be ignored. (This
is a reasonable assumption, but we will not attempt to justify it.) We will consider
only systems of equations with
coefficient matrices, so, if the coefficient
matrix is n X n, the number of input variables is n. Thus, our task is to find the
number of operations performed by Gaussian and Gauss-Jordan elimination as
a function of n. Furthermore, we will not worry about special cases that may
arise, but rather establish the
that can arise-when the algorithm takes
as long as possible. Since this will give us an estimate of the time it will take a
computer to perform the algorithm (if we know how long it takes a computer to
perform a single operation) , we will denote the number of operations performed
by an algorithm by T ( n ) . We will typically be interested in T ( n ) for large values of
n, so comparing this function for different algorithms will allow us to determine
which will take less time to execute.
com­
plexity
0
00
j
780-850)
Abu Ja'far Muhammad ibn Musa
al-Khwarizmi (c.
was
a Persian mathematician whose
book Hisab al-jabr w'al muqabalah
(c.
described the use of Hindu­
Arabic numerals and the rules
of basic arithmetic. The second
word of the book's title gives rise
to the English word algebra, and
the word algorithm is derived from
al-Khwarizmi's name.
825)
9
square
worst case
1.
Consider the augmented matrix
[A l h J
9
4
85
I
Count the number of operations required to bring [A b] to the row echelon
form
4]
[:
2 3
1 -1 0
0 1 1
(By "operation" we mean a multiplication or a division.) Now count the number
of operations needed to complete the back substitution phase of Gaussian elimi­
nation. Record the total number of operations.
2. Count the number of I operations needed to perform Gauss-Jordan
elimination-that is, to reduce [A b ] to its reduced row echelon form
(where the zeros are introduced into each column immediately after the leading 1 is
created in that column) . What do your answers suggest about the relative efficiency
of the two algorithms?
We will now attempt to analyze
I the algorithms in a general, systematic way. Sup­
pose the augmented matrix
[A
b ] arises from a linear system with n equations and
I
n variables; thus, [A b ] is n X (n + 1 ):
I
[A b ]
=
[
all
�
a 1
an ]
We will assume that row interchanges are never needed-that we can always create a
leading 1 from a pivot by dividing by the pivot.
3. (a) Show that n operations are needed to create the first leading 1 :
�
86
l
�
l�
:,
l
l l
*
*
a I
a z2
a2 n
an l
an2
a nn b n
*
*
*
*
an2
a nn
a.,
a 12
aln b,
a2 1
a 22
a2 n
an l
a n2
a nn b n
2
�
(Why don't we need to count an operation for the creation of the leading 1 ?) Now
show that n operations are needed to obtain the first zero in column 1 :
la�,
iJ
....,..
(Why don't we need to count an operation for the creation of the zero itself?) When
the first column has been "swept out;' we have the matrix
[� : : :]
* *
0 *
Show that the total number of operations needed up to this point is n + (n - 1) n.
(b) Show that the total number of operations needed to reach the row
echelon form
is
[!
*
*
* *
0
1 *
'
]
[ n + (n - l)n ] + [ (n - 1) + (n - 2)(n - 1 ) ] + [ (n - 2) + (n - 3)(n - 2 ) ]
+ . . . + [2 + 1 . 2 ] + 1
which simplifies to
n 2 + (n - 1 ) 2 +
·
·
·
+ 22 + 1 2
( c) Show that the number of operations needed to complete the back substi­
tution phase is
1 +2+
·
·
·
+ (n - 1)
(d) Using summation formulas for the sums in parts (b) and (c) (see
Exercises 51 and 52 in Section 2.4 and Appendix B), show that the total number of
operations, T(n), performed by Gaussian elimination is
T(n) = tn 3 + n 2 - t n
Since every polynomial function is dominated by its leading term fo r large values of
the variable, we see that T (n) = t n 3 for large values of n.
4. Show that Gauss-Jordan elimination has T(n) = ! n 3 total operations if we
create zeros above and below the leading l s as we go. (This shows that, for large
systems of linear equations, Gaussian elimination is faster than this version of Gauss­
Jordan elimination.)
81
88
Chapter
2 Systems of Linear Equations
�
S p a n n i n g se1s a n d l i n e a r I n d e p e n d e n c e
The second of the three roads in our "trivium" is concerned with linear combina­
tions of vectors. We have seen that we can view solving a system of linear equations
as asking whether a certain vector is a linear combination of certain other vectors.
We explore this idea in more detail in this section. It leads to some very important
concepts, which we will encounter repeatedly in later chapters.
Spanning Sets of Vectors
We can now easily answer the question raised in Section 1 . 1 : When is a given vector
a linear combination of other given vectors?
Exa m p l e 2 . 1 8
(o) fa th, wdoc
(b) fa
m
m
m [ � �}
m [ � �}
' linm rnmbinotion of the vodorn
' linemomb;mtion of the vectorn
Solulion
and
and
(a) We want to find scalars x and y such that
Expanding, we obtain the system
whose augmented matrix is
x- y= 1
y=2
3x - 3y = 3
[� �� �]
(Observe that the columns of the augmented matrix are just the given vectors; notice
the order of the vectors-in particular, which vector is the constant vector.)
The reduced echelon form of this matrix is
[� : �]
�
(Verify this.) So the solution is x = 3, y
combination is
2, and the corresponding linear
Section
2. 3 Spanning Sets and Linear Independence
89
(b) Utilizing our observation in part (a), we obtain a linear system whose augmented
matrix is
which reduces to
m [J
;eve.ling that th, 'Y'"m h"' no ,oJotion. Th°', in thi, ""·
bination of
and
[ �]
4
i' not a linm com-
The notion of a spanning set is intimately connected with the solution of linear
systems. Look back at Example 2 . 1 8. There we saw that a system with augmented
matrix [A I b] has a solution precisely when b is a linear combination of the columns
of A. This is a general fact, summarized in the next theorem.
Theorem 2 . 4
A system of linear equations with augmented matrix [A I b] is consistent if and
only if b is a linear combination of the columns of A.
Let's revisit Example 2.4, interpreting it in light of Theorem 2.4.
(a) The system
x-y= l
x+y=3
has the unique solution x = 2, y = 1 . Thus,
See Figure 2.S(a).
(b) The system
x- y=2
2 x - 2y = 4
has infinitely many solutions of the form x = 2 + t, y = t. This implies that
for all values of t. Geometrically, the vectors
[ � ], [ = � ], [! ]
and
are all parallel and
so all lie along the same line through the origin [see Figure 2.S(b) ] .
Chapter
90
2 Systems of Linear Equations
y
y
y
5
5
5
4
4
-2
2
-1
3
-1
-2
-2
-3
-3
( a)
-3
(b)
(c)
Figure 2 . 8
(c) The system
x-y= I
x-y= 3
has no solutions, so there are no values of x and y that satisfy
In this case,
[�] [=�]
and
are parallel, but
through the origin [see Figure 2.S(c) ] .
[ �]
does not lie along the same line
We will often b e interested in the collection of all linear combinations of a given
set of vectors.
D e f i n i t i o n If S = {v1 , v2 , . . . , vk} is a set of vectors in u;g n , then the set of all
linear combinations of v1 , v2 , . . . , vk is called the span of v1 , v2 , . . . , vk and is de, vk) or span(S ). If span(S ) = W, then S is called a span­
noted by span(v1 , v2 ,
ning set for u;g n .
.
Exa m p l e 2 . 1 9
Show that u;g z = span
Solulion
•
( [ _ � l [ �] ) .
We need to show that an arbitrary vector
combination of
1 [�] [ : ]
=
.
[ _ � ] [� }
and
[:]
can be written as a linear
that is, we must show that the equation x
[ _�] +
can always be solved for x and y (in terms of a and b ), regardless of the
values of a and b.
Section
The augmented matrix is
2. 3 Spanning Sets and Linear Independence
91
[ _ � � I �] '
1] [ 1 ]
and row reduction produces
b
3 b �1 - 1 3
0 7 a + 2b
1 a
at which point it is clear that the system has a (unique) solution. (Why?) If we con­
tinue, we obtain
---'-----+
b R1
] [ I
[ 1
R 1 - 3R, - 1 O (b - 3 a )/ 7
b
3
�
0 1 (a + 2b)/7
0 1 (a + 2b)/7
-1
]
from which we see that x = ( 3a - b)/7 and y = (a + 2b)/7. Thus, for any choice of
a and b, we have
�
(Check this.)
Remark
( [ _�l [�l [�] } [:l
[ _ �] y [�] [ : l
[ _ �] [�]
It is also true that IR 2 = span
find x and y such that x
[�] [:] .
+
=
If, given
we can
then we also have x
+y
+
In fact, any set of vectors that contains a spanning set for IR 2 will also be
a spanning set for IR 2 (see Exercise 20).
The next example is an important (easy) case of a spanning set. We will encounter
versions of this example many times.
0
Exa m p l e 2 . 2 0
=
Let e , , e, . and e; be the standMd unit vedocs in HI'. Then I°' any mtn<
=
[;] � x[ � ] m m �
+y
+z
xe , + ye, + ze.,
[;J
Thus, IR 3 span(e 1 , e2 , e 3 ) .
You should have n o difficulty seeing that, i n general, !R n = span(e 1 , e2 ,
.
•
.
we haw
, en
4
When the span of a set of vectors in !R n is not all of !R n , it is reasonable to ask for a
description of the vectors' span.
Exa m p l e 2 . 2 1
92
Chapter
2 Systems of Linear Equations
z
Solulion
m [J
and
m [ �: ]
[�l m { ]
m [J
Thinking geometrically, we can see that the set of all linear combinations of
i' iu't the plane thrnugh the migm wiili
and
� s
vecto" ( Figme 2. 9). The vectm equotion of this pfane is
which i*t anothe; way of "ying that
Figure 2 . 9
Two nonparallel vectors span a
plane
[� l
OS
is m th, ,pan of
dimtion
+
and
Suppose we want to obtain the general equation of this plane. There are several
ways to proceed. One is to use the fact that the equation ax + by + cz = 0 must be
satisfied by the points ( 1 , 0, 3) and ( - 1, 1, - 3) determined by the direction vectors.
Substitution then leads to a system of equations in a, b, and c. (See Exercise 1 7.)
Another method is to use the system of equations arising from the vector equation:
s- t=x
t=y
3s - 3t = z
If we row reduce the augmented matrix, we obtain
N�� we know that this ry'1em is consistent, since
[; ]
is in the span of
[�]
and
[ _�]
by ossumption. So we must have z - 3x � 0 (o< 3x - z � 0, in mo<e standa<d
Remark
A normal vector to the plane in this example is also given by the cross
form), giving us the general equation we seek.
product
Linear Independence
ill
E"mple 2 18, wdound that 3
m [ �: l m
+2
�
Let's abb<e�ate thi"qua­
tion as 3u + 2v = w. The vector w "depends" on u and v in the sense that it is a
linear combination of them. We say that a set of vectors is linearly dependent if one
Section
2. 3 Spanning Sets and Linear Independence
93
of them can be written as a linear combination of the others. Note that we also have
�v + t w and v = � u + ! w. To get around the question of which vector to
express in terms of the rest, the formal definition is stated as follows:
u=
-
-
Definition
scalars c 1 , c2 ,
A set of vectors v1 , v2 ,
, vk is linearly dependent if there are
, ck, at least one of which is not zero, such that
.
.
•
•
•
.
A set of vectors that is not linearly dependent is called linearly independent.
R e m arks
In the definition of linear dependence, the requirement that at least one of the
scalars c 1 , c2 , , ck must be nonzero allows for the possibility that some may be zero.
In the example above, u, v, and w are linearly dependent, since 3u + 2v - w = 0 and,
in fact, all of the scalars are nonzero. On the other hand,
•
so
.
.
•
[!l [� l [�]
are linearly dependent, since at least one (in fact, two) of the
and
three scalars 1 , - 2, and 0 is nonzero. (Note that the actual dependence arises simply
from the fact that the first two vectors are multiples.) (See Exercise 44.)
•
Since Ov1 + Ov2 + . . . + Ovk = 0 for any vectors v1 , v2 , . . . , vk, linear de­
pendence essentially says that the zero vector can be expressed as a nontrivial linear
combination of v1 , v2 , . . . , vk · Thus, linear independence means that the zero vector
can be expressed as a linear combination of v1 , v2 ,
, vk only in the trivial way: c 1 v1
+ c2v2 + + ckvk = 0 only if c 1 = 0, c2 = 0, . . . , ck = 0.
•
·
·
•
.
·
The relationship between the intuitive notion of dependence and the formal defi­
nition is given in the next theorem. Happily, the two notions are equivalent!
Theorem 2 . 5
Vectors v1 , v2 , , vm in !R n are linearly dependent if and only if at least one of the
vectors can be expressed as a linear combination of the others.
•
.
.
Proof
If one of the vectors-say, v1 -is a linear combination of the others, then
there are scalars c2 , , c m such that v1 = c2v2 + . . . + c mvm . Rearranging, we obtain
v1 c2v2
cmvm = 0, which implies that v1 , v2 , . . . , vm are linearly dependent,
since at least one of the scalars (namely, the coefficient 1 of v1 ) is nonzero.
Conversely, suppose that v1 , v2 ,
, vm are linearly dependent. Then there are
scalars c 1 , c2 , . . . , cm , not all zero, such that c 1 v 1 + c2v2 + . . . + c mvm = 0. Suppose
c 1 -=!= 0. Then
.
-
-
·
·
·
•
•
-
•
.
.
and we may multiply both sides by 1/ c 1 to obtain v1 as a linear combination of the
other vectors:
( C1 )
v1 = - � v2
-
·
·
·
-
( cCm1 )
vm
94
Chapter
2 Systems of Linear Equations
It may appear as if we are cheating a bit in this proof. After all, we cannot
be sure that v1 is a linear combination of the other vectors, nor that c1 is nonzero.
However, the argument is analogous for some other vector V; or for a different scalar
cj . Alternatively, we can just relabel things so that they work out as in the above proof.
In a situation like this, a mathematician might begin by saying, "without loss of gen­
erality, we may assume that v1 is a linear combination of the other vectors" and then
proceed as above.
Note
Exa m p l e 2 . 2 2
Any set of vectors containing the zero vector is linearly dependent. For if 0, v2, , vm
are in IJ�r, then we can find a nontrivial combination of the form c10 + c2v2 + . . . +
Cm Vm = 0 by setting c1 = 1 and c2 = c3 = . . . = C m = 0 .
Exa m p l e 2 . 2 3
Determine whether the following sets of vectors are linearly independent:
.
(a)
(c)
.
•
[�] [ �]
[ -i ] . [ J n l
and
-
and
In answering any question of this type, it is a good idea to see if you can
determine by inspection whether one vector is a linear combination of the others. A
little thought may save a lot of computation!
(a) The only way two vectors can be linearly dependent is if one is a multiple of
the other. (Why?) These two vectors are clearly not multiples, so they are linearly
independent.
(b) There is no obvious dependence relation here, so we try to find scalars c1, c2, c3
such that
Solution
�
The corresponding linear system is
C3 = 0
C1 +
C 1 + Cz
and the augmented matrix is
[i
=O
0 1 O
0 0
1 0
J
Once again, we make the fundamental observation that the columns of the coefficient
matrix are just the vectors in question!
Section
2. 3 Spanning Sets and Linear Independence
The reduced row echelon form is
�
�
[i
95
OJ
0 0
1 0 0
0 1 0
(check this), so c 1 = 0, c2 = 0, c 3 = 0. Thus, the given vectors are linearly independent.
( c) A little reflection reveals that
so the three vectors are linearly dependent. [Set up a linear system as in part (b) to
check this algebraically.]
( d) Once again, we observe no obvious dependence, so we proceed directly to reduce
a homogeneous linear system whose augmented matrix has as its columns the given
vectors:
[ � � : � 1 �I [ �
1 1 0
-1 2 0
0 -1 2 0
l �: �:
0 -1 2 0
If we let the scalars be c 1 , c2 , and c 3 , we have
c1 +
3c 3 = 0
c2 - 2c3 = 0
�
_
R
,
[ -� �1
1 0
0 1
0 0
0 0
from which we see that the system has infinitely many solutions. In particular, there
must be a nonzero solution, so the given vectors are linearly dependent.
If we continue, we can describe these solutions exactly: c 1 = - 3c3 and c2 = 2c3 .
Thus, for any nonzero value of c3 , we have the linear dependence relation
�
(Once again, check that this is correct.)
We summarize this procedure for testing for linear independence as a theorem.
Theorem 2 . 6
Let v 1 , v2 , . . . , vm be (column) vectors in IK£ n and let A be the n X m matrix
[v1 v2 vm l with these vectors as its columns. Then v1 , v2 , , vm are linearly
dependent if and only if the homogeneous linear system with augmented matrix
[A I O ] has a nontrivial solution.
·
·
.
·
[ :: ]
•
•
, vm are linearly dependent ifand only if there are scalars c 1 , c2 , , cm ,
not all zero, such that c 1 v1 + c2v2 + . . . + cmvm = 0. By Theorem 2.4, this is equivalent
Proof
v1 , v2 ,
. . •
to sa'.i� g that the nonzero vector
matnx is [v1 v2 vm I OJ .
.
•
.
.
·
cm
• • .
is a solution of the system whose augmented
96
Chapter
2 Systems of Linear Equations
Exa m p l e 2 . 2 4
The standard unit vectors e 1 , e2 , and e 3 are linearly independent in IR 3 , since the sys­
tem with augmented matrix [ e1 e2 e3 I OJ is already in the reduced row echelon form
[:
0 0 O
1 0 0
0 1 0
J
and so clearly has only the trivial solution. In general, we see that e1 , e2 , . . . , en will be
linearly independent in !R n .
Performing elementary row operations on a matrix constructs linear combina­
tions of the rows. We can use this fact to come up with another way to test vectors for
linear independence.
Exa m p l e 2 . 2 5
Consider the three vectors of Example 2.23(d) as row vectors:
[ 1 , 2, 0] , ( 1 , 1 , - 1 ] , and [ 1 , 4, 2]
We construct a matrix with these vectors as its rows and proceed to reduce it to eche­
lon form. Each time a row changes, we denote the new row by adding a prime symbol:
[:
2
1
4
2
-1
2
From this we see that
O
-1
2
J
�
R; = R; + ZR;
l -�J
[
2
0 -1
0
0
or, in terms of the original vectors,
- 3 ( 1 , 2, 0 ] + 2 [ 1 , 1 , - 1 ] + ( 1 , 4, 2 ] = [ O, O, O ]
[Notice that this approach corresponds to taking c 3 = 1 in the solution to
Example 2.23(d).]
Thus, the rows of a matrix will be linearly dependent if elementary row opera­
tions can be used to create a zero row. We summarize this finding as follows:
Theorem 2 . 1
Let v, , v, . . . . . vm be ( rnw) ve<to" in �" and kt A be t he m X n motrix
these vectors as its rows. Then v1 , v2 ,
rank(A) < m .
•
.
•
[J
with
, vm are linearly dependent if and only if
Proof Assume that v1 , v2 ,
, vm are linearly dependent. Then, by Theorem 2.2,
at least one of the vectors can be written as a linear combination of the others.
•
•
.
Section
2. 3 Spanning Sets and Linear Independence
91
We relabel the vectors, if necessary, so that we can write vm = c 1 v1 + c 2v2 +
+
cm - iYm - l · Then the elementary row operations R m - c 1 R 1 , R m - c 2 R2 ,
,
R m - cm - i R m - I applied to A will create a zero row in row m. Thus, rank(A) < m.
Conversely, assume that rank(A) < m. Then there is some sequence of row opera­
tions that will create a zero row. A successive substitution argument analogous to that
used in Example 2.25 can be used to show that 0 is a nontrivial linear combination of
v1 , v2 , , vm . Thus, v1 , v2 , , vm are linearly dependent.
·
•
•
.
.
.
•
·
·
•
•
.
In some situations, we can deduce that a set of vectors is linearly dependent with­
out doing any work. One such situation is when the zero vector is in the set (as in
Example 2.22). Another is when there are "too many" vectors to be independent. The
following theorem summarizes this case. (We will see a sharper version of this result
in Chapter 6.)
Theorem 2 . 8
Any set of m vectors in !R n is linearly dependent if m > n.
Let v1 , v2 ,
, vm be (column) vectors in !R n and let A be the n X m matrix
[v1 v2 vm l with these vectors as its columns. By Theorem 2.6, v1 , v2 , , vm are lin­
early dependent if and only if the homogeneous linear system with augmented matrix
[A I O] has a nontrivial solution. But, according to Theorem 2.6, this will always be the
case if A has more columns than rows; it is the case here, since number of columns m
is greater than number of rows n.
Proof
.
.
Exa m p l e 2 . 2 6
...
I
•
•
•
•
.
The vectors
[ � ], [ !] [ � ]
, and
.
.
are linearly dependent, since there cannot be more
than two linearly independent vectors in IR 2 • (Note that if we want to find the ac­
tual dependence relation among these three vectors, we must solve the homogeneous
system whose coefficient matrix has the given vectors as columns. Do this! )
Exercises 2 . 3
In Exercises 1 -6, determine if the vector v is a linear combi­
nation of the remaining vectors.
� m. u, m. u, [ : J
u, � m
3[ .2 ] [ 0.4 ] [ 31..44 ]
-[ �:� ]
5. v
GAS
�
�
6. V =
1 .0
, ll2 =
2.0 , U 1 =
48
- 2.6
.
U3
=
- 1 .0
-
64
.
,
Chapter
98
2 Systems of Linear Equations
In Exercises 7 and 8, determine if the vector b is in the span
of the columns of the matrix A.
[� ! l [ : J
8. [ � ! � ] , [ : l
([�] [ �])
( [ � l [ �] ).
'P""( [ H [J [ : ] )
'P"" ( [J m. [ _ : ] )
7. A =
A=
(b) In part (a), suppose in addition that each vj is also
a linear combination of u 1 , . . . , uk. Prove that
span(u 1 , . . . , uk) = span(v1 , . . . , vm ) .
(c ) Use the result o f part (b) t o prove that
b=
9 10 1 1
b=
9 . Show that IR 2 = span
10. Show that IR 2 = span
12
, _
.
_
1 1 . Show thot � ' �
[Hint: We know that IR 3 = span(e 1 , e2 , e3 ).]
Use the method of Example 2.23 and Theorem 2. 6 to deter­
mine if the sets of vectors in Exercises 22-31 are linearly in­
dependent. If, for any of these, the answer can be determined
by inspection (i.e., without calculation), state why. For any
sets that are linearly dependent, find a dependence relation­
ship among the vectors.
[ : ] . m. [ - � ]
[ :H�lE
m. [ !J Hl
[ -; i . [ -!J [ : J m [ ;] . [ H [ � :
23.
12. Show thot � ' �
24 •
25
::r!fFr ::�fa [:J
m. u i
[ _ � i r n [ - : i 28r�H �J u1
;�:�;�; �� �rt��� � ��� �;���:� �;::! � ��:
In Exercises 1 3- 1 6, describe the span of the given vectors
d (b) alg,bm
15.
26.
16.
17.
1
r
f
n
f
,
r
i
,
1
form ax + by + cz = 0. Solve for a, b, and c.
18. Prove that u, v, and w are all in span(u, v, w) .
19. Prove that u, v, and w are all in span(u, u + v, u +
� ; ;:�
ve that if u 1 , . . . , um are vectors in !R n , S =
{ U 1 , U2 , . . . , Uk } , an d T = { U 1 , . . . , Uk , Uk + !> . . . ,
um }, then span (S ) � span ( T) . [Hint: Rephrase this
question in terms of linear combinations.]
(b) Deduce that if [R n = span (S), then [R n = span ( T)
also.
2 1 . (a) Suppose that vector w is a linear combination
of vectors u 1 , . . . , uk and that each u; is a linear
combination of vectors v1 , . . . , vm . Prove that w is
a linear combination ofv1 , . . . , vm and therefore
span(u 1 , . . . , uk) � span(v 1 , . . . , vm ) .
20 . a
[ l [ -1 1 [ l [ o l
�[ l [ � l [ � l [ : l
3
�[ l [ - � 1 [ � l , [ -- 3� 1
l
29 .
30 ·
31.
27.
-1
,
1
0
l
1
1
0
,
,
-1
1
0
1
1
-1
0 ' 2 ' 2 ' 2
1
1
1
1
1
l
-
,
,
-l
l
-l
l
In Exercises 32- 41, determine if the sets of vectors in the
given exercise are linearly independent by converting the
Section
vectors to row vectors and using the method of Example 2.25
and Theorem 2.7. For any sets that are linearly dependent,
find a dependence relationship among the vectors.
32. Exercise 22
33. Exercise 23
34. Exercise 24
35. Exercise 25
36. Exercise 26
37. Exercise 27
38. Exercise 28
39. Exercise 29
40. Exercise 30
41. Exercise 3 1
42. (a) If the columns of an n X n matrix A are linearly in­
dependent as vectors in !R n , what is the rank of A?
Explain.
(b) If the rows of an n X n matrix A are linearly inde­
pendent as vectors in !R n , what is the rank of A?
Explain.
43. (a) If vectors u, v, and w are linearly independent, will
u + v, v + w, and u + w also be linearly indepen dent? Justify your answer.
11 f
2.4 Applications
99
(b) If vectors u, v, and w are linearly independent, will
u - v, v - w, and u - w also be linearly indepen­
dent? Justify your answer.
44. Prove that two vectors are linearly dependent if
and only if one is a scalar multiple of the other.
[Hint: Separately consider the case where one of the
vectors is O.]
45. Give a "row vector proof" of Theorem 2.8.
46. Prove that every subset of a linearly independent set is
linearly independent.
47. Suppose that S = {v1 , , vk> v} is a set of vectors in
some !R n and that v is a linear combination of v 1 , . . . ,
vk. If S' = {v1 , . . . , vd, prove that span (S ) = span (S').
[Hint: Exercise 2 l (b) is helpful here.]
48. Let {v1 , . . . , vd be a linearly independent set of vec­
tors in !R n , and let v be a vector in !R n . Suppose that
v = c 1 v1 + c 2 v2 + · · · + ck vk with c 1 * 0. Prove that
{v, v2 , , vd is linearly independent.
.
.
•
•
.
.
A p p l icati o n s
There are too many applications of systems of linear equations to do them justice in a
single section. This section will introduce a few applications, to illustrate the diverse
settings in which they arise.
Allocation o f Resources
A great many applications of systems of linear equations involve allocating limited
resources subject to a set of constraints.
Exa m p l e 2 . 2 1
A biologist has placed three strains of bacteria (denoted I, II, and III) in a test tube,
where they will feed on three different food sources (A, B, and C). Each day 2300 units
of A, 800 units of B, and 1 500 units of C are placed in the test tube, and each bacte­
rium consumes a certain number of units of each food per day, as shown in Table 2.2.
How many bacteria of each strain can coexist in the test tube and consume all of the
food?
Ta b l e 2 . 2
Food A
Food B
Food C
Bacteria
Strain I
2
1
1
Bacteria
Strain II
2
2
3
Bacteria
Strain III
4
0
100
Chapter
2 Systems of Linear Equations
Let x 1 , x 2 , and x3 b e the numbers of bacteria of strains I , II, and III,
respectively. Since each of the x 1 bacteria of strain I consumes 2 units of A per day,
strain I consumes a total of 2x 1 units per day. Similarly, strains II and III consume a
total of 2x 2 and 4x 3 units of food A daily. Since we want to use up all of the 2300 units
of A, we have the equation
2x 1 + 2x2 + 4x3 = 2300
Solulion
Likewise, we obtain equations corresponding to the consumption of B and C:
= 800
X 1 + 2X 2
X 1 + 3X 2 + X3 = 1 500
Thus, we have a system of three linear equations in three variables. Row reduction of
the corresponding augmented matrix gives
[:
2 4 2300
2 0 800
3 1 1 500
l
] [
�
0 0 1 00
0
0 350
0 0 1 350
]
Therefore, x 1 = 1 00, x 2 = 350, and x 3 = 350. The biologist should place 100 bacteria
of strain I and 350 of each of strains II and III in the test tube if she wants all the food
to be consumed.
Exa m p l e 2 . 2 8
Repeat Example 2.27, using the data on daily consumption of food (units per day)
shown in Table 2.3. Assume this time that 1 500 units of A, 3000 units of B, and 4500
units of C are placed in the test tube each day.
Ta b l e 2 . 3
Bacteria
Strain II
Bacteria
Strain I
Food A
Food B
Food C
2
3
Bacteria
Strain III
1
3
5
Let x 1 , x2 , and x3 again be the numbers of bacteria of each type. The aug­
mented matrix for the resulting linear system and the corresponding reduced echelon
form are
Solulion
[:
� ���� i
1
2
3 5 4500
�
�[ � O J
-1
2 1 500
0 0
0
0
We see that in this case we have more than one solution, given by
0
- X3 =
X 2 + 2X 3 = 1 500
Letting x3 = t, we obtain x 1 = t, x2 = 1 500 - 2t, and x3 = t. In any applied problem,
we must be careful to interpret solutions properly. Certainly the number of bacteria
X1
Section
2.4 Applications
101
cannot be negative. Therefore, t 2: 0 and 1 500 2t 2: 0. The latter inequality implies
that t :::::: 750, so we have 0 :::::: t :::::: 750. Presumably the number of bacteria must be a
whole number, so there are exactly 75 1 values of t that satisfy the inequality. Thus, our
75 1 solutions are of the form
-
one for each integer value of t such that 0 :::::: t :::::: 750. (So, although mathematically
this system has infinitely many solutions, physically there are only finitely many.)
Balancing Chemical Equations
.+
When a chemical reaction occurs, certain molecules (the reactants) combine to form
new molecules (the products). A balanced chemical equation is an algebraic equation
that gives the relative numbers of reactants and products in the reaction and has the
same number of atoms of each type on the left- and right-hand sides. The equation is
usually written with the reactants on the left, the products on the right, and an arrow
in between to show the direction of the reaction.
For example, for the reaction in which hydrogen gas (H 2 ) and oxygen (0 2 ) com­
bine to form water (H 2 0), a balanced chemical equation is
2H 2 + 0 2 -----+ 2H 2 0
indicating that two molecules of hydrogen combine with one molecule of oxygen to
form two molecules of water. Observe that the equation is balanced, since there are
four hydrogen atoms and two oxygen atoms on each side. Note that there will never
be a unique balanced equation for a reaction, since any positive integer multiple of
a balanced equation will also be balanced. For example, 6H 2 + 30 2 -----+ 6H 2 0 is also
balanced. Therefore, we usually look for the simplest balanced equation for a given
reaction.
While trial and error will often work in simple examples, the process of balancing
chemical equations really involves solving a homogeneous system oflinear equations,
so we can use the techniques we have developed to remove the guesswork.
Exa m p l e 2 . 2 9
The combustion of ammonia (NH 3 ) in oxygen produces nitrogen (N 2 ) and water.
Find a balanced chemical equation for this reaction.
Solution
If we denote the numbers of molecules of ammonia, oxygen, nitrogen, and
water by w, x, y, and z, respectively, then we are seeking an equation of the form
wNH 3 + x0 2 -----+ yN 2 + zH 2 0
Comparing the numbers of nitrogen, hydrogen, and oxygen atoms in the reactants
and products, we obtain three linear equations:
Nitrogen: w = 2y
Hydrogen: 3w = 2z
Oxygen: 2x = z
Rewriting these equations in standard form gives us a homogeneous system of three
linear equations in four variables. [Notice that Theorem 2.3 guarantees that such a
102
Chapter
2 Systems of Linear Equations
system will have (infinitely many) nontrivial solutions.] We reduce the corresponding
augmented matrix by Gauss-Jordan elimination.
w
3w
- 2y
2x
0O -----+ [
= 0
- 2z =
-
z
=
1 0
3 0
0 2
-2
0
0
- � �] -----+ [ � �
-1 0
0
0
0 0 1
0]
-t
-! 0
-t 0
Thus, w = t z, x = ! z, and y = t z . The smallest positive value of z that will produce
integer values for all four variables is the least common denominator of the fractions
t, !, and t-namely, 6-which gives w = 4, x = 3, y = 2, and z = 6. Therefore, the
balanced chemical equation is
Network Analysis
j20
f
+2
i 30
Many practical situations give rise to networks: transportation networks, communi­
cations networks, and economic networks, to name a few. Of particular interest are
the possible flows through networks. For example, vehicles flow through a network
of roads, information flows through a data network, and goods and services flow
through an economic network.
For us, a network will consist of a finite number of nodes (also called junctions
or vertices) connected by a series of directed edges known as branches or arcs. Each
branch will be labeled with a flow that represents the amount of some commodity
that can flow along or through that branch in the indicated direction. (Think of cars
traveling along a network of one-way streets.) The fundamental rule governing flow
through a network is conservation offlow:
At each node, the flow in equals the flow out.
Figure 2 . 1 0
Flow at a node: f1 + f2 =
50
Exa m p l e 2 . 3 0
Figure 2 . 1 0 shows a portion of a network, with two branches entering a node and two
leaving. The conservation of flow rule implies that the total incoming flow, f1 + f
2
units, must match the total outgoing flow, 20 + 30 units. Thus, we have the linear
equation f1 + f = 50 corresponding to this node.
2
We can analyze the flow through an entire network by constructing such equa­
tions and solving the resulting system of linear equations.
Describe the possible flows through the network of water pipes shown in Figure 2. 1 1 ,
where flow is measured in liters per minute.
At each node, we write out the equation that represents the conservation
of flow there. We then rewrite each equation with the variables on the left and the
constant on the right, to get a linear system in standard form.
Solution
Node A : 1 5 = !1 + 14
Node B : !1 = !2 + 1 0
Node C: !2 + f3 + 5 = 3 0
Node D: f4 + 20 = f3
-----+
+ f4 = 1 5
!1
= 10
!1 - !2
= 25
!2 + f3
f, - f4 = 20
Section
5!
jQ. A
20
-
!1
J4 !
h
-
D
Figure 2 . 1 1
10
1d
] 1
[
c
5
+--
Using Gauss-Jordan elimination, we reduce the augmented matrix:
.......,..
[i
1 15
0 0
0 10
-1 0
0 25
1
0 1 - 1 20
�
103
-
B
3 0!
2.4 Applications
0 0
0
0
0 0 1
0 0 0
1 15
1 5
- 1 20
0 0
]
(Check this.) We see that there is one free variable, f4 , so we have infinitely many
solutions. Settingf4 = t and expressing the leading variables in terms off4 , we obtain
f, = 1 5 - t
!2 = 5 - t
f3 = 20 + t
!4 =
These equations describe all possible flows and allow us to analyze the network. For
example, we see that if we control the flow on branch AD so that t = 5 L/min, then
the other flows are f1 = 1 O,f2 = 0, and f3 = 25.
We can do even better: We can find the minimum and maximum possible flows
on each branch. Each of the flows must be nonnegative. Examining the first and sec­
ond equations in turn, we see that t :s 15 (otherwise f1 would be negative) and t :s 5
(otherwisef2 would be negative). The second of these inequalities is more restrictive
than the first, so we must use it. The third equation contributes no further restrictions
on our parameter t, so we have deduced that 0 :s t :s 5. Combining this result with
the four equations, we see that
1 0 :s f, :s 1 5
0 :S f2 :S 5
20 :s f3 :s 25
0 :S .fi :S 5
We now have a complete description of the possible flows through this network.
4
104
Chapter
2 Systems of Linear Equations
Electrical Networks
Electrical networks are a specialized type of network providing information about
power sources, such as batteries, and devices powered by these sources, such as light
bulbs or motors. A power source "forces" a current of electrons to flow through the
network, where it encounters various resistors, each of which requires that a certain
amount of force be applied in order for the current to flow through it.
The fundamental law of electricity is Ohm's law, which states exactly how much
force E is needed to drive a current I through a resistor with resistance R.
force = resistance X current
O h m 's Law
or
E = RI
Force is measured in volts, resistance in ohms, and current in amperes (or amps, for
short). Thus, in terms of these units, Ohm's law becomes "volts = ohms X amps;' and
it tells us what the "voltage drop" is when a current passes through a resistor-that is,
how much voltage is used up.
Current flows out of the positive terminal of a battery and flows back into the
negative terminal, traveling around one or more closed circuits in the process. In
a diagram of an electrical network, batteries are represented by � I- (where the
positive terminal is the longer vertical bar) and resistors are represented by -'VV'v- .
The following two laws, whose discovery we owe to Kirchhoff, govern electrical net­
works. The first is a "conservation of flow" law at each node; the second is a "balancing
of voltage" law around each circuit.
Kirchh off 's Laws
Current Law (nodes)
The sum of the currents flowing into any node is equal to the sum of the currents
flowing out of that node.
Voltage Law (circuits)
The sum of the voltage drops around any circuit is equal to the total voltage around
the circuit (provided by the batteries) .
Figure 2 . 1 2 illustrates Kirchhoff's laws. I n part (a), the current law gives 11 = 12 + 13
(or 11 - 12 - 13 = 0, as we will write it); part (b) gives 41 = 10, where we have used
Ohm's law to compute the voltage drop 41 at the resistor. Using Kirchhoff's laws, we
can set up a system of linear equations that will allow us to determine the currents in
an electrical network.
Exa m p l e 2 . 3 1
Determine the currents 11 , 12 , and 13 in the electrical network shown in Figure 2 . 1 3 .
This network has two batteries and four resistors. Current 11 flows through
the top branch BCA, current 12 flows across the middle branch AB, and current 13
flows through the bottom branch BDA.
At node A, the current law gives 11 + 13 = 12 , or
11 - 12 + 13 = 0
(Observe that we get the same equation at node B.)
Solution
Section
1
lY',,
11
c
+---
1
+---
+---
2. 4 Applications
12
2 ohms
10 volts
--+
A
1
�
8 volts
h
4 ohms
D
+---
4 ohms
(b) 41 = 1 0
11
+---
lz
--+
I ohm
105
2 ohms
B
h
+---
1 6 volts
Figure 2 . 1 3
Figure 2 . 1 2
Next we apply the voltage law for each circuit. For the circuit CABC, the voltage
drops at the resistors are 2Ii , I2 , and 2Ii . Thus, we have the equation
4Ii + I2
Similarly, for the circuit DABD, we obtain
I2 + 4I3
=
=
8
16
(Notice that there i s actually a third circuit, CADBC, if we "go against the flow:' In this
case, we must treat the voltages and resistances on the "reversed" paths as negative.
Doing so gives 2Ii + 2Ii - 4I3 = 8
= - 8 or 4I - 4I3 = - 8, which we observe
i
is just the difference of the voltage equations for the other two circuits. Thus, we can
omit this equation, as it contributes no new information. On the other hand, includ­
ing it does no harm.)
We now have a system of three linear equations in three variables:
- 16
0
16
- 1 1 o 01 o1 � 41
[� � 1! l [ 0 o 1 3 l
1
I1 - I2 + I3
4I1 + I2
I2 + 4I3
Gauss-Jordan elimination produces
=
8
=
�
Hence, the currents are Ii = amp, I2 = 4 amps, and I3 = 3 amps.
In some electrical networks, the currents may have fractional values or may
even be negative. A negative value simply means that the current in the correspond­
ing branch flows in the direction opposite that shown on the network diagram.
Remark
CAS
Exa m p l e 2 . 3 2
The network shown in Figure 2 . 1 4 has a single power source A and five resistors. Find
the currents I, Ii , . . . , I5 • This is an example of what is known in electrical engineering
as a Wheatstone bridge circuit.
106
Chapter
2 Systems of Linear Equations
1i
i
13
14
--
B
I ohm
-1
l ohm
c
2 ohms
!Ii
2 ohms
15
! -E
D
2 ohms
A
1 0 volts
-1
Figure 2 . 1 4
A bridge circuit
Solulion
nodes:
Kirchhoff's current law gives the following equations at the four
Node B: I - I1 - I4 = 0
Node C: I1 - I2 - I3 = 0
Node D: I - I2 - Is = 0
Node E : I3 + I4 - Is = 0
For the three basic circuits, the voltage law gives
Circuit ABEDA: I4 + 2Is = 1 0
Circuit BCEB: 2I1 + 2I3 - I4 = 0
Circuit CDEC: I2 - 2Is - 2I3 = 0
(Observe that branch DAB has no resistor and therefore no voltage drop; thus, there
is no I term in the equation for circuit ABEDA. Note also that we had to change signs
three times because we went "against the current:' This poses no problem, since we
will let the sign of the answer determine the direction of current flow.)
We now have a system of seven equations in six variables. Row reduction gives
1
0
1
0
0
0
0
�
-1
1
0
0
0
2
0
0
-1
-1
0
0
0
0
-1
0
-1
0
0
0
2
-2
-1
0
0 0
0 0
-1 0
-1 0
2 10
0 0
-2 0
�
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
7
0
3
0
4
0 -1
0
4
1
3
0
0
(Use your calculator or CAS to check this.) Thus, the solution (in amps) is I = 7, I1 =
Is = 3, I2 = I4 = 4, and I3 = - 1 . The significance of the negative value here is that
the current through branch CE is flowing in the direction opposite that marked on
the diagram.
-+
There is only one power source in this example, so the single 1 0-volt battery sends a current of 7 amps through the network. If we substitute these values into
Remark
Section
2.4 Applications
101
Ohm's law, E = RI, we get 1 0 = 7R or R = 1?-. Thus, the entire network behaves as if
there were a single 1?--ohm resistor. This value is called the effective resistance (Reff) of
the network.
linear Economic Models
An economy is a very complex system with many interrelationships among the vari­
ous sectors of the economy and the goods and services they produce and consume.
Determining optimal prices and levels of production subject to desired economic
goals requires sophisticated mathematical models. Linear algebra has proven to be a
powerful tool in developing and analyzing such economic models.
In this section, we introduce two models based on the work of Harvard econo­
mist Wassily Leontief in the 1 930s. His methods, often referred to as input-output
analysis, are now standard tools in mathematical economics and are used by cities,
corporations, and entire countries for economic planning and forecasting.
We begin with a simple example.
Exa m p l e 2 . 3 3
The economy of a region consists of three industries, or sectors: service, electricity,
and oil production. For simplicity, we assume that each industry produces a single
commodity (goods or services) in a given year and that income (output) is gener­
ated from the sale of this commodity. Each industry purchases commodities from the
other industries, including itself, in order to generate its output. No commodities are
purchased from outside the region and no output is sold outside the region. Further­
more, for each industry, we assume that production exactly equals consumption (out­
put equals input, income equals expenditure). In this sense, this is a closed economy
that is in equilibrium. Table 2.4 summarizes how much of each industry's output is
consumed by each industry.
Ta b l e 2 . 4
(1906-1999)
Wassily Leontief
was
born in St. Petersburg, Russia. He
studied at the University of Lenin­
grad and received his Ph.D. from the
University of Berlin. He emigrated
to the United States in
teach­
ing at Harvard University and later
at New York University. In
Leontiefbegan compiling data for the
monumental task of conducting an
input-output analysis of the United
States economy, the results of which
were published in
He was also
an early user of computers, which he
needed to solve the large-scale linear
systems in his models. For his pio­
neering work, Leontief was awarded
the Nobel Prize in Economics in
1931,
1932,
1941.
1973.
Consumed by
(input)
Service
Electricity
Oil
Service
Produced by (output)
Electricity
Oil
1/4
1/4
1/2
1/3
1/3
1/3
1 /2
1 /4
1 /4
From the first column of the table, we see that the service industry consumes 1 I4
of its own output, electricity consumes another 1/4, and the oil industry uses 1/2 of
the service industry's output. The other two columns have similar interpretations.
Notice that the sum of each column is 1 , indicating that all of the output of each
industry is consumed.
Let X i , x2 , and x3 denote the annual output (income) of the service, electricity,
and oil industries, respectively, in millions of dollars. Since consumption corresponds
to expenditure, the service industry spends � X i on its own commodity, .!.3 x2 on electricity, and t x3 on oil. This means that the service industry's total annual expenditure is � X i + .!.3 x2 + t x3 . Since the economy is in equilibrium, the service industry's
108
Chapter
2 Systems of Linear Equations
expenditure must equal its annual income x 1 . This gives the first of the following
equations; the other two equations are obtained by analyzing the expenditures of the
electricity and oil industries.
Service:
Electricity:
Oil:
......
"]
"]
Rearranging each equation, we obtain a homogeneous system of linear equations,
which we then solve. (Check this!)
-� x 1 + l3 x2 + t x3 = 0
i x1 - � x2 - i x3 = 0
3
t x1 + l3 x2 - � x3 = 0
---7
n
I
3
2
l3
-3
l
l 0
-i 0
�
[�
0 1 - :4 0
0
0 0
Setting x3 = t, we find that x 1 = t and x2 = � t. Thus, we see that the relative outputs of
the service, electricity, and oil industries need to be in the ratios x 1 : x2 : x3 = 4 : 3 : 4
for the economy to be in equilibrium.
4
R e m a rks
The last example illustrates what is commonly called the Leontief closed model.
Since output corresponds to income, we can also think of x 1 , x2 , and x3 as the
prices of the three commodities.
•
•
We now modify the model in Example 2.33 to accommodate an open economy, one
in which there is an external as well as an internal demand for the commodities that
are produced. Not surprisingly, this version is called the Leontief open model.
Exa m p l e 2 . 3 4
Consider the three industries of Example 2.33 but with consumption given by
Table 2.5. We see that, of the commodities produced by the service industry, 20% are
consumed by the service industry, 40% by the electricity industry, and 1 0% by the oil
industry. Thus, only 70% of the service industry's output is consumed by this econ­
omy. The implication of this calculation is that there is an excess of output (income)
over input (expenditure) for the service industry. We say that the service industry is
productive. Likewise, the oil industry is productive but the electricity industry is non­
productive. (This is reflected in the fact that the sums of the first and third columns
are less than 1 but the sum of the second column is equal to 1 ) . The excess output may
be applied to satisfy an external demand.
Tab l e 2 . 5
Consumed by
(input)
Service
Electricity
Oil
Service
Produced by (output)
Electricity
Oil
0.20
0.40
0. 1 0
0.50
0.20
0.30
0. 1 0
0.20
0.30
Section
2.4 Applications
109
For example, suppose there is an annual external demand (in millions of dollars)
for 10, 1 0, and 30 from the service, electricity, and oil industries, respectively. Then,
equating expenditures (internal demand and external demand) with income (out­
put), we obtain the following equations:
output
internal demand
external demand
Service
Xi
= 0.2X 1 + 0.5X2 + O.lX3
Electricity
X2
X3
= 0.4X 1 + 0.2X2 + 0.2X3
+ 10
+ 10
+ 30
Oil
= O.lx 1 + 0.3x2 + 0.3x3
Rearranging, we obtain the following linear system and augmented matrix:
0.8X 1 - O.SX2 - O. lX3 = 1 0
- 0.4X 1 + 0.8X2 - 0.2X3 = 1 0 ---+
- 0. lx 1 - 0.3X2 + 0.7X3 = 30
CAS
[
08
- 0.4
-0.l
Row reduction yields
[�
0
1
0
0
0
6 1 .74
63.04
78.70
- 0.5
0.8
- 0.3
-0.1
- 0.2
0.7
10
10
30
]
l
from which we see that the service, electricity, and oil industries must have an an­
nual production of $6 1 . 7 4, $63.04, and $78. 70 (million), respectively, in order to meet
both the internal and external demand for their commodities.
We will revisit these models in Section 3.7.
Finile Linear Games
There are many situations in which we must consider a physical system that has only a
finite number of states. Sometimes these states can be altered by applying certain pro­
cesses, each of which produces finitely many outcomes. For example, a light bulb can be
on or off and a switch can change the state of the light bulb from on to off and vice versa.
Digital systems that arise in computer science are often of this type. More frivolously, many
computer games feature puzzles in which a certain device must be manipulated by various
switches to produce a desired outcome. The finiteness of such situations is perfectly suited
to analysis using modular arithmetic, and often linear systems over some ZP play a role.
Problems involving this type of situation are often called finite linear games.
Exa m p l e 2 . 3 5
A row of five lights is controlled by five switches. Each switch changes the state (on or
off) of the light directly above it and the states of the lights immediately adjacent to
the left and right. For example, if the first and third lights are on, as in Figure 2 . l S (a),
then pushing switch A changes the state of the system to that shown in Figure 2 . l S (b).
If we next push switch C, then the result is the state shown in Figure 2 . l S (c).
110
Chapter
B
2 Systems of Linear Equations
c
D
E
A
B
( a)
c
D
E
'-
A
"
B
c
D
IH
E
(c)
(b)
Figure 2 . 1 5
Suppose that initially all the lights are off. Can we push the switches in some order
so that only the first, third, and fifth lights will be on? Can we push the switches in
some order so that only the first light will be on?
The on/off nature of this problem suggests that binary notation will be helpful
and that we should work with 2 2 • Accordingly, we represent the states of the five lights by
a vector in Z�, where 0 represents off and 1 represents on. Thus, for example, the vector
Solution
0
1
0
0
corresponds to Figure 2 . I S (b).
We may also use vectors in Z� to represent the action of each switch. If a switch
changes the state of a light, the corresponding component is a 1 ; otherwise, it is 0.
With this convention, the actions of the five switches are given by
a =
1
0 , b= 1 , c=
0
0
0
0
0
0
0
1
0
0
1 , d=
0
, e =
1
0
The situation depicted in Figure 2 . I S(a) corresponds to the initial state
s =
1
0
1
0
0
followed by
a =
1
0
0
0
Section
2.4 Applications
111
It is the vector sum (in Z� )
0
1
1
s+a=
0
0
Observe that this result agrees with Figure 2 . 1 5 (b).
Starting with any initial configuration s, suppose we push the switches in the order
A, C, D, A, C, B. This corresponds to the vector sum s + a + c + d + a + c + b. But
in Z�, addition is commutative, so we have
s + a + c + d + a + c + b = s + 2a + b + 2c + d
= s + b + d
.-
where we have used the fact that 2 = 0 in z:'. 2 . Thus, we would achieve the same result
by pushing only B and D-and the order does not matter. (Check that this is correct.)
Hence, in this example, we do not need to push any switch more than once.
So, to see if we can achieve a target configuration t starting from an initial
configuration s, we need to determine whether there are scalars x 1 , , x5 in z:'. 2 such that
s + x 1 a + x2 b + · · · + x5 e = t
. • •
In other words, we need to solve (if possible) the linear system over z:'. 2 that corre­
sponds to the vector equation
x 1 a + x2 b + · · · + x5 e = t
-
s=t+s
In this case, s = 0 and our first target configuration is
1
0
t =
1
0
The augmented matrix of this system has the given vectors as columns:
1
1
0
0
1
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
1
1
1
0
0
0
0
0
0
1
1
We reduce it over z:'. 2 to obtain
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1 1
0
112
Chapter
2 Systems of Linear Equations
Thus, x5 i s a free variable. Hence, there are exactly two solutions (corresponding to
x5 = 0 and x5 = 1 ) . Solving for the other variables in terms of x5 , we obtain
X1 =
X2 =
X3 =
X4 =
X5
+ X5
+ X5
So, when x5 = 0 and x5 = 1 , we have the solutions
X1
X2
X3
X4
X5
_...
0
1
0
and
1
0
1
0
X1
X2
X3
X4
X5
respectively. (Check that these both work.)
Similarly, in the second case, we have
1
0
t= 0
0
0
The augmented matrix reduces as follows:
1
0
0
0
1
1
0
0
0 0 0 1
1 0 0 0
1
0 0
1
0
0
0
-----+
1
0
0
0
0
0 0
0
0 1
0 0
0 0
0
0
0
0 0
1 1
0 0
showing that there is no solution in this case; that is, it is impossible to start with all
of the lights off and turn only the first light on.
Example 2.35 shows the power of linear algebra. Even though we might have
found out by trial and error that there was no solution, checking all possible ways to
push the switches would have been extremely tedious. We might also have missed the
fact that no switch need ever be pushed more than once.
Exa m p l e 2 . 3 6
Consider a row with only three lights, each of which can be off, light blue, or dark blue.
Below the lights are three switches, A, B, and C, each of which changes the states of
particular lights to the next state, in the order shown in Figure 2 . 1 6. Switch A changes
the states of the first two lights, switch B all three lights, and switch C the last two
Section
Dark blue
2.4 Applications
113
j
Light blue
�
B
c
Figure 2.11
Figure 2 . 1 6
lights. If all three lights are initially off, is it possible to push the switches in some order
so that the lights are off, light blue, and dark blue, in that order (as in Figure 2 . 1 7) ?
Whereas Example 2 . 3 5 involved "1!.. 2 , this one clearly (is it clear?) involves
"1!.. 3 • Accordingly, the switches correspond to the vectors
Solution
in Zl. and th, final wnfigmation w' "' <riming fod, t �
[n
( Offo 0, light bl"' i' 1 ,
and dark blue i s 2 . ) We wish t o find scalars X i , x2 , x3 i n "1!.. 3 such that
X 1 a + X 2 b + X3 C = t
(where X; represents the number of times the ith switch is pushed) . This equation
gives rise to the augmented matrix [a b c I t] , which reduces over "1!.. 3 as follows:
..
I
Hence, there is a unique solution: X i = 2, x2 = 1 , x3 = 1 . In other words, we must push
switch A twice and the other two switches once each. (Check this.)
Exercises 2 . 4
1. Suppose that, in Example 2.27, 400 units of food A,
A l l o c a t i o n of R e s o u rces
600 units of B, and 600 units of C are placed in the test
tube each day and the data on daily food consump­
tion by the bacteria (in units per day) are as shown
in Table 2.6. How many bacteria of each strain can
coexist in the test tube and consume all of the food?
2. Suppose that in Example 2.27, 400 units of food A,
500 units of B, and 600 units of C are placed in
the test tube each day and the data on daily food
Tab l e 2 . 6
Food A
Food B
Food C
Bacteria
Strain I
1
2
1
Bacteria
Strain II
2
1
Bacteria
Strain III
0
1
2
consumption by the bacteria (in units per day) are
as shown in Table 2.7. How many bacteria of each
114
Chapter
2 Systems of Linear Equations
Tab l e 2 . 1
Food A
Food B
Food C
Bacteria
Strain I
2
1
Bacteria
Strain II
2
Bacteria
Strain III
0
3
1
the special blend produces a profit of $ 1 .50, and one bag
of the gourmet blend produces a profit of $2.00. How
many bags of each type should the merchant prepare
if he wants to use up all of the beans and maximize his
profit? What is the maximum profit?
Balancing Chemical Equations
In Exercises 7-14, balance the chemical equation for each
reaction.
7. FeS 2 + 0 2 -----+ Fe2 0 3 + S0 2
8. C0 2 + H 2 0 -----+ C6H 1 2 06 + 0 2 (This reaction takes
strain can coexist in the test tube and consume all
of the food?
place when a green plant converts carbon dioxide and
3. A florist offers three sizes of flower arrangements
water to glucose and oxygen during photosynthesis.)
containing roses, daisies, and chrysanthemums. Each
small arrangement contains one rose, three daisies,
9. C4H 1 0 + 0 2 -----+ C0 2 + H 2 0 (This reaction occurs
and three chrysanthemums. Each medium arrange­
when butane, C 4 H 1 0 , burns in the presence of oxygen
ment contains two roses, four daisies, and six chry­
to form carbon dioxide and water.)
santhemums. Each large arrangement contains four
10. C 7 H60 2 + 0 2 -----+ H 2 0 + C0 2
roses, eight daisies, and six chrysanthemums. One
1 1 . C 5 H 1 1 0H + 0 2 -----+ H 2 0 + C0 2 (This equation rep­
day, the florist noted that she used a total of 24 roses,
resents the combustion of amyl alcohol.)
50 daisies, and 48 chrysanthemums in filling orders
12.
HC10
4 + P 4 0 1 0 ----+ H 3 P0 4 + Cl2 0 7
for these three types of arrangements. How many
arrangements of each type did she make?
13. Na2 C0 3 + C + N2 -----+ NaCN + CO
4. (a) In your pocket you have some nickels, dimes, and
cA5 14. C 2 H 2 Cl4 + Ca(OH) z -----+ C 2 HC13 + CaC12 + H 2 0
quarters. There are 20 coins altogether and exactly
twice as many dimes as nickels. The total value of the
N e t w o r k A n alvsis
coins is $3.00. Find the number of coins of each type.
15. Figure 2 . 1 8 shows a network of water pipes with flows
(b) Find all possible combinations of 20 coins (nickels,
measured in liters per minute.
dimes, and quarters) that will make exactly $3.00.
(a) Set up and solve a system of linear equations to find
5. A coffee merchant sells three blends of coffee. A bag
the possible flows.
of the house blend contains 300 grams of Colombian
(b) If the flow through AB is restricted to 5 L/min, what
beans and 200 grams of French roast beans. A bag of the
will the flows through the other two branches be?
special blend contains 200 grams of Colombian beans,
(c) What are the minimum and maximum possible
200 grams of Kenyan beans, and 100 grams of French
flows through each branch?
roast beans. A bag of the gourmet blend contains
(
d)
We have been assuming that flow is always posi­
100 grams of Colombian beans, 200 grams of Kenyan
tive. What would negative flow mean, assum­
beans, and 200 grams of French roast beans. The mer­
ing we allowed it? Give an illustration for this
chant has on hand 30 kilograms of Colombian beans,
example.
15 kilograms of Kenyan beans, and 25 kilograms of French
roast beans. Ifhe wishes to use up all of the beans, how
many bags of each type of blend can be made?
!1
--+
A
6. Redo Exercise 5, assuming that the house blend contains
300 grams of Colombian beans, 50 grams of Kenyan
beans, and 1 50 grams of French roast beans and the
gourmet blend contains 100 grams of Colombian beans,
c
350 grams of Kenyan beans, and 50 grams of French roast
beans. This time the merchant has on hand 30 kilograms of
Colombian beans, 1 5 kilograms of Kenyan beans, and
B
1 5 kilograms of French roast beans. Suppose one bag of
the house blend produces a profit of $0.50, one bag of
Fioure 2 . 1 8
Section
16. The downtown core of Gotham City consists of
one-way streets, and the traffic flow has been
measured at each intersection. For the city block
shown in Figure 2 . 1 9, the numbers represent the
average numbers of vehicles per minute entering and
leaving intersections A, B, C, and D during business
hours.
(a) Set up and solve a system oflinear equations to find
the possible flows f1 , . . . ,f4 .
(b) If traffic is regulated on CD so that f4 = 1 0 vehi­
cles per minute, what will the average flows on the
other streets be?
(c) What are the minimum and maximum possible
flows on each street?
(d) How would the solution change if all of the direc­
tions were reversed?
!
10
J.Q.
A
h
.£
!
D
f
!1
(b) Suppose DC is closed. What range of flow will need
to be maintained through DB?
( c ) From Figure 2.20 it is clear that DB cannot be
closed. (Why not?) How does your solution in part
(a) show this?
(d) From your solution in part (a), determine the mini­
mum and maximum flows through DB.
18. (a) Set up and solve a system of linear equations to
find the possible flows in the network shown in
Figure 2.2 1 .
(b) Is it possible forf1 = 1 0 0 andf6 = 1 50? [Answer
this question first with reference to your solution
in part (a) and then directly from Figure 2.2 1 . ]
( c) I ff4 = 0 , what will the range o f flow b e o n each of
the other branches?
f4
+---
!
10
B
13
t
c
is
f
1 50
1 0o
200
.2....
13
.£
t
200
f
f
Ji
A
--
D
+---
1 0o
Figure 2 . 1 9
16
f4
1 00
!
!
h
f
20o
B
--
E
+---
!
fs
h
t
l
c
F
l.QQ..
.l2Q.
1 0o
Figure 2 . 2 1
17. A network of irrigation ditches is shown in Figure 2.20,
with flows measured in thousands of liters per day.
l.QQ.. A
E l e c t r i c a l Networks
For Exercises 19 and 20, determine the currents for the
given electrical networks.
19.
c
/1
/1
+---
+---
8 volts
1 ohm
.l2Q.
A
"
/2
c
Figure 2 . 2 0
/2
--
!
..!22.
--
1 ohm
D
200
115
(a) Set up and solve a system oflinear equations to find
the possible flows f1 , . . . ,f5 •
20
--
2.4 Applications
4 ohms
/3
+---
D
1 3 volts
/3
+---
B
Chapter
116
2 Systems of Linear Equations
c
Ii
20.
-
11
R 1
1
R1
Rz
eff
-
5 volts
-+­
1 ohm
h
12
--
A
---
--
B
2 ohms
4 ohms
13
-
D
13
E
-
( a)
8 volts
21. (a) Find the currents I, I1 , . . . , I5 in the bridge circuit
in Figure 2.22.
(b) Find the effective resistance of this network.
O?
(c) Can you change the resistance in branch BC (but
leave everything else unchanged) so that the cur­
rent through branch CE becomes
l
Ii
i
B
c
ohm
2 ohms
/4
13
2 ohms
I
-
! 15
E
A
Ri
12
R2
1
f
--
E
(b)
!h
1 ohm
--
11
--
--
14 volts
1 ohm
D
I
-
Figure 2 . 2 2
22. The networks in parts (a) and (b) of Figure 2.23
show two resistors coupled in series and in parallel,
respectively. We wish to find a general formula for the
effective resistance of each network-that is, find Reff
such that E = Refl
(a) Show that the effective resistance Reff of a network
with two resistors coupled in series [Figure 2.23(a) J
is given by
(b) Show that the effective resistance Reff of a net­
work with two resistors coupled in parallel
[Figure 2.23(b)] is given by
Figure 2 . 2 3
Resistors in series and in parallel
l i n e a r E c o n o m i c M o d e ls
23. Consider a simple economy with just two industries:
farming and manufacturing. Farming consumes 1/2 of
the food and 1/3 of the manufactured goods. Manufac­
turing consumes 1/2 of the food and 2/3 of the manu­
factured goods. Assuming the economy is closed and
in equilibrium, find the relative outputs of the farming
and manufacturing industries.
24. Suppose the coal and steel industries form a closed
economy. Every $1 produced by the coal industry
requires $0.30 of coal and $0.70 of steel. Every $ 1
produced by steel requires $0.80 o f coal and $0.20 of
steel. Find the annual production (output) of coal and
steel if the total annual production is $20 million.
25. A painter, a plumber, and an electrician enter into a
cooperative arrangement in which each of them agrees
to work for himself/herself and the other two for a
total of 1 0 hours per week according to the schedule
shown in Table 2.8. For tax purposes, each person must
establish a value for his/her services. They agree to do
this so that they each come out even-that is, so that the
Section
total amount paid out by each person equals the amount
he/she receives. What hourly rate should each person
charge if the rates are all whole numbers between $30
and $60 per hour?
Ta b l e 2 . 8
Supplier
Painter Plumber Electrician
2
4
4
Painter
Consumer Plumber
Electrician
5
1
4
5
4
26. Four neighbors, each with a vegetable garden, agree to
share their produce. One will grow beans (B), one will
grow lettuce (L), one will grow tomatoes (T), and one
will grow zucchini (Z). Table 2.9 shows what fraction
of each crop each neighbor will receive. What prices
should the neighbors charge for their crops if each
person is to break even and the lowest-priced crop has
a value of $50?
2.4 Applications
111
they produce, each department uses a certain amount
of the services produced by the other departments
and itself, as shown in Table 2 . 1 0. Suppose that, dur­
ing the year, other city departments require $ 1 million
in Administrative services, $ 1 .2 million in Health
services, and $0.8 million in Transportation services.
What does the annual dollar value of the services
produced by each department need to be in order to
meet the demands?
Ta b l e 2 . 1 0
Buy
A
H
T
Department
H
A
T
$0.20
0. 1 0
0.20
0.20
0.20
0.30
0. 1 0
0. 1 0
0.40
Finite linear Games
29. (a) In Example 2.35, suppose all the lights are initially
Ta b l e 2 . 9
Consumer
B
L
T
z
B
Producer
L
T
z
0
1 /2
1 /4
1 /4
1/4
1/4
1/4
1/4
1/8
1/4
1/2
1/8
1 /6
1 /6
1/3
1/3
27. Suppose the coal and steel industries form an open
economy. Every $1 produced by the coal industry
requires $0. 1 5 of coal and $0.20 of steel. Every $ 1
produced by steel requires $0.25 of coal and $0. 1 0 of
steel. Suppose that there is an annual outside demand
for $45 million of coal and $ 1 24 million of steel.
(a) How much should each industry produce to satisfy
the demands?
(b) If the demand for coal decreases by $5 million
per year while the demand for steel increases by
$6 million per year, how should the coal and steel
industries adjust their production?
28. In Gotham City, the departments of Administra­
tion (A), Health (H), and Transportation (T) are
interdependent. For every dollar's worth of services
off. Can we push the switches in some order so
that only the second and fourth lights will be on?
(b) Can we push the switches in some order so that
only the second light will be on?
30. (a) In Example 2.35, suppose the fourth light is
initially on and the other four lights are off. Can
we push the switches in some order so that only
the second and fourth lights will be on?
(b) Can we push the switches in some order so that
only the second light will be on?
31. In Example 2.35, describe all possible configurations
of lights that can be obtained if we start with all the
lights off.
32. (a) In Example 2.36, suppose that all of the lights
are initially off. Show that it is possible to
push the switches in some order so that the
lights are off, dark blue, and light blue, in that order.
(b) Show that it is possible to push the switches in
some order so that the lights are light blue, off,
and light blue, in that order.
(c) Prove that any configuration of the three lights
can be achieved.
33. Suppose the lights in Example 2.35 can be off, light
blue, or dark blue and the switches work as described
Chapter
118
2 Systems of Linear Equations
in Example 2.36. (That is, the switches control the same
lights as in Example 2.35 but cycle through the colors as
in Example 2.36.) Show that it is possible to start with
all of the lights off and push the switches in some order
so that the lights are dark blue, light blue, dark blue,
light blue, and dark blue, in that order.
34. For Exercise 33, describe all possible configurations
of lights that can be obtained, starting with all the
lights off.
cAs 35. Nine squares, each one either black or white, are ar­
ranged in a 3 X 3 grid. Figure 2.24 shows one possible
how the state changes work. (Touching the square
whose number is circled causes the states of the
squares marked * to change.) The object of the game
is to turn all nine squares black. [Exercises 35 and 36
are adapted from puzzles that can be found in
the interactive CD-ROM game The Seventh Guest
(Trilobyte Software/Virgin Games, 1 992) .]
(a) If the initial configuration is the one shown in
Figure 2.24, show that the game can be won and
describe a winning sequence of moves.
(b) Prove that the game can always be won, no matter
what the initial configuration.
CAS 36. Consider a variation on the nine squares puzzle. The
game is the same as that described in Exercise 35
except that there are three possible states for each
square: white, gray, or black. The squares change as
shown in Figure 2.25, but now the state changes follow
the cycle white � gray � black � white. Show how
the winning all-black configuration can be achieved
from the initial configuration shown in Figure 2.26.
Figure 2 . 2 4
The nine squares
puzzle
arrangement. When touched, each square changes
its own state and the states of some of its neighbors
(black � white and white � black) . Figure 2.25 shows
CD*
2
4
5
*
*
*
8
7
@*
3
4
5
7
8
3
l
6
9
*
1
2
6
4
5
9
7
8
*
*
*
®*
6
*
Figure 2.26
The nine squares puzzle
with more states
9
Miscellaneous Problems
l
2
3
l
@*
5
6
4
7
8
9
2
*
*
l
4
*
(j)*
5
8
*
*
2
*
3
1
2
3
4
5
®*
G)*
6
7
8
9
7
8
9
3
l
2
3
l
2
3
6
4
5
6
4
5
9
7
®*
9
7
8
*
*
*
Figure 2 . 2 5
State changes for the nine squares puzzle
*
*
*
*
6
*
*
*
®*
In Exercises 37-53, set up and solve an appropriate system
of linear equations to answer the questions.
37. Grace is three times as old as Hans, but in 5 years she
will be twice as old as Hans is then. How old are they
now?
38. The sum of Annie's, Bert's, and Chris's ages is 60.
Annie is older than Bert by the same number of years
that Bert is older than Chris. When Bert is as old as
Annie is now, Annie will be three times as old as Chris
is now. What are their ages?
The preceding two problems are typical of those found in
popular books of mathematical puzzles. However, they have
their origins in antiquity. A Babylonian clay tablet that sur­
vives from about 300 B. c. contains the following problem.
Section
Over 2000 years ago, the Chinese developed methods for
solving systems of linear equations, including a version of
Gaussian elimination that did not become well known in
Europe until the 1 9th century. (There is no evidence that
Gauss was aware of the Chinese methods when he devel­
oped what we now call Gaussian elimination. However, it is
clear that the Chinese knew the essence of the method, even
though they did not justify its use.) The following problem
is taken from the Chinese text Jiuzhang suanshu (Nine
Chapters in the Mathematical Art), written during the early
Han Dynasty, about 200 B . C .
(a) (O, 1), ( - 1, 4), and (2, 1 )
(b) ( - 3, 1), ( - 2, 2), and ( - 1, 5)
46. Through any three noncollinear points there also
passes a unique circle. Find the circles (whose general
equations are of the form x2 + y2 + ax + by + c = 0)
that pass through the sets of points in Exercise 45. (To
check the validity of your answer, find the center and
radius of each circle and draw a sketch.)
40. There are three types of corn. Three bundles of the
first type, two of the second, and one of the third
make 39 measures. Two bundles of the first type, three
of the second, and one of the third make 34 measures.
And one bundle of the first type, two of the second,
and three of the third make 26 measures. How many
measures of corn are contained in one bundle of
each type?
41. Describe all possible values of a, b, c, and d that
will make each of the following a valid addition
table. [Problems 4 1 -44 are based on the article
''An Application of Matrix Theory" by Paul Glaister in
The Mathematics Teacher, 85 ( 1 992), pp. 220-223.]
(b)
The process of adding rationalfunctions (ratios ofpolyno­
mials) by placing them over a common denominator is
the analogue of adding rational numbers. The reverse
process of taking a rationalfunction apart by writing it as
a sum ofsimpler rationalfunctions is useful in several
areas of mathematics; for example, it arises in calculus
when we need to integrate a rationalfunction and in dis­
crete mathematics when we use generatingfunctions to
solve recurrence relations. The decomposition of a rational
function as a sum ofpartialfractions leads to a system of
linear equations. In Exercises 47-50,find the partial
fraction decomposition of the givenform. (The capital
letters denote constants.)
�
b
c 3 6
d 4 5
42. What conditions on w, x, y, and z will guarantee that
we can find a, b, c, and d so that the following is a valid
addition table?
3x + 1
A
B
+ -=
x + 2x - 3 x - 1 x + 3
C
B
x2 - 3x + 3 A
=48. 3
+
+
2
x + 2x + x x x + 1 (x + 1) 2
x-1
49.
(X + l )(x 2 + l )(x 2 + 4 )
A
Bx + C Dx + E
=
+ 2
+ --x+ 1 x + 1
x2 + 4
x3 + x + 1
A
B
50 •
=-+
x (x - l) (x 2 + x + l) (x 2 + 1 ) 3 x x - 1
Cx + D
Ex + F Gx + H
Ix + J
+
+ x2
+
+
2
2
2
+ x + l x + 1 (x + 1 )
(x 2 + 1 ) 3
47. 2
-
--
--
43. Describe all possible values of a, b, c, d, e, and f
that will make each of the following a valid addition
table.
( a)
+
d
e
f
a b c
3 2 1
5 4 3
4 3
(b)
+
d
e
f
a b c
1 2 3
3 4 5
4 5 6
119
44. Generalizing Exercise 42, find conditions on the en­
tries of a 3 X 3 addition table that will guarantee that
we can solve for a, b, c, d, e, and f as previously.
45. From elementary geometry we know that there
is a unique straight line through any two points
in a plane. Less well known is the fact that there is a
unique parabola through any three noncollinear
points in a plane. For each set of points below, find
a parabola with an equation of the form y = ax 2 +
bx + c that passes through the given points. (Sketch
the resulting parabola to check the validity of your
answer.)
39. There are two fields whose total area is 1 800 square
yards. One field produces grain at the rate of � bushel
per square yard; the other field produces grain at the
rate of t bushel per square yard. If the total yield is
1 1 00 bushels, what is the size of each field?
( a)
2.4 Applications
CAS
---
------
--
CAS
--
----
120
Chapter
2 Systems of Linear Equations
Following are two useful formulas for the sums ofpowers of
consecutive natural numbers:
n(n + 1 )
1 + 2 + · · · + n = ---2
and
n(n + 1 ) (2n + 1)
1 2 + 2 2 + · . · + n 2 = ----6
The validity of these formulas for all values of n 2:: 1 (or
even n 2:: OJ can be established using mathematical induc­
tion (see Appendix B) . One way to make an educated guess
as to what the formulas are, though, is to observe that we
can rewrite the two formulas above as
respectively. This leads to the conjecture that the sum ofpth
powers of the first n natural numbers is a poly nomial of
degree p + 1 in the variable n.
5 1 . Assuming that 1 + 2 + · · · + n = an 2 + bn + c,
find a, b, and c by substituting three values for n and
thereby obtaining a system of linear equations in a,
b, and c.
52. Assume that 1 2 + 2 2 + · · · + n 2 = an 3 + bn 2 + en + d.
Find a, b, c, and d. [Hint: It is legitimate to use n = 0.
What is the left-hand side in that case?]
53. Show that 1 3 + 2 3 + · · · + n 3 = (n(n + 1)/2) 2 .
Vi gnette
T h e Glob al Po s it i o n i n g System
This application is based on the
article "An Underdetermined
Linear System for GPS" by
Dan Kalman in The College
33 (2002),
pp. 384-390. For a more in-depth
Mathematics Journal,
treatment of the ideas introduced
here, see G. Strang and K. Borre,
Linear Algebra, Geodesy, and GPS
1997).
(Wellesley-Cambridge Press, MA,
The Global Positioning System (GPS) i s used i n a variety o f situations fo r determin­
ing geographical locations. The military, surveyors, airlines, shipping companies,
and hikers all make use of it. GPS technology is becoming so commonplace that
some automobiles, cellular phones, and various handheld devices are now equipped
with it.
The basic idea of GPS is a variant on three-dimensional triangulation: A point
on Earth's surface is uniquely determined by knowing its distances from three other
points. Here the point we wish to determine is the location of the GPS receiver, the
other points are satellites, and the distances are computed using the travel times of
radio signals from the satellites to the receiver.
We will assume that Earth is a sphere on which we impose an xyz-coordinate
system with Earth centered at the origin and with the positive z-axis running through
the north pole and fixed relative to Earth.
For simplicity, let's take one unit to be equal to the radius of Earth. Thus Earth's
surface becomes the unit sphere with equation x 2 + y 2 + z 2 = 1 . Time will be
measured in hundredths of a second. GPS finds distances by knowing how long it
takes a radio signal to get from one point to another. For this we need to know the
speed of light, which is approximately equal to 0.47 (Earth radii per hundredths of
a second) .
Let's imagine that you are a hiker lost in the woods at point (x, y, z) at some time
t. You don't know where you are, and furthermore, you have no watch, so you don't
know what time it is. However, you have your G PS device, and it receives simultaneous
signals from four satellites, giving their positions and times as shown in Table 2 . 1 1 .
(Distances are measured in Earth radii and time in hundredths of a second past
midnight.)
Ta b l e 2 . 1 1 sa1em1e Dara
Satellite
2
3
4
Position
Time
( 1 . 1 1 , 2.55, 2 . 1 4)
(2.87, 0.00, 1 .43)
(0.00, 1 .08, 2.29)
( 1 .54, 1 . 0 1 , 1 .23)
1 .29
1.31
2.75
4.06
121
Let (x, y, z) be your position, and let t be the time when the signals arrive. The
goal is to solve for x, y, z, and t. Your distance from Satellite 1 can be computed as
follows. The signal, traveling at a speed of 0.47 Earth radii/ 1 0 - 2 sec, was sent at time
1 .29 and arrived at time t, so it took t - 1 .29 hundredths of a second to reach you.
Distance equals velocity multiplied by (elapsed) time, so
d = 0.47(t - 1 .29)
We can also express d in terms of (x, y, z) and the satellite's position ( 1 . 1 1 , 2.55, 2. 14)
using the distance formula:
d = V( x - 1 . 1 1 ) 2 + (y - 2.55 ) 2 + ( z - 2 . 1 4 ) 2
Combining these results leads to the equation
(x - 1 . 1 1 ) 2 + (y - 2.55) 2 + (z - 2 . 1 4) 2 = 0.47 2 (t - 1 .29) 2
Expanding, simplifying, and rearranging, we find that Equation ( 1 ) becomes
2.22x + 5 . l Oy + 4.28z - 0.57t = x 2 + y 2 + z 2 - 0.22t 2 + 1 1 .95
(1)
Similarly, we can derive a corresponding equation for each of the other three satel­
lites. We end up with a system of four equations in x, y, z, and t:
2.22x + 5 . l Oy + 4.28z - 0.57t = x 2 + y 2 + z 2 - 0.22t 2 + 1 1 .95
5.74x
+ 2.86z - 0.58t = x 2 + y 2 + z 2 - 0.22t 2 + 9.90
2 . 1 6y + 4.58z - 1 .2 1 t = x 2 + y 2 + z 2 - 0.22t 2 + 4.74
3.08x + 2.02y + 2.46z - 1 .79t = x 2 + y 2 + z 2 - 0.22t 2 + 1 .26
These are not linear equations, but the nonlinear terms are the same in each equation.
If we subtract the first equation from each of the other three equations, we obtain a
linear system:
3.52x - 5 . l Oy - 1 .42z - O.O l t = 2.05
- 2.22x - 2.94y + 0.30z - 0.64t = 7.2 1
0.86x - 3.08y - 1 .82z - 1 .22t = - 1 0.69
The augmented matrix row reduces as
122
[
3.52
- 2.22
0.86
- 5. 1 0
- 2.94
- 3.08
- 1 .42
0.30
- 1 .82
= �:��1 [� � � �:�: ]
- 0.01
- 0.64
- 1 .22 - 1 0.69
�
0 0
2.97
0.81
1 0.79 5.91
from which we see that
x = 2.97 - 0.36t
y = 0.81 - 0.03t
z = 5.91 - 0.79t
(2)
with t free. Substituting these equations into ( 1 ), we obtain
(2.97 - 0.36t - 1 . 1 1 ) 2 + (0.81 - 0.03t - 2.55) 2
+ (5.9 1 - 0.79t - 2. 14) 2 = 0.47 2 (t - 1 .29) 2
which simplifies to the quadratic equation
0.54t 2 - 6.65t + 20.32 = 0
There are two solutions:
t = 6.74 and t = 5.60
Substituting into (2), we find that the first solution corresponds to (x, y, z) = (0.55,
0.6 1 , 0.56) and the second solution to (x, y, z) = (0.96, 0.65, 1 .46). The second solution
is clearly not on the unit sphere (Earth), so we reject it. The first solution produces
x 2 + y 2 + z 2 = 0.99, so we are satisfied that, within acceptable roundoff error, we
have located your coordinates as (0.55, 0.6 1 , 0.56).
In practice, GPS takes significantly more factors into account, such as the fact
that Earth's surface is not exactly spherical, so additional refinements are needed in­
volving such techniques as least squares approximation (see Chapter 7). In addition,
the results of the GPS calculation are converted from rectangular (Cartesian) coor­
dinates into latitude and longitude, an interesting exercise in itself and one involving
yet other branches of mathematics.
123
124
Chapter
'�
2 Systems of Linear Equations
�
llerative M e l h o d s f o r S o l v i n g l i n e a r svste m s
The direct methods for solving linear systems, using elementary row operations, lead
to exact solutions in many cases but are subject to errors due to roundoff and other
factors, as we have seen. The third road in our "trivium" takes us down quite a different
path indeed. In this section, we explore methods that proceed iteratively by succes­
sively generating sequences of vectors that approach a solution to a linear system. In
many instances (such as when the coefficient matrix is sparse-that is, contains many
zero entries), iterative methods can be faster and more accurate than direct methods.
Also, iterative methods can be stopped whenever the approximate solution they gen­
erate is sufficiently accurate. In addition, iterative methods often benefit from inac­
curacy: Roundoff error can actually accelerate their convergence toward a solution.
We will explore two iterative methods for solving linear systems: Jacobi's method
and a refinement of it, the Gauss-Seidel method. In all examples, we will be consid­
ering linear systems with the same number of variables as equations, and we will
assume that there is a unique solution. Our interest is in finding this solution using
iterative methods.
Exa m p l e 2 . 3 1
Consider the system
7x 1 - x2 = 5
3x 1 - 5x2 = - 7
Jacobi's method begins with solving the first equation for x1 and the second equation
for x2, to obtain
5 + Xz
X1 = 7
(1)
7 + 3x 1
Xz =
5
We now need an initial approximation to the solution. It turns out that it does not
matter what this initial approximation is, so we might as well take x1 = 0, x2 = 0. We
use these values in Equations ( 1 ) to get new values of x1 and x2:
(1804-1851)
Carl Gustav Jacobi
was
a German mathematician who made
important contributions to many
fields of mathematics and physics,
including geometry, number theory,
analysis, mechanics, and fluid
dynamics. Although much of his
work was in applied mathematics,
Jacobi believed in the importance of
doing mathematics for its own sake.
A fine teacher, he held positions
at the Universities of Berlin and
Konigsberg and was one of the most
famous mathematicians in Europe.
-- = - = 0.714
5 + 0 5
7
7
7 + 3.0 7
Xz =
5
5
1 .400
Now we substitute these values into ( 1 ) to get
5 + 1 .4
= 0.9 14
7
7 + 3 27 = 1 .829
Xz = --5
X1 =
---
.
(written to three decimal places) . We repeat this process (using the old values of x2
and x1 to get the new values of x1 and x2 ) , producing the sequence of approximations
given in Table 2 . 1 2 .
Section
1
0
Ta b l e 2 . 1 2
2
4
3
125
5
6
00 0.1.47001 4 0.1.982914 0.1.994769 0.1.999385 0.1.999698 0.1.999999
4,
[::]
[ 01.·999385 ] .
[�l
n
.......
2. 5 Iterative Methods for Solving Linear Systems
The successive vectors
are called iterates, so, for example, when n =
the fourth iterate is
We can see that the iterates in this example are
approaching
which is the exact solution of the given system. (Check this.)
4
We say in this case that Jacobi's method converges.
2. 1 3.
Jacobi's method calculates the successive iterates in a two-variable system accord­
ing to the crisscross pattern shown in Table
1
0
Ta b l e 2 . 1 3
n
The Gauss-Seidel method is named
after C. F. Gauss and Philipp Ludwig
von Seidel
Seidel
worked in analysis, probability
theory, astronomy, and optics.
Unfortunately, he suffered from
eye problems and retired at a young
age. The paper in which he described
the method now known as Gauss­
Seidel was published in
Gauss,
it seems, was unaware of the
method!
(1821-1896).
1874.
2
3
Before we consider Jacobi's method in the general case, we will look at a
modification of it that often converges faster to the solution. The Gauss-Seidel method
is the same as the Jacobi method except that we use each new value as soon as we can.
=*
So in our example, we begin by calculating x1 = +
as before, but
we now use this value of x 1 to get the next value of x 2 :
(5 0)/7 0.7 14
7 53. 1.829
=
Xz = ---7
+
2. 1 4.
2.
=
We then use this value of x2 to recalculate x 1 , and so on. The iterates this time are
shown in Table
We observe that the Gauss-Seidel method has converged faster to the solu­
tion. The iterates this time are calculated according to the zigzag pattern shown in
Table
2. 1 5.
0
Ta b l e 2 . 1 4
n
1
2
3
4
5
00 0.1.872914 0.1.997685 0.1.999998 2.1.000000 2.1.000000
126
Chapter
2 Systems of Linear Equations
1
0
Ta b l e 2 . 1 5
n
2
3
Xz
The Gauss-Seidel method also has a nice geometric interpretation in the case of
two variables. We can think of x 1 and x2 as the coordinates of points in the plane. Our
starting point is the point corresponding to our initial approximation, (0, O). Our first
calculation gives x 1 = � , so we move to the point ( �, O) (0.714, 0). Then we compute
x2 = � 1 .829, which moves us to the point ( � , �) (0.714, 1 .829). Continuing in
this fashion, our calculations from the Gauss-Seidel method give rise to a sequence
of points, each one differing from the preceding point in exactly one coordinate. If
we plot the lines 7x 1 - x2 = 5 and 3x 1 - 5x2 = - 7 corresponding to the two given
equations, we find that the points calculated above fall alternately on the two lines,
as shown in Figure 2.27. Moreover, they approach the point of intersection of the
lines, which corresponds to the solution of the system of equations. This is what
convergence means!
=
=
=
2
05
.
0.2
-
0.4
05
.
-1
Figure 2 . 2 1
Converging iterates
The general cases of the two methods are analogous. Given a system of n linear
equations in n variables,
a i 1X1 + a i 2X2 + · · · + a i nXn = b i
a z1 X1 + a z 2X2 + · · · + a z nXn = b z
(2)
we solve the first equation for x 1 , the second for x2 , and so on. Then, beginning
with an initial approximation, we use these new equations to iteratively update each
Section
2. 5 Iterative Methods for Solving Linear Systems
121
variable. Jacobi's method uses all of the values at the kth iteration to compute the
(k + l )st iterate, whereas the Gauss-Seidel method always uses the most recent value
of each variable in every calculation. Example 2.39 later illustrates the Gauss-Seidel
method in a three-variable problem.
At this point, you should have some questions and concerns about these iterative
methods. (Do you?) Several come to mind: Must these methods converge? If not,
when do they converge? If they converge, must they converge to the solution? The
answer to the first question is no, as Example 2.38 illustrates.
Apply the Gauss-Seidel method to the system
Exa m p l e 2 . 3 8
X1 - Xz = 1
2x 1 + Xz = 5
with initial approximation
Solution
[ � ].
We rearrange the equations to get
X1 = 1 + X2
X2 = 5 - 2x 1
�
[:: ] [ � ].
The first few iterates are given in Table 2 . 1 6. (Check these.)
The actual solution to the given system is
=
Clearly, the iterates in
Table 2 . 1 6 are not approaching this point, as Figure 2.28 makes graphically clear in an
example of divergence.
0
Ta b l e 2 . 1 6
n
Xi
X2
0
0
1
3
2
4
-3
3
-2
9
4
10
- 15
5
- 14
33
-4
Fioure 2.28
Diverging iterates
128
Chapter
2 Systems of Linear Equations
S o when do these iterative methods converge? Unfortunately, the answer to this
question is rather tricky. We will answer it completely in Chapter 7, but for now we
will give a partial answer, without proof.
Let A be the n X n matrix
[ ""
a12
a
a
A = .2 1 22
a n) a n2
a,,
a2 n
a nn
l
We say that A is strictly diagonally dominant if
l I l I l I
l l
a 11 > a 1 2 + a 13 + · · · + a 1n
l I l I l I
l l
a22 > a2 1 + a 2 3 + · · · + a2 n
That is, the absolute value of each diagonal entry a11 , aw . . . , ann is greater than the
sum of the absolute values of the remaining entries in that row.
Theorem 2 . 9
If a system of n linear equations in n variables has a strictly diagonally domi­
nant coefficient matrix, then it has a unique solution and both the Jacobi and the
Gauss-Seidel method converge to it.
Remark Be warned! This theorem is a one-way implication. The fact that a
system is not strictly diagonally dominant does not mean that the iterative meth­
ods diverge. They may or may not converge. (See Exercises 1 5 - 1 9.) Indeed, there are
examples in which one of the methods converges and the other diverges. However, if
either of these methods converges, then it must converge to the solution-it cannot
converge to some other point.
Theorem 2 . 1 0
If the Jacobi or the Gauss-Seidel method converges for a system of n linear
equations in n variables, then it must converge to the solution of the system.
Proof We will illustrate the idea behind the proof by sketching it out for the case of
Jacobi's method, using the system of equations in Example 2.37. The general proof
is similar.
Convergence means that "as iterations increase, the values of the iterates get
closer and closer to a limiting value:' This means that x1 and x2 converge to r and s,
respectively, as shown in Table 2 . 1 7.
We must prove that
[:: ] [:]
is the solution of the system of equations. In
other words, at the ( k + l ) st iteration, the values of x1 and x2 must stay the same as at
Section
2. 5 Iterative Methods for Solving Linear Systems
k+ I
k
Ta b l e 2 . 1 1
n
r
s
+
+
k+2
r
s
r
s
the kth iteration. But the calculations give x1 = (5
(7 3x1)/5 = (7 3r)/5. Therefore,
Rearranging, we see that
5
129
+ x2 )/7 = (5 + s)/7 and x2 =
+ s = r and 7 + 3r = s
---
--
5
7
7r - s = 5
3r - 5s = - 7
Thus, x1 = r, x2 = s satisfy the original equations, as required.
By now you may be wondering: If iterative methods don't always converge to the
solution, what good are they? Why don't we just use Gaussian elimination? First, we
have seen that Gaussian elimination is sensitive to roundoff errors, and this sensitiv­
ity can lead to inaccurate or even wildly wrong answers. Also, even if Gaussian elimi­
nation does not go astray, we cannot improve on a solution once we have found it. For
example, if we use Gaussian elimination to calculate a solution to two decimal places,
there is no way to obtain the solution to four decimal places except to start over again
and work with increased accuracy.
In contrast, we can achieve additional accuracy with iterative methods simply by
doing more iterations. For large systems, particularly those with sparse coefficient
matrices, iterative methods are much faster than direct methods when implemented
on a computer. In many applications, the systems that arise are strictly diagonally
dominant, and thus iterative methods are guaranteed to converge. The next example
illustrates one such application.
Exa m p l e 2 . 3 9
Suppose we heat each edge of a metal plate to a constant temperature, as shown in
Figure 2 . 2 9 .
50°
Figure 2 . 2 9
A heated metal plate
oo
130
Chapter
2 Systems of Linear Equations
Eventually the temperature at the interior points will reach equilibrium, where the
following property can be shown to hold:
The temperature at each interior point P on a plate is the average of the tempera­
tures on the circumference of any circle centered at P inside the plate (Figure 2.30).
figure 2 . 3 0
To apply this property in an actual example requires techniques from calculus. As
an alternative, we can approximate the situation by overlaying the plate with a grid,
or mesh, that has a finite number of interior points, as shown in Figure 2.3 1 .
Figure 2 . 3 1
The discrete version of the heated
plate problem
The discrete analogue of the averaging property governing equilibrium tempera­
tures is stated as follows:
The temperature at each interior point P is the average of the temperatures at the
points adjacent to P.
For the example shown in Figure 2.3 1 , there are three interior points, and each is
adjacent to four other points. Let the equilibrium temperatures of the interior points
Section
2. 5 Iterative Methods for Solving Linear Systems
131
be t 1 , t2 , and t3 , as shown. Then, by the temperature-averaging property, we have
t! =
1 00 + 1 00 + G + 50
------
4
t + t + 0 + 50
t1 = 1 3
4
1 00 + 1 00 + o + G
t3 =
4
(3)
_
_
_
_
_
_
_
or
250
- t1 + 4t2 - t3 = 50
- t1 + 4t3 = 200
Notice that this system is strictly diagonally dominant. Notice also that Equa­
tions (3) are in the form required for Jacobi or Gauss-Seidel iteration. With an initial
approximation of t i = 0, t2 = 0, t3 = 0, the Gauss-Seidel method gives the following
iterates.
t1 =
Iteration 2:
�
1 00 + 1 00 + 0 + 50
= 62.5
4
62.5 + 0 + 0 + 50
= 28. 125
t1 =
4
1 00 + 1 00 + 0 + 28. 125
= 57.03 1
t3 =
4
100 + 1 00 + 28. 125 + 50
= 69.5 3 1
t1 =
4
69.53 1 + 57.03 1 + 0 + 50
= 44. 141
t1 =
4
100 + 1 00 + 0 + 44. 141
= 6 1 .035
t3 =
4
Iteration 1 :
Continuing, we find the iterates listed in Table 2 . 1 8 . We work with five-significant­
digit accuracy and stop when two successive iterates agree within 0.00 1 in all
variables.
Thus, the equilibrium temperatures at the interior points are (to an accuracy of
0.00 1 ) t i = 74. 1 08, t2 = 46.430, and t3 = 6 1 .607. (Check these calculations.)
By using a finer grid (with more interior points), we can get as precise informa­
tion as we like about the equilibrium temperatures at various points on the plate.
0
Ta b l e 2 . 1 8
n
ti
t1
t3
0
0
0
1
62.500
28. 125
57.03 1
2
69.53 1
44 . 1 4 1
6 1 .035
3
73.535
4 6. 1 4 3
6 1 .536
7
74. 107
46.429
6 1 .607
8
74. 107
46.429
6 1 .607
4
132
..
GAS
I
Chapter
2 Systems of Linear Equations
Exercises 2 . 5
In Exercises 1 -6, apply Jacobi's method to the given system.
Take the zero vector as the initial approximation and work
with four-significant-digit accuracy until two successive
iterates agree within 0.001 in each variable. In each case,
compare your answer with the exact solution found using
any direct method you like.
1 . 7X 1 - X2 = 6
2. 2X 1 + X2 = 5
X 1 - 5X2 = - 4
X 1 - X2 = 1
3. 4.5X 1 - 0.5X2 = 1
X 1 - 3.5X2 = - 1
4. 20X 1 + X2 - X3 = 1 7
X1 - 10x2 + x3 = 1 3
-x 1 + X2 + 10x3 = 1 8
5 . 3X 1 + X2
=1
X 1 + 4X2 + X3 = 1
X2 + 3X3 = 1
=1
6. 3X 1 - X2
=0
-X 1 + 3X2 - X3
-X2 + 3X3 - X4 =
-x3 + 3X4 = 1
In Exercises 7-12, repeat the given exercise using the Gauss­
Seidel method. Take the zero vector as the initial approxi­
mation and work with four-significant-digit accuracy until
two successive iterates agree within 0.001 in each variable.
Compare the number of iterations required by the Jacobi
and Gauss-Seidel methods to reach such an approximate
solution.
7. Exercise 1
8. Exercise 2
9. Exercise 3
10. Exercise 4
1 1 . Exercise 5
12. Exercise 6
In Exercises 13 and 14, draw diagrams to illustrate the con­
vergence of the Gauss-Seidel method with the given system.
13. The system in Exercise 1
14. The system in Exercise 2
In Exercises 15 and 1 6, compute the first four iterates,
using the zero vector as the initial approximation, to show
that the Gauss-Seidel method diverges. Then show that
the equations can be rearranged to give a strictly diagonally
dominant coefficient matrix, and apply the Gauss-Seidel
method to obtain an approximate solution that is accurate
to within 0.001.
16. X 1 - 4X2 + 2X3 = 2
15. X 1 - 2X2 = 3
3x 1 + 2x2 = 1
2x2 + 4x3 = 1
6X 1 - X2 - 2X3 = 1
17. Draw a diagram to illustrate the divergence of the
Gauss-Seidel method in Exercise 15.
In Exercises 18 and 1 9, the coefficient matrix is not strictly
diagonally dominant, nor can the equations be rearranged
to make it so. However, both the Jacobi and the Gauss-Seidel
method converge anyway. Demonstrate that this is true of
the Gauss-Seidel method, starting with the zero vector as
the initial approximation and obtaining a solution that is
accurate to within 0.01.
18. - 4x 1 + 5X2 = 14
X 1 - 3x2 = - 7
19. 5x 1 - 2x2 + 3x3 = - 8
X1 + 4X2 - 4X3 = 102
- 2x1 - 2x2 + 4x3 = - 90
20. Continue performing iterations in Exercise 18 to
obtain a solution that is accurate to within 0.00 1 .
2 1 . Continue performing iterations in Exercise 19 to
obtain a solution that is accurate to within 0.00 1 .
In Exercises 22-24, the metal plate has the constant tem­
peratures shown on its boundaries. Find the equilibrium
temperature at each of the indicated interior points by
setting up a system of linear equations and applying either
the Jacobi or the Gauss-Seidel method. Obtain a solution
that is accurate to within 0.001.
22.
50
50
Section
oo
oo
23.
t1
oo
100°
t2
t3
24.
oo
40°
100°
!00°
100°
oo
20°
f1
t2
f3
t4
20°
100°
100°
40°
In Exercises 25 and 26, we refine the grids used in Exer­
cises 22 and 24 to obtain more accurate information about
the equilibrium temperatures at interior points of the plates.
Obtain solutions that are accurate to within 0.001, using
either the Jacobi or the Gauss-Seidel method.
25.
oo
oo
t2
oo
t4
ts
50
50
50
oo
oo
20°
20°
oo
f1
t2
t3
t4
20°
oo
15
16
I]
lg
20°
40°
19
110
11 1
112
100°
40°
t1 3
t14
t15
t1 6
100°
26.
40°
133
Exercises 27 and 28 demonstrate that sometimes, if we are
lucky, the form of an iterative problem may allow us to use a
little insight to obtain an exact solution.
27. A narrow strip of paper 1 unit long is placed along a
oo
t4
2. 5 Iterative Methods for Solving Linear Systems
number line so that its ends are at 0 and 1 . The paper
is folded in half, right end over left, so that its ends
are now at 0 and i. Next, it is folded in half again, this
time left end over right, so that its ends are at ! and t .
Figure 2.32 shows this process. We continue folding
the paper in half, alternating right-over-left and left­
over-right. If we could continue indefinitely, it is clear
that the ends of the paper would converge to a point. It
is this point that we want to find.
(a) Let x 1 correspond to the left-hand end of the paper
and x2 to the right-hand end. Make a table with the
first six values of [x 1 , x2 ] and plot the correspond­
ing points on x 1 , x2 coordinate axes.
(b) Find two linear equations of the form x2 = ax1 + b
and x 1 = cx 2 + d that determine the new values
of the endpoints at each iteration. Draw the corre­
sponding lines on your coordinate axes and show
that this diagram would result from applying the
Gauss-Seidel method to the system of linear equa­
tions you have found. (Your diagram should resem ble Figure 2.27 on page 1 2 6.)
(c) Switching to decimal representation, continue
applying the Gauss-Seidel method to approximate
the point to which the ends of the paper are con­
verging to within 0.00 1 accuracy.
(d) Solve the system of equations exactly and compare
your answers.
28. An ant is standing on a number line at point A. It
walks halfway to point B and turns around. Then it
walks halfway back to point A, turns around again,
and walks halfway to point B. It continues to do this
indefinitely. Let point A be at 0 and point B be at 1 .
40° 100° 100°
The ant's walk is made up of a sequence of overlap­
ping line segments. Let x 1 record the positions of
the left-hand endpoints of these segments and x2
their right-hand endpoints. (Thus, we begin with
x 1 = 0 and x2 = t . Then we have x 1 = ! and x2 = t,
and so on. ) Figure 2.33 shows the start of the ant's
walk.
(a) Make a table with the first six values of [x1 , x2 ] and
plot the corresponding points on x 1 , x2 coordinate
axes.
(b) Find two linear equations of the form x2 = ax1 + b
and x 1 = cx2 + d that determine the newvalues ofthe
endpoints at each iteration. Draw the corresponding
Chapter
134
2 Systems of Linear Equations
�
0
I
........----..
!
I
0
D
0
l
4
l
2
0
!"tf
l
4
I
3
4
2
)
l
2
0
I
3
4
�
l
4
l
2
_5
0
l
4
�
l
2
Figure 2 . 3 2
Figure 2 . 3 3
Folding a strip of paper
The ant's walk
5
8
3
4
lines on your coordinate axes and show that this dia­
gram would result from applying the Gauss-Seidel
method to the system of linear equations you have
found. (Your diagram should resemble Figure 2.27
on page 126.)
(c) Switching to decimal representation, continue
applying the Gauss-Seidel method to approximate
the values to which x 1 and x2 are converging to
within 0.001 accuracy.
(d) Solve the system of equations exactly and compare
your answers. Interpret your results.
3
4
Chapter Review
Kev Defi nitions and concepts
augmented matrix, 61, 64
back substitution, 61
coefficient matrix, 64
consistent system, 60
convergence, 125- 126
divergence, 127
elementary row operations, 66
free variable, 71
Gauss-Jordan elimination, 73
Gauss-Seidel method, 124
Gaussian elimination, 68-69
homogeneous system, 76
inconsistent system, 60
iterate, 125
Jacobi's method, 124
leading variable (leading 1), 71 -73
linear equation, 58
linearly dependent vectors, 93
linearly independent
vectors, 93
1. Mark each of the following statements true or false:
Review Questions
(a) Every system of linear equations has a solution.
(b) Every homogeneous system of linear equations
has a solution.
(c) If a system of linear equations has more vari­
ables than equations, then it has infinitely many
solutions.
(d) If a system of linear equations has more equations
than variables, then it has no solution.
pivot, 66
rank of a matrix, 72
Rank Theorem, 72
reduced row echelon form, 73
row echelon form, 65
row equivalent matrices, 68
span of a set of vectors, 90
spanning set, 90
system of linear
equations, 59
(e) Determining whether b is in span(a 1 , . . . , an ) is
equivalent to determining whether the system
[A I b] is consistent, where A = [a 1 . . . an l ·
(f) In IR 3 , span( u, v) is always a plane through the origin.
( g) In IR 3 , if nonzero vectors u and v are not parallel,
then they are linearly independent.
(h) In IR 3 , if a set of vectors can be drawn head to tail,
one after the other so that a closed path (polygon)
is formed, then the vectors are linearly dependent.
Chapter Review
(i) If a set of vectors has the property that no
two vectors in the set are scalar multiples of
one another, then the set of vectors is linearly
independent.
(j) If there are more vectors in a set of vectors than
the number of entries in each vector, then the set
of vectors is linearly dependent.
2. Find the rnAA of the ma1'U
]
[ i �; �
_
3. Solve the linear system
3
3
-3
6
2
4
.
2
2
x + y - 2z = 4
x + 3y - z = 7
2x + y - Sz = 7
4. Solve the linear system
-
3w + 8x - 18y + z = 35
w + 2x 4y = 1 1
w + 3x - 7y + z = 10
5. Solve the linear system
2x + 3y = 4
x + 2y = 3
over 2 7 .
6. Solve the linear system
over 2 5 .
3x + 2y = 1
x + 4y = 2
[� I � ]
7. For what value(s) of k is the linear system with
augmented matrix
2
2k
inconsistent?
8. Find parametric equations for the line of intersection
of the planes x + 2y + 3z = 4 and Sx + 6y + 7z = 8.
9. Find the point of intersection of the following lines, if
it exists.
135
[} m
1 1 . Find the general equation of the plane spanned by
nd
[ � ] , [ -- � ] , [ ! ]
w)
u � [ } � [ Hw � [ : ]
u � [ -} � [ -n w � [ -:]
12. ? etermine whether
mdependent.
-3
2
13. Determine whether IR 3 = span(u, v,
( a)
(b)
are linearly
-2
if:
14. Let a1, a2 , a3 be linearly independent vectors in IR 3 , and
let A = [ a1 a2 a3 ] . Which of the following statements
are true?
( a) The reduced row echelon form of A is 13 .
(b) The rank of A is 3.
(c) The system [A I b] has a unique solution for any
vector b in IR 3 .
(d) (a), (b), and (c) are all true.
(e) (a) and (b) are both true, but not (c) .
15. Let a1, a2 , a3 be linearly dependent vectors in IR 3 , not
all zero, and let A = [ a1 a2 a3 ] . What are the possible
values of the rank of A?
16. What is the maximum rank of a 5 X 3 matrix? What is
the minimum rank of a 5 X 3 matrix?
17. Show that if u and v are linearly independent vectors,
then so are u + v and u - v.
18. Show that span(u, v) = span(u, u + v) for any vectors
u and v.
19. In order for a linear system with augmented matrix
[A I b] to be consistent, what must be true about the
ranks of A and [A I b] ?
20. Are ilie matci=
O
U ! l [i -:i
-
row equivalent? Why or why not?
nd
M atri ces
We [Halmos and KaplanskyJ share
a philosophy about linear algebra:
we think basisjree, we write
basis-free, but when the chips are
down we close the office door and
compute with matrices like fury.
-Irving Kaplansky
In Paul Halmos: Celebrating
J. H. Ewing and Gehring,
eds. Springer-Verlag, 1991, p. 88
50 Years of Mathematics
F. W
3.0
I n t ro d u ctio n : M atrices i n Acti o n
In this chapter, we will study matrices in their own right. We have already used
matrices-in the form of augmented matrices-to record information about and to
help streamline calculations involving systems of linear equations. Now you will see
that matrices have algebraic properties of their own, which enable us to calculate
with them, subject to the rules of matrix algebra. Furthermore, you will observe that
matrices are not static objects, recording information and data; rather, they represent
certain types of functions that "act" on vectors, transforming them into other vectors.
These "matrix transformations" will begin to play a key role in our study of linear
algebra and will shed new light on what you have already learned about vectors and
systems of linear equations. Furthermore, matrices arise in many forms other than
augmented matrices; we will explore some of the many applications of matrices at the
end of this chapter.
In this section, we will consider a few simple examples to illustrate how matri­
ces can transform vectors. In the process, you will get your first glimpse of "matrix
arithmetic:'
Consider the equations
Y 1 = X 1 + 2Xz
3x2
Yz =
We can view these equations as describing a transformation of the vector x =
[;:]
[ � � ],
( 1)
[:: ]
into the vector y =
. If we denote the matrix of coefficients of the right-hand side
by F, then F =
and we can rewrite the transformation as
or, more succinctly, y = Fx. [Think of this expression as analogous to the functional
notation y = f( x) you are used to: x is the independent "variable" here, y is the depen­
dent "variable;' and F is the name of the "function:']
136
Section
Thus, if x =
[ - � ],
3. 0 Introduction: Matrices in Action
then the Equations ( 1) give
3 3
[ � ] [ � � ] [ - � ].
y1 = - 2 + 2 · 1 = 0
or y =
·1 =
y2 =
We can write this expression as
Problem 1
131
[ o3 ]
=
Compute Fx for the following vectors x:
(a) x =
[�]
(b) x =
[ �]
(c) x =
_
[ = �J
Problem 2 The heads of the four vectors x in Problem 1 locate the four corners
of a square in the x 1 x2 plane. Draw this square and label its corners A, B, C, and D,
corresponding to parts (a), (b), (c), and (d) of Problem 1 .
O n separate coordinate axes (labeled y 1 and y2 ), draw the four points determined
by Fx in Problem 1 . Label these points A', B', C', and D'. Let's make the (reasonable)
assumption that the line segment AB is transformed into the line segment A ' B ', and
likewise for the other three sides of the square ABCD. What geometric figure is rep­
resented by A 'B' C'D'?
Problem 3 Th e center o f square ABCD i s the origin 0 =
A' B' C' D ' ? What algebraic calculation confirms this?
Now consider the equations
Z1 =
that transform a vector y =
[ �: ]
Y1 - Yz
Zz = - 2y l
into the vector z =
transformation as z = Gy, where
G=
[ �]. What i s the center of
[ 1 -1]
-2
(2)
[:: J.
We can abbreviate this
0
Problem 4 We are going to find out how G transforms the figure A 'B' C'D'.
Compute Gy for each of the four vectors y that you computed in Problem 1 . [That
is, compute z = G ( Fx). You may recognize this expression as being analogous to
the composition of functions with which you are familiar.] Call the corresponding
points A " , B " , C " , and D " , and sketch the figure A "B " C "D " on z 1 z2 coordinate axes.
Problem 5 By substituting Equations ( 1 ) into Equations ( 2 ) , obtain equations for
z 1 and z2 in terms of x1 and x2 • If we denote the matrix of these equations by H, then
we have z = Hx. Since we also have z = GFx, it is reasonable to write
H = GF
Can you see how the entries of H are related to the entries of F and G?
Problem 6 Let's do the above process the other way around: First transform the
square ABCD, using G, to obtain figure A * B * C * D * . Then transform the resulting
figure, using F, to obtain A ** B ** C ** D ** . [Note: Don't worry about the "variables" x,
138
Chapter
3 Matrices
y, and z here. Simply substitute the coordinates of A, B, C, and D into Equations
(2)
and then substitute the results into Equations ( l ).] Are A * *B* *C**D** and A"B"C"D"
the same? What does this tell you about the order in which we perform the transfor­
mations and G?
Problem 1 Repeat Problem 5 with general matrices
F
F = [ ] = [ggll21 gg2212 ] ,
(2) F
=
i !
!ll !1 2 , G
2 1 22
and H
= [hh2ll1 hh2212 ]
F
That is, if Equations ( 1) and Equations
have coefficients as specified by and G,
find the entries of H in terms of the entries of and G. The result will be a formula
for the "product" H GF.
Problem 8 Repeat Problems 1 -6 with the following matrices. (Your formula from
Problem 7 may help to speed up the algebraic calculations.) Note any similarities or
differences that you think are significant.
F = [� -�l = [� �] F = �l = [� � ]
(c) F= [ � � ],c = [ _� - � ] F = - 2 ] = [ 2 � ]
( a)
(b)
G
G
( d)
4
,G
1
M atrix O p e rati o n s
Although we have already encountered matrices, we begin by stating a formal
definition and recording some facts for future reference.
A matrix is a rectangular array of numbers called the entries, or
elements, of the matrix.
D e fi n it i o n
Although numbers will usually be
chosen from the set � of real numbers, they may also be taken from
the set C of complex numbers or
from "ll_P' where p is prime.
Technically, there is a distinction
between row/column matrices
and vectors, but we will not be­
labor this distinction. We will,
however, distinguish between
row matrices/vectors and column
matrices/vectors. This distinction
is important-at the very least­
for algebraic computations, as we
will demonstrate.
The following are all examples of matrices:
[� � l [ �
�l Ul 1 1 1 1
-1
'TT
[
],
[ 1.2
51
6.9 0
- 7.3 9
-1
4.4 , [ 7 ]
8.5
l
The size of a matrix is a description of the numbers of rows and columns it has. A
matrix is called m X n (pronounced " m by n") if it has m rows and n columns. Thus,
the examples above are matrices of sizes X
X 3, 3 X 1, 1 X 4, 3 X 3, and 1 X 1,
respectively. A 1 X m matrix i s called a row matrix (or row vector), and an n X 1
matrix is called a column matrix (or column vector) .
We use double-subscript notation to refer to the entries of a matrix A. The entry of
A in row i and column j is denoted by a ij · Thus, if
=
2 2, 2
=
A = [�
9
5
then a 1 3 - 1 and a 22 5. (The notation A ij is sometimes used interchangeably with
a ij .) We can therefore compactly denote a matrix A by [a;j ] (or [a;j l m x n if it is impor­
tant to specify the size of A, although the size will usually be clear from the context) .
Section
[
3.1 Matrix Operations
139
With this notation, a general m X n matrix A has the form
A=
a ., a 12
a 21 a zz
a ml a m 2
If the columns of A are the vectors a1, a2 , . . . , an , then we may represent A as
A = [ a 1 a2 · · · a" ]
If the rows of A are A 1 , A2 , . . . , Am , then we may represent A as
A�
[]J
The diagonal entries of A are a11, aw a 33 , . . . , and if m = n (that is, if A has the same
number of rows as columns), then A is called a square matrix. A square matrix whose
nondiagonal entries are all zero is called a diagonal matrix. A diagonal matrix all
of whose diagonal entries are the same is called a scalar matrix. If the scalar on the
diagonal is 1, the scalar matrix is called an identity matrix.
For example, let
[
0
6
0
2 5
A=
-1 4
The diagonal entries of A are 2 and 4, but A is not square; B is a square matrix of size
2 X 2 with diagonal entries 3 and 5; C is a diagonal matrix; D is a 3 X 3 identity ma­
trix. The n X n identity matrix is denoted by In (or simply I if its size is understood).
Since we can view matrices as generalizations of vectors (and, indeed, matrices
can and should be thought of as being made up of both row and column vectors),
many of the conventions and operations for vectors carry through (in an obvious
way) to matrices.
Two matrices are equal if they have the same size and if their corresponding
entries are equal. Thus, if A = [a ij l m x n and B = [b ij lrxs' then A = B if and only if
m = r and n = s and a
ij = b;j for all i and j.
Exa m p l e 3 . 1
Consider the matrices
OJ
3
, and C =
[
2 0 x
5 3 y
]
Neither A nor B can be equal to C (no matter what the values of x and y), since A and
B are 2 X 2 matrices and C is 2 X 3. However, A = B if and only if a = 2, b = 0, c = 5,
and d = 3.
Exa m p l e 3 . 2
Consider the matrices
R � I 1 4 3 ] •nd c �
[r
4
140
Chapter
3 Matrices
Despite the fact that R and C have the same entries in the same order, R of- C since
R is 1 X 3 and C is 3 X 1 . (If we read R and C aloud, they both sound the same:
"one, four, three:') Thus, our distinction between row matrices/vectors and column
matrices/vectors is an important one.
Matrix Addilion a n d scalar M u ltiolicalion
Generalizing from vector addition, we define matrix addition componentwise. If A =
[a;) and B = [b;) are m X n matrices, their sum A + B is the m X n matrix obtained
by adding the corresponding entries. Thus,
[We could equally well have defined A + B in terms of vector addition by specifying
that each column (or row) of A + B is the sum of the corresponding columns (or
rows) of A and B.] If A and B are not the same size, then A + B is not defined.
Exa m p l e 3 . 3
Let
A =
Then
[
OJ [
]
- �]
[ ]
1 4
-1
4 3
-3
, and C =
, B=
2 1
3 0 2
-2 6 5
but neither A + C nor B + C is defined.
The componentwise definition o f scalar multiplication will come a s n o surprise. If
A is an m X n matrix and c is a scalar, then the scalar multiple cA is the m X n matrix
obtained by multiplying each entry of A by c. More formally, we have
[In terms of vectors, we could equivalently stipulate that each column (or row) of
cA is c times the corresponding column (or row) of A.]
Exa m p l e 3 . 4
For matrix A in Example 3.3,
2A =
[
2 8
- 4 12
-4
-6
-�J
4
The matrix ( - l )A is written as -A and called the negative of A. As with vectors, we
can use this fact to define the difference of two matrices: If A and B are the same size,
then
A - B = A + ( - B)
Section
Exa m p l e 3 . 5
For matrices A and B in Example 3.3,
A-B=
[l
1
0
4
-2 6
3.1 Matrix Operations
-�]
3
6
141
�]
A matrix all of whose entries are zero is called a zero matrix and denoted by 0 (or
Om x n if it is important to specify its size) . It should be clear that if A is any matrix and
0 is the zero matrix of the same size, then
A+O=A=O+A
and
A - A = 0 = -A + A
M alrix M u ltiplication
Mathematicians are sometimes like
Lewis Carroll's Humpty Dumpty:
"When I use a word;' Humpty
Dumpty said, "it means just what
I choose it to mean-neither more
nor less" (from Through the Look­
ing Glass).
The Introduction in Section 3.0 suggested that there is a "product" of matrices that is
analogous to the composition of functions. We now make this notion more precise.
The definition we are about to give generalizes what you should have discovered in
Problems 5 and 7 in Section 3.0. Unlike the definitions of matrix addition and scalar
multiplication, the definition of the product of two matrices is not a componentwise
definition. Of course, there is nothing to stop us from defining a product of matrices
in a componentwise fashion; unfortunately such a definition has few applications and
is not as "natural" as the one we now give.
If A is an m x n matrix and B is an n x r matrix, then the product
C = AB is an m X r matrix. The (i, j) entry of the product is computed as
D e fi n it i o n
follows:
R e m a rks
Notice that A and B need not be the same size. However, the number of col­
umns of A must be the same as the number of rows of B. If we write the sizes of A, B,
•
and AB in order, we can see at a glance whether this requirement is satisfied. More­
over, we can predict the size of the product before doing any calculations, since the
number of rows of AB is the same as the number of rows of A, while the number of
columns of AB is the same as the number of columns of B, as shown below:
A
m X n
B
n X r
Size of AB
AB
m X r
142
Chapter
3 Matrices
The formula for the entries of the product looks like a dot product, and indeed
it is. It says that the (i, j) entry of the matrix AB is the dot product of the ith row of A
and the jth column of B:
•
a 11 a 1 2
a ln
a i1
a i2
a in
a ml a m 2
a mn
[
b"
b2 1
b l)
b2j
b,,
b2 r
b nl
b nj
b nr
:
Notice that, in the expression c iJ = a i 1 b11 + a i2 b 21 + · · · + a in b n) ' the "outer subscripts"
on each ab term in the sum are always i and j whereas the "inner subscripts" always
agree and increase from 1 to n. We see this pattern clearly if we write c iJ using sum­
mation notation:
n
Ci) = � a ikbkj
k� I
Exa m p l e 3 . 6
Compute AB if
[
1
3
A=
-2 - 1
-[ : �
-�]
3
- -1
and B =
-1 2 0
Since A is 2 X 3 and B is 3 X 4, the product AB is defined and will be a
2 X 4 matrix. The first row of the product C = AB is computed by taking the dot
product of the first row of A with each of the columns of B in turn. Thus,
C11 = 1 ( - 4) + 3(5) + ( - 1) ( - 1 ) = 12
C1 2 = 1 (0) + 3( - 2) + ( - 1 ) (2) = - 8
C1 3 = 1 ( 3) + 3( - 1) + ( - 1) (0) = 0
C1 = l ( - 1) + 3( 1) + ( - 1) (6) = - 4
The second row of C is computed by taking the dot product of the second row of A
with each of the columns of B in turn:
Solulion
4
Cz1 = ( - 2)( - 4) + ( - 1)(5) + ( 1) ( - 1) = 2
C22 = ( - 2) ( 0) + ( - 1 ) ( - 2) + ( 1) ( 2 ) = 4
Cz 3 = ( - 2) ( 3) + ( - 1 ) ( - 1 ) + ( 1) ( 0) = - 5
c2 4 = ( - 2)( - l) + ( - l ) ( l) + ( 1 ) ( 6 ) = 7
Thus, the product matrix is given by
AB =
[
12 - 8 0
2 4 -5
- �]
(With a little practice, you should b e able t o d o these calculations mentally without
writing out all of the details as we have done here. For more complicated examples, a
calculator with matrix capabilities or a computer algebra system is preferable.)
4
Section
3.1 Matrix Operations
143
Before we go further, we will consider two examples that justify our chosen
definition of matrix multiplication.
Exa m p l e 3 . 1
Ann and Bert are planning to go shopping for fruit for the next week. They each want
to buy some apples, oranges, and grapefruit, but in differing amounts. Table 3.1 lists
what they intend to buy. There are two fruit markets nearby-Sam's and Theo's-and
their prices are given in Table 3.2. How much will it cost Ann and Bert to do their
shopping at each of the two markets?
Ta b l e 3 . 2
Ta b l e 3 . 1
Ann
Bert
Apples
6
4
Grapefruit
3
8
Oranges
10
5
Solution
Theo's
$0. 15
$0.30
$0.20
Sam's
$0. 10
$0.40
$0. 10
Apple
Grapefruit
Orange
If Ann shops at Sam's, she will spend
6(0.10) + 3(0.40) + 10(0. 10) = $2.80
If she shops at Theo's, she will spend
6(0. 1 5) + 3(0.30) + 10(0.20) = $3.80
Bert will spend
4(0. 10) + 8(0.40) + 5(0. 10) = $4. 10
at Sam's and
4(0. 15) + 8(0.30) + 5(0.20) = $4.00
at Theo's. (Presumably, Ann will shop at Sam's while Bert goes to Theo's.)
The "dot product form" of these calculations suggests that matrix multiplication
is at work here. If we organize the given information into a demand matrix D and a
price matrix P, we have
D=
[
[6 3
0. 10 0. 15
and P = 0.40 0.30
4 8
0. 10 0.20
]
[ 6 3 10 ] [ 0.40
0. 10 0. 15 ] [
]
2.80
3.80
0.30 -
The calculations above are equivalent to computing the product
Ta b l e 3 . 3
Ann
Bert
Sam's
$2.80
$4. 10
Theo's
$3.80
$4.00
DP =
4 8
5
0. 10 0.20
- 4. 10 4.00
Thus, the product matrix DP tells us how much each person's purchases will cost at
each store (Table 3.3).
144
Chapter
3 Matrices
Exa m p l e 3 . 8
Consider the linear system
X 1 - 2X2 + 3X3 = 5
-X 1 + 3X2 + X3 = 1
2x 1 - x2 + 4x3 = 14
(1)
Observe that the left-hand side arises from the matrix product
so the system ( 1) can be written as
or A x = b, where A is the coefficient matrix, x is the (column) vector of variables, and
b is the (column) vector of constant terms.
You should have no difficulty seeing that every linear system can be written in the
form Ax = b. In fact, the notation [A I b] for the augmented matrix of a linear system
is just shorthand for the matrix equation A x = b. This form will prove to be a tre­
mendously useful way of expressing a system of linear equations, and we will exploit
it often from here on.
Combining this insight with Theorem 2.4, we see that Ax = b has a solution if
and only if b is a linear combination of the columns of A.
There is another fact about matrix operations that will also prove to be quite use­
ful: Multiplication of a matrix by a standard unit vector can be used to "pick out" or
]
[
4 2
l
and consider the
0 5 -1
products Ae 3 and e2A, with the unit vectors e 3 and e2 chosen so that the products
"reproduce" a column or row of a matrix. Let A =
make sense. Thus,
2
5
[ �]
_
and e2A = [ 0 l ]
=
[O 5
[� - � J
2
5
-
1]
Notice that Ae 3 gives us the third column of A and e2A gives us the second row of A.
We record the general result as a theorem.
Theorem 3 . 1
Let A be an m X n matrix, e; a 1 X m standard unit vector, and ej an n X 1 standard
unit vector. Then
a. e; A is the ith row of A and
b. Aej is the jth column of A.
Section
3. 1 Matrix Operations
145
We prove (b) and leave proving (a) as Exercise 4 1 . If a1, . . . , an are the columns
of A, then the product Aej can be written
Proof
Aej = Oa 1 + Oa2 +
[
·
·
·
+ laj +
·
·
·
We could also prove (b) by direct calculation:
Ae. =
}
+ Oan = aj
0
a1 1
a2 1
.
aml
0
since the 1 in ej is the jth entry.
Partitioned Matrices
It will often be convenient to regard a matrix as being composed of a number of
smaller submatrices. By introducing vertical and horizontal lines into a matrix, we
can partition it into blocks. There is a natural way to partition many matrices, par­
ticularly those arising in certain applications. For example, consider the matrix
1
0
A= 0
0
0
0
1
0
0
0
0
0
1
0
0
2 -1
3
4
1
7
0
7
2
It seems natural to partition A as
1 0 0 ! 2 -1
0 1 0i 1
3
� � � L� �
o 0 0 ! 1
7
0 0 0! 7 2
-
-----
-----
-
--------
-
[ � �]
where I is the 3 X 3 identity matrix, B is 3 X 2, 0 is the 2 X 3 zero matrix, and C is 2 X 2.
In this way, we can view A as a 2 X 2 matrix whose entries are themselves matrices.
When matrices are being multiplied, there is often an advantage to be gained by
viewing them as partitioned matrices. Not only does this frequently reveal underly­
ing structures, but it often speeds up computation, especially when the matrices are
large and have many blocks of zeros. It turns out that the multiplication of partitioned
matrices is just like ordinary matrix multiplication.
We begin by considering some special cases of partitioned matrices. Each gives
rise to a different way of viewing the product of two matrices.
Suppose A is m X n and B is n X r, so the product AB exists. If we partition B in
terms of its column vectors, as B = [b 1 : b 2 : . . . : b r ] , then
146
Chapter
3 Matrices
This result is an immediate consequence of the definition of matrix multiplication.
The form on the right is called the matrix-column representation of the product.
Exa m p l e 3 . 9
If
A=
then
Ab I =
[� - 1 �]
3
[ 1 - 1 :t ] � [ ' ]
�1 : 5
[3 ]
3
and B =
and Ab 2 =
Q
Therefore, AB = [Ab 1 : Ah2 ] =
.
2 : -2
4[ � - �1
i
[� - 1 :J [ -� ] � [ _�]
3
. (Check by ordinary matrix multiplication.)
�
Remark
Observe that the matrix-column representation of AB allows us to
write each column of AB as a linear combination of the columns of A with entries
from B as the coefficients. For example,
-1
3
(See Exercises 23 and 26.)
Suppose A is m X n and B is n X r, so the product AB exists. If we partition A in
terms of its row vectors, as
then
Once again, this result is a direct consequence of the definition of matrix multiplication.
The form on the right is called the row-matrix representation of the product.
Exa m p l e 3 . 1 0
Using the row-matrix representation, compute AB for the matrices in Example 3.9.
Section
Solution
3. 1 Matrix Operations
141
We compute
A ,B � [ 1 3 2 ]
Therefore ' AB =
[ : - �]
� [ 13
[-���-] [-�-�- - - - �-]
A2 B
=
2 -2 '
5
and A2 B = [ 0 - 1 l ]
[� �]
4 -1
[2 -2]
as before.
The definition of the matrix product AB uses the natural partition of A into rows
and B into columns; this form might well be called the row-column representation of
the product. We can also partition A into columns and B into rows; this form is called
the column-row representation of the product.
In this case, we have
so
(2)
Notice that the sum resembles a dot product expansion; the difference is that the in­
dividual terms are matrices, not scalars. Let's make sure that this makes sense. Each
term a;B; is the product of an m X 1 and a 1 X r matrix. Thus, each a;B; is an m X r
matrix-the same size as AB. The products a;B; are called outer products, and (2) is
called the outer product expansion of AB.
Exa m p l e 3 . 1 1
Compute the outer product expansion of AB for the matrices in Example 3.9.
Solution
We have
The outer products are
and
148
Chapter
3 Matrices
(Observe that computing each outer product is exactly like filling in a multiplication
table.) Therefore, the outer product expansion of AB is
] [
] 4
�- J [ : ] [
-1
3
+
0
-1
0
13 5
=
= AB
0
2 -2
+
We will make use of the outer product expansion in Chapters 5 and 7 when we
discuss the Spectral Theorem and the singular value decomposition, respectively.
Each of the foregoing partitions is a special case of partitioning in general. A ma­
trix A is said to be partitioned if horizontal and vertical lines have been introduced,
subdividing A into submatrices called blocks. Partitioning allows A to be written as a
matrix whose entries are its blocks.
For example,
1
0
A= 0
0
0
0 0 ! 2 -1
0i 1
3
0 l ..i 4 0
0 0 1
7
0 0 7 2
________________
_____________
4
and B =
-1
---------
1
0
3! 1 2
2!2 1
!
= � - _ _ } _ _ _ } _ _: _ }_
0!0 0i2
1 i0 0i3
are partitioned matrices. They have the block structures
and B =
[ BB2ll1
If two matrices are the same size and have been partitioned in the same way, it is clear
that they can be added and multiplied by scalars block by block. Less obvious is the
fact that, with suitable partitioning, matrices can be multiplied blockwise as well. The
next example illustrates this process.
Exa m p l e 3 . 1 2
Consider the matrices A and B above. If we ignore for the moment the fact that their
entries are matrices, then A appears to be a 2 X 2 matrix and B a 2 X 3 matrix. Their
product should thus be a 2 X 3 matrix given by
[
[
][
]
A 11 A 1 2 B 11 B 1 2 B 13
A 2 1 A 22 B2 1 B22 B2 3
A 11 B 11 + A , 2B2 1 A 11 B 1 2 + A 1 2B 22 A ll B 13 + A , 2B2 3
A 2 1 B 1 1 + A 22B2 1 A 2 1 B 1 2 + A 22B 22 A 2 1 B 13 + A 22B2 3
But all of the products in this calculation are actually matrix products, so we need to
AB =
]
make sure that they are all defined. A quick check reveals that this is indeed the case,
since the numbers of columns in the blocks of A ( 3 and 2) match the numbers of rows
in the blocks of B. The matrices A and B are said to be partitioned conformably for
block multiplication.
Carrying out the calculations indicated gives us the product AB in partitioned form:
A , , B , , + A ,, B,, � I, B ,, + A ,, I, � B ,, + A" �
6
]
�
[-: � i [ [
-1
3 = 0
+
4
0
5
-5
Section
3. 1 Matrix Operations
149
(When some of the blocks are zero matrices or identity matrices, as is the case here,
these calculations can be done quite quickly.) The calculations for the other five
blocks of AB are similar. Check that the result is
6
0
2 : 1 2 2
5 i 2 1 12
5 -5 i 3 3 ' 9
---------7-r-0----o-T-i3
'
�
'
2 i 0 0 i 20
7
(Observe that the block in the upper-left corner is the result of our calculations above.)
Check that you obtain the same answer by multiplying A by B in the usual way.
M atrix Powers
When A and B are two n X n matrices, their product AB will also be an n X n matrix.
A special case occurs when A = B. It makes sense to define A 2 = AA and, in general,
to define A k as
k
A = AA · · · A
k factors
�
if k is a positive integer. Thus, A 1 = A, and it is convenient to define A 0 = Iw
Before making too many assumptions, we should ask ourselves to what extent
matrix powers behave like powers of real numbers. The following properties follow
immediately from the definitions we have just given and are the matrix analogues of
the corresponding properties for powers of real numbers.
If A is a square matrix and r and s are nonnegative integers, then
1. A rAs = A r+s
2. (A r ) s = A'"
In Section 3.3, we will extend the definition and properties to include negative integer
powers.
Exa m p l e 3 . 1 3
(a) If A =
A2 =
[� �]
[ � � ] [ � � ] [� ]
, then
[ ] [ � ] [! :J
2 2 1
2 3
, A = A 2A =
2 2 1
2
and, in general,
2n - l
2n - 1
]
for all n 2: 1
The above statement can be proved by mathematical induction, since it is an
infinite collection of statements, one for each natural number n. (Appendix B gives a
150
Chapter
3 Matrices
1.
brief review of mathematical induction.) The basis step is to prove that the formula
holds for n = In this case,
A1 =
[ 22 11 -- 11 22 11 -- 1J J
as required.
The induction hypothesis is to assume that
1.
1.
[� �J
=A
for some integer k 2: The induction step is to prove that the formula holds for =
k + Using the definition of matrix powers and the induction hypothesis, we compute
n
�J
[ 22(k+(k+ JJ)) -- 1J
1
[ 01 - o1 J ' [ o1 - o1 J [ o1 - o1 J [ - o1 - 01 J.
[ - 01 - o1 J [ o1 - 01 J [ _ � � J
[ - 01 0l J [ O1 -l0 J [ 0l O1 J
Thus, the formula holds for all n 2: by the principle of mathematical induction.
(b) If B =
we find
then B 2 =
=
Continuing,
B 3 = B 2B =
and
B 4 = B 3B =
=
5
Thus, B = B, and the sequence of powers of B repeats in a cycle of four:
[ o1 -1J0 ' [ -10 - oJ1 ' [ - o1 0l J ' [ 01 o1 J ' [ o1 -1J0 '
The Tra nspose of a M atrix
Thus far, all of the matrix operations we have defined are analogous to operations on
real numbers, although they may not always behave in the same way. The next opera­
tion has no such analogue.
Section
3. 1 Matrix Operations
151
The transpose of an m x n matrix A is the n x m matrix A T
obtained by interchanging the rows and columns of A. That is, the ith column of
A T is the ith row of A for all i.
Definition
Exa m p l e 3 . 1 4
Let
A=
[ � � � l [ : �l
B=
and C = [ 5
-
1 2]
Then their transposes are
The transpose is sometimes used to give an alternative definition of the dot prod­
uct of two vectors in terms of matrix multiplication. If
then
[ �: j
u2 . . . u n l .
vn
A useful alternative definition of the transpose is given componentwise:
(A T) ij = Aji for all i and j
In words, the entry in row i and column j of A T is the same as the entry in row j and
column i of A.
The transpose is also used to define a very important type of square matrix: a
symmetric matrix.
Definition
A square matrix A is symmetric if A T = A-that is, if A is equal to
its own transpose.
Exa m p l e 3 . 1 5
Let
and B =
[
1 2
1
- 3
]
Chapter
152
3 Matrices
Then A is symmetric, since A T = A; but B is not symmetric, since B T =
[� -�] *
4
B.
A symmetric matrix has the property that it is its own "mirror image" across its
main diagonal. Figure 3.1 illustrates this property for a 3 X 3 matrix. The correspond­
ing shapes represent equal entries; the diagonal entries (those on the dashed line) are
arbitrary.
A componentwise definition of a symmetric matrix is also useful. It is simply the
algebraic description of the "reflection" property.
Figure 3 . 1
A symmetric matrix
A square matrix A is symmetric if and only if A ij = Aji for all i and j.
..
Let
I
Exercises 3 . 1
c
[ _ � �l = [�
[ � :J
0 -3
] E = (4 2 ] , F = [ -�]
D = [_
2
-2
2
B
A=
l
�
,
In Exercises 1 - 1 6, compute the indicated matrices (if
possible).
2. 3D - 2A
1 . A + 2D
4. c - B T
3. B - C
5. AB
6. BD
7. D + BC
8. BB T
10. F (DF)
9. E (AF)
1 1 . FE
12. EF
14. DA - AD
13. B TCT - ( CB ) T
15. A 3
16. U2 - D) 2
17. Give an example of a nonzero 2 X 2 matrix A such
[� �] .
that A 2 = 0.
18. Let A =
Find 2 X 2 matrices B and C such
that AB = AC but B *- C.
19. A factory manufactures three products (doohickies,
gizmos, and widgets) and ships them to two ware­
houses for storage. The number of units of each prod­
uct shipped to each warehouse is given by the matrix
[
200 75
A = 1 50 100
100 125
]
(where a ij is the number of units of product i sent to
warehouse j and the products are taken in alphabetical
order) . The cost of shipping one unit of each product
by truck is $ 1 .50 per doohickey, $1 .00 per gizmo, and
$2.00 per widget. The corresponding unit costs to ship
by train are $ 1 .75, $1 .50, and $1 .00. Organize these
costs into a matrix B and then use matrix multiplica­
tion to show how the factory can compare the cost of
shipping its products to each of the two warehouses by
truck and by train.
20. Referring to Exercise 19, suppose that the unit cost
of distributing the products to stores is the same for
each product but varies by warehouse because of the
distances involved. It costs $0.75 to distribute one unit
from warehouse 1 and $1 .00 to distribute one unit
from warehouse 2. Organize these costs into a matrix
C and then use matrix multiplication to compute the
total cost of distributing each product.
Section
In Exercises 21 -22, write the given system of linear equa­
tions as a matrix equation of the form Ax = b.
X 1 - 2X2 + 3X3 = 0
2x 1 + x2 - 5x3 = 4
- X1
+ 2X3 = 1
21.
22.
= -2
X 1 - Xz
Xz + x3 = - 1
In Exercises 23-28, let
A=
H -�i
u :J
0
B=
25.
26.
27.
28.
3
-1
6
to write each column of AB as a linear combination of
the columns of A.
Use the row-matrix representation of the product to
write each row of AB as a linear combination of the
rows of B.
Compute the outer product expansion of AB.
Use the matrix-column representation of the product
to write each column of BA as a linear combination of
the columns of B.
Use the row-matrix representation of the product to
write each row of BA as a linear combination of the
rows of A.
Compute the outer product expansion of BA.
In Exercises 29 and 30, assume that the product AB makes
sense.
29. Prove that if the columns of B are linearly dependent,
then so are the columns of AB.
30. Prove that if the rows of A are linearly dependent, then
so are the rows of AB.
In Exercises 31 -34, compute AB by block multiplication,
using the indicated partitioning.
31. A �
o
[ � H i � J n � [ o of 1 l
1 - 1 :: 0 0
2 3:
33. A �
:1
B
-2 : 3 2
[! �L! J·
34· L:jH '
]
[
35.
[ ]
B=
0 0 Oi4
O 1
.
Let A =
-1 1
23. Use the matrix-column representation of the product
24.
[
153
o o
�
� ] . [ +!t -� l
A=
0 -1
and
32. A = 42 53 0
3. 1 Matrix Operations
:
-1 1 : 0
0 0: 1
(a) Compute A 2 , A 3 , . • . , A 7 •
(b) What is A 2 0 1 5 ? Why?
36. Let B =
37. Let A =
0 0.
[[� ] ]
�
]
[
[
-
v2
Find, with justification, B 2 0 1 5 .
v2
. Find a formula for A n ( n 2 1) and
verify your formula using mathematical induction.
38. Let A = cossin tJtJ -cossin etJ .
39.
]
cos W - sin W
(a) Show that A 2 = .
.
cos W
sm W
(b) Prove, by mathematical induction, that
cos n tJ - sin n tJ c
An =
1or n 2 1
cos n tJ
sin n tJ
In each o f the following, find the 4 X 4 matrix A = [ a;j ]
that satisfies the given condition:
(b) a u = j - i
(a) a ij = ( - l) i +j
(i + j - 1 )7T
(c) a ij = (i - lY
(d) aiJ = sin
[
]
(
)
40. In each of the following, find the 6 X 6 matrix A = [aij]
41.
{
{
that satisfies the given condition:
i + j if i ::; j
(a) a ij =
(b) a ij =
0 if i > j
1 if 6 ::; i + j ::; 8
(c) a11 =
0 otherwise
Prove Theorem 3 . l (a) .
4
{ 01 II ii - jj ll > 11
if
if
-
:::::
154
Chapter
3 Matrices
M atrix A l g e b ra
In some ways, the arithmetic of matrices generalizes that of vectors. We do not expect
any surprises with respect to addition and scalar multiplication, and indeed there are
none. This will allow us to extend to matrices several concepts that we are already
familiar with from our work with vectors. In particular, linear combinations, span­
ning sets, and linear independence carry over to matrices with no difficulty.
However, matrices have other operations, such as matrix multiplication, that vec­
tors do not possess. We should not expect matrix multiplication to behave like multi­
plication of real numbers unless we can prove that it does; in fact, it does not. In this
section, we summarize and prove some of the main properties of matrix operations
and begin to develop an algebra of matrices.
Prooerlies of Addilion and scalar M u lliolicalion
All of the algebraic properties of addition and scalar multiplication for vectors
(Theorem 1 . 1 ) carry over to matrices. For completeness, we summarize these proper­
ties in the next theorem.
Theorem 3 . 2
Algebraic Properties of Matrix Addition and Scalar Multiplication
Let A, B, and C be matrices of the same size and let c and d be scalars. Then
a. A + B = B + A
b. (A + B) + C = A + (B + C )
c. A + 0 = A
d. A + ( -A) = 0
e. c (A + B) = cA + cB
f. ( c + d )A = cA + dA
g. c (dA) = (cd )A
h. IA = A
Commutativity
Associativity
Distributivity
Distributivity
The proofs of these properties are direct analogues of the corresponding proofs
of the vector properties and are left as exercises. Likewise, the comments following
Theorem 1 . 1 are equally valid here, and you should have no difficulty using these
properties to perform algebraic manipulations with matrices. (Review Example 1.5
and see Exercises 17 and 18 at the end o f this section.)
The associativity property allows us to unambiguously combine scalar multiplica­
tion and addition without parentheses. IfA, B, and C are matrices of the same size, then
( 2A + 3B ) - C = 2A + ( 3B - C)
and so we can simply write 2A + 3B - C. Generally, then, if A 1 , A 2 , , Ak are matri­
ces of the same size and c 1 , c2 , . . . , ck are scalars, we may form the linear combination
•
.
.
c 1 A 1 + c 2A 2 + · · · + ckA k
We will refer to c 1 , c2 , . . . , ck as the coefficients of the linear combination. We can now
ask and answer questions about linear combinations of matrices.
Section
Exa m p l e 3 . 1 6
Let A 1 =
[ _ � �l [ � �l
[ � �]
[� !]
A2 =
and A 3 =
3. 2 Matrix Algebra
155
[ � � ].
(a) Is B =
a linear combination ofA 1 , A 2 , and A 3 ?
(b) Is C =
a linear combination of A 1 , A 2 , and A 3 ?
Solution
(a) We want to find scalars c 1 , c2 , and c 3 such that c 1A 1 + c2A 2 + c 3A 3 = B. Thus,
The left-hand side of this equation can be rewritten as
Comparing entries and using the definition of matrix equality, we have four linear
equations:
C1
- cl
1
C2 + c 3 =
+ c3 = 4
+ c3 = 2
C2 + C 3 =
1
[ � � � :j ----+ [ � 0 � � j
Gauss-Jordan elimination easily gives
�
1,
-1 0 1 2
0
1
-
0 0 1 3
0 0 0 0
(check this!), so c 1 = c 2 = - 2, and c3 = 3. Thus, A 1 - 2A 2 + 3A 3 = B, which can
be easily checked.
(b) This time we want to solve
Proceeding as in part (a), we obtain the linear system
C1
- c1
1
c2 + C3 =
+ c3 = 2
+ c3 = 3
C2 + C 3 = 4
156
Chapter
3 Matrices
Row reduction gives
We need go no further: The last row implies that there is no solution. Therefore, in
this case, C is not a linear combination of Ai, A 2 , and A 3 •
Observe that the columns of the augmented matrix contain the entries
of the matrices we are given. If we read the entries of each matrix from left to right
and top to bottom, we get the order in which the entries appear in the columns of
the augmented matrix. For example, we read Ai as " O, 1, - 1 , O," which corresponds
to the first column of the augmented matrix. It is as if we simply "straightened out"
the given matrices into column vectors. Thus, we would have ended up with exactly
the same system of linear equations as in part (a) if we had asked
Remark
{]
a linemombillation of
[-l}[ � l [ '. }
and
We will encounter such parallels repeatedly from now on. In Chapter 6, we will
explore them in more detail.
We can define the span of a set of matrices to be the set of all linear combinations
of the matrices.
Exa m p l e 3 . 1 1
Describe the span of the matrices Ai, A 2 , and A 3 in Example 3 . 1 6.
One way to do this is simply to write out a general linear combination of
Ai, A 2 , and A 3 • Thus,
Solulion
�]
[ ; :J
(which is analogous to the parametric representation of a plane) . But suppose we
want to know when the matrix
is in span (Ai, A 2 , A 3 ) . From the representa­
tion above, we know that it is when
for some choice of scalars Ci, c2 , c3 . This gives rise to a system of linear equations
whose left-hand side is exactly the same as in Example 3 . 1 6 but whose right-hand side
Section
3.2 Matrix Algebra
151
is general. The augmented matrix of this system is
and row reduction produces
__.
[ -l � : fl [�
0
�
--
0 0 ! x tJ
0 - ! x tJ + w
0 1 !x + h
0 0 w-z
l
(Check this carefully.) The only restriction comes from the last row, where clearly we
must have w - z = in order to have a solution. Thus, the span of A 1 , A 2 , and A 3 con-
[ ; :]
sists of all matrices
for which w = z. That is, span (A 1 , A 2 , A 3 ) =
[ ]
[� �]
{ [ ; :] } .
4
If we had known this before attempting Example 3.16, we would have seen
Nole
is a linear combination of A 1 , A 2 , and A 3 , since it has
1 2
the necessary form (take w = 1, x = 4, and y = 2), but C =
cannot be a linear
3 4
combination of A 1 , A 2 , and A 3 , since it does not have the proper form ( 1 i= 4).
immediately that B =
Linear independence also makes sense for matrices. We say that matrices
A 1 , A2 ,
, Ak of the same size are linearly independent if the only solution of the
equation
(1)
•
•
•
0.
is the trivial one: c 1 = c2 = · · · = ck = If there are nontrivial coefficients that satisfy
( 1 ) , then A 1 , A 2 , , Ak are called linearly dependent.
•
Exa m p l e 3 . 1 8
.
.
Determine whether the matrices A 1 , A 2 , and A 3 in Example 3 . 1 6 are linearly
independent.
We want to solve the equation c 1A 1 + c2A 2 + c3A 3 = 0. Writing out the
matrices, we have
Solulion
This time we get a homogeneous linear system whose left-hand side is the same as
in Examples 3 . 1 6 and 3.17. (Are you starting to spot a pattern yet?) The augmented
matrix row reduces to give
158
Chapter
3 Matrices
Thus, c1 = c2 = c 3 = 0 , and we conclude that the matrices A1, A 2 , and A 3 are linearly
independent.
4
Prooenies of M alrix M ulliolicalion
Whenever we encounter a new operation, such as matrix multiplication, we must
be careful not to assume too much about it. It would be nice if matrix multiplication
behaved like multiplication of real numbers. Although in many respects it does, there
are some significant differences.
Exa m p l e 3 . 1 9
Consider the matrices
[ � �] [ � �]
[ � �] [ � �] [ : �]
[ � �] [ � �]
[� �]
A=
_
and B =
_
Multiplying gives
AB =
_
_
_
_
and BA =
_
_
Thus, AB -=fa BA. So, in contrast to multiplication of real numbers, matrix multiplica­
tion is not co m m uta tive the order of the factors in a product matters!
-
It is easy to check that A 2 =
[� �]
(do so!). So, for matrices, the equation
A 2 = 0 does not imply that A = 0 (unlike the situation for real numbers, where the
equation x2 = 0 has only x = 0 as a solution) .
However gloomy things might appear after the last example, the situation is not
really bad at all-you just need to get used to working with matrices and to constantly
remind yourself that they are not numbers. The next theorem summarizes the main
properties of matrix multiplication.
Theorem 3 . 3
Properties of Matrix Multiplication
Let A, B, and C be matrices (whose sizes are such that the indicated operations can
be performed) and let k be a scalar. Then
a. A (B C ) = (AB) C
b. A ( B + C ) = AB + AC
c. (A + B)C = AC + BC
d. k (AB ) = ( kA ) B = A ( kB )
e. Im A = A = Aln if A is m X n
Associativity
Left distributivity
Right distributivity
Multiplicative identity
We prove (b) and half of (e) . We defer the proof of property (a) until
Section 3.6. The remaining properties are considered in the exercises.
Proof
Section
3. 2 Matrix Algebra
159
(b) To prove A(B + C ) = AB + AC, we let the rows of A be denoted by A; and the
columns of B and C by bj and cf Then the jth column of B + C is bj + cj (since addi­
tion is defined componentwise), and thus
[A ( B + C) J u = A; · ( bj + c)
= A; . bj + A; . CJ
= (AB ) ;1 + (AC) ;1
= (AB + A C) ;j
Since this is true for all i and j, we must have A(B + C ) = AB + AC.
(e) To prove Ain = A, we note that the identity matrix In can be column-partitioned as
where e; is a standard unit vector. Therefore,
Ain = [Ae 1 : Ae2 : : Aen l
: an ]
= [ a 1 : az :
=A
•
·
·
•
·
·
by Theorem 3 . 1 (b) .
We can use these properties to further explore how closely matrix multiplication
resembles multiplication of real numbers.
Exa m p l e 3 . 2 0
If A and B are square matrices of the same size, is (A + B) 2 = A 2 + 2AB + B 2 ?
Solution
Using properties of matrix multiplication, we compute
( A + B ) 2 = ( A + B ) (A + B )
by left distributivity
= (A + B ) A + (A + B ) B
by right distributivity
= A 2 + BA + AB + B 2
Therefore, (A + B) 2 = A 2 + 2AB + B 2 if and only if A 2 + BA + AB + B 2 = A 2 +
2AB + B 2 . Subtracting A 2 and B 2 from both sides gives BA + AB = 2AB. Subtracting
AB from both sides gives BA = AB. Thus, (A + B) 2 = A 2 + 2AB + B2 if and only if A
and B commute. (Can you give an example of such a pair of matrices? Can you find
two matrices that do not satisfy this property?)
Properties of the Transpose
Theorem 3 . 4
Properties of the Transpose
Let A and B be matrices (whose sizes are such that the indicated operations can be
performed) and let k be a scalar. Then
a. (A T ) T = A
b. (A + B) T = A T + B T
T
d. (AB) T = B TA T
c. (kAl = k (A )
r
r
e. (A l = (A T ) for all nonnegative integers r
160
Chapter
3 Matrices
Proof
Properties (a) -(c) are intuitively clear and straightforward to prove (see Exercise
30). Proving property ( e) is a good exercise in mathematical induction (see Exercise 3 1).
We will prove (d), since it is not what you might have expected. [Would you have sus­
pected that (AB) T = A TB T might be true?]
First, ifA is m X n and B is n X r, then B T is r X n and A T is n X m. Thus, the product
B TA T is defined and is r X m. Since AB is m X r, (AB) T is r X m, and so (AB) T and B TA T
have the same size. We must now prove that their corresponding entries are equal.
We denote the ith row of a matrix X by row; (X) and its jth column by col/X) .
Using these conventions, we see that
[ (AB ) T ] ;j = (AB )ji
= row/A ) · col; ( B )
= col/A T) · row; ( B T)
= row; ( B T) · col/A T) = [ B TA T ] ij
(Note that we have used the definition of matrix multiplication, the definition of the
transpose, and the fact that the dot product is commutative.) Since i and j are arbi­
trary, this result implies that (AB) T = B TA T.
Remark
Properties (b) and (d) of Theorem 3.4 can be generalized to sums and
products of finitely many matrices:
(A 1 + A 2 + · · · + A k f = A[ + Af + · · · + A [ and (A 1 A 2 · · A k f
•
= A [ · · AfA [
assuming that the sizes o f the matrices are such that all o f the operations can be per­
formed. You are asked to prove these facts by mathematical induction in Exercises 32
and 33.
-
Exa m p l e 3 . 2 1
Let
Then A T =
We have
[� ]
A =
3
4
'
so A + AT =
[
�]
[
�
:]
and B =
2
5
-1
3
, a symmetric matrix.
H �l
[� � i [ - r n � [ ': l: J
H : J [� � i � n � ]
BT =
so
BBT =
and
[� !]
B TB =
-1
3
-1
3
2
10
3
Thus, both BB T and B TB are symmetric, even though B is not even square! (Check
that AA T and A TA are also symmetric.)
Section
3.2 Matrix Algebra
161
The next theorem says that the results of Example 3.2 1 are true in general.
Theorem 3 . 5
a. If A is a square matrix, then A + A T is a symmetric matrix.
b. For any matrix A, AA T and A TA are symmetric matrices.
Proof
We prove (a) and leave proving (b) as Exercise 34. We simply check that
(A + A Tf = A T + (A Tf = A T + A = A + AT
(using properties of the transpose and the commutativity of matrix addition). Thus,
A + A T is equal to its own transpose and so, by definition, is symmetric.
I
Exercises 3 . 2
[� !] [ - � �] .
- -
In Exercises 1 -4, solve the equation for X, given that
and B =
A=
1. X
2A + 3B = 0
In Exercises 9- 12, find the general form of the span of the
indicated matrices, as in Example 3. 1 7.
9. span(A 1 , A 2 ) in Exercise 5
10. span(A 1 , A 2 , A 3 ) in Exercise 6
1 1 . span(A 1 , A 2 , A 3 ) in Exercise 7
12. span(A 1 , A 2 , A 3 , A 4 ) in Exercise 8
2. 2X = A B
3. 2 (A + 2B ) = 3X
4. 2 (A - B + X) = 3 (X - A )
In Exercises 5-8, write B as a linear combination of the
other matrices, ifpossible.
5. B =
6. B =
A3 =
7. B =
A2 =
8. B �
A, �
[� :l [ _ � �l [� � ]
[ � l [ l �l [ � ]
[� � ]
[ � �l [ l ]
[ �l [� �]
[: - n [ � n
[: n n i
A1 =
2
-4
Az =
AI =
Q
AI =
-1 2
0
Az =
Q
A3 =
-2
1
0
0
0
0
A, �
A3 =
0 -1
,
0
1
1
-1
0
In Exercises 1 3- 1 6, determine whether the given matrices
are linearly independent.
,
13.
14.
15.
0
0
1
0
0 -1
0 ,
0 -1
I •.
[� !l [� � ]
[� �l [ - � �l [ � � ]
,
[ � � ] , [ � � ] , [ � � i [ � - !]
[� -� m: ! m� : n
-[ � � i
-1 0
1
-1
0
0 -4
1
0
2
4
5
Chapter
162
3 Matrices
17. Prove Theorem 3.2(a) -(d).
18. Prove Theorem 3.2(e) -(h).
19. Prove Theorem 3.3(c).
20. Prove Theorem 3.3(d) .
21. Prove the half o f Theorem 3.3(e) that was not proved
in the text.
22. Prove that, for square matrices A2 and B,2 AB = BA if
and only if (A - B) (A + B) = A - B •
[ : �] ,find conditions on a, b,
In Exercises 23-25, if B =
c, and d such that AB = BA.
23. A =
[ � ] 24. [ l - ] 25. A [ ]
1
1
A=
1
1
-1
=
26. Find conditions on a, b, c, and d such that B =
commutes with both
2
3 4
0
o
o
and
o
o
[ ]
27. Find conditions on a, b, c, and d such that B = ac db
commutes with every 2 X 2 matrix.
28. Prove that if AB and BA are both defined, then AB and
BA are both square matrices.
A square matrix is called upper triangular if all of the en­
tries below the main diagonal are zero. Thus, the form of an
upper triangular matrix is
*
0
0
0
*
*
*
*
*
*
*
*
0
0
0
*
where the entries marked are arbitrary. A more formal
definition of such a matrix A = [ a ij ] is that a ij = 0 if i > j.
Prove that the product of two upper triangular n X n
*
then so is A + B.
36.
37.
(b) Prove that if A is a symmetric n X n matrix, then
so is kA for any scalar k.
(a) Give an example to show that if A and B are
symmetric n X n matrices, then AB need not be
symmetric.
(b) Prove that if A and B are symmetric n X n matrices,
then AB is symmetric if and only if AB = BA.
A square matrix is called skew-symmetric if A r = -A.
Which of the following matrices are skew-symmetric?
(a)
1
[ ] [
1
35. (a) Prove that ifA and B are symmetric n X n matrices,
29.
matrices is upper triangular.
30. Prove Theorem 3.4(a) - (c) .
31. Prove Theorem 3.4(e).
32. Using induction, prove that for all n 2: 1,
(A 1 + A 2 + · · · + A n ) r = A f + Af + · · · + A�.
33. Using induction,
prove that for all n 2: 1 ,
T
(A 1 A 1 · · · A n ) = A�· · · AfA f.
34. Prove Theorem 3.S(b).
(<)
[� -�]
[ _� �]
-[ : -� - �] [ - : � �]
(b)
(d)
38. Give a componentwise definition of a skew-symmetric
matrix.
39. Prove that the main diagonal of a skew-symmetric ma­
trix must consist entirely of zeros.
40. Prove that if A and B are skew-symmetric n X n
matrices, then so is A + B.
41. If A and B are skew-symmetric 2 X 2 matrices, under
what conditions is AB skew-symmetric?
42. Prove that if A is an n X n matrix, then A - A r is
skew-symmetric.
43. (a) Prove that any square matrix A can be written
as the sum of a symmetric matrix and a skew­
symmetric matrix. [Hint: Consider Theorem 3.5
and Exercise 42.]
2
5
(b) Illu,tmte pact (o) fo, the nrnt'ix A �
[�
8
The trace of an n X n matrix A = [ a ij ] is the sum of the en­
tries on its main diagonal and is denoted by tr(A). That is,
tr (A ) = a 11 + a 22 + · · · + a ""
If A and B are n X n matrices, prove the following
44.
45.
46.
47.
properties of the trace:
(a) tr (A + B) = tr (A) + tr (B)
(b) tr (kA) = ktr (A), where k is a scalar
Prove that if A and B are n X n matrices, then
tr (AB) = tr (BA).
If A is any matrix, to what is tr (AA T ) equal?
Show that there are no 2 X 2 matrices A and B such
that AB - BA = 12 •
Section
3. 3 The Inverse of a Matrix
163
T h e Inverse of a M atrix
In this section, we return to the matrix description Ax = b of a system of linear equa­
tions and look for ways to use matrix algebra to solve the system. By way of analogy,
consider the equation ax = b, where a, b, and x represent real numbers and we want
to solve for x. We can quickly figure out that we want x = b/a as the solution, but we
must remind ourselves that this is true only if a * 0. Proceeding more slowly, assum­
ing that a * 0, we will reach the solution by the following sequence of steps:
( )
1
1
1
b
b
b
= -( =:> -( a) x = - =:> l · x = - =:> x = ax = b =:> -(
a ax) a b)
a
a
a
a
(This example shows how much we do in our head and how many properties of arith­
metic and algebra we take for granted! )
To imitate this procedure for the matrix equation Ax = b, what do we need? We
need to find a matrix A ' (analogous to 1 /a) such that A ' A = I, an identity matrix
(analogous to 1 ) . If such a matrix exists (analogous to the requirement that a * O),
then we can do the following sequence of calculations:
Ax = b =:> A ' (Ax) = A 'b =:> (A 'A)x = A 'b =:> Ix = A 'b =:> x = A 'b
�
�
(Why would each of these steps be justified?)
Our goal in this section is to determine precisely when we can find such a matrix
A ' . In fact, we are going to insist on a bit more: We want not only A ' A = I but also
AA' = I. This requirement forces A and A ' to be square matrices. (Why?)
If A is an n x n matrix, an inverse of A is an n x n matrix A ' with
the property that
AA ' = I and A ' A = I
where I = In is the n X n identity matrix. If such an A ' exists, then A is called
invertible.
Definition
Exa m p l e 3 . 2 2
If A =
AA , =
Exa m p l e 3 . 2 3
[ � 53 ] , [ 3 -52 ]
[ 2 35 ] [ 3 -52 ] [ � ]
, then A =
1
-1
-1
o
1
is an inverse of A, since
[ 3 - 25 ] [ 2 35 ]
and A 'A = - 1
Show that the following matrices are not invertible:
(a) 0 =
[� �]
(b) B =
1
[� �]
Solution
(a) It is easy to see that the zero matrix 0 does not have an inverse. If it did, then there
would be a matrix O ' such that 00' = I = O' O. But the product of the zero matrix
with any other matrix is the zero matrix, and so 00' could never equal the identity
164
Chapter
3 Matrices
matrix I. (Notice that this proof makes no reference to the size of the matrices and so
is true for n X n matrices in general.)
[; :]
[ � ! ][; :J [ � � ]
(b) Suppose B has an inverse B' =
. The equation BB' = I gives
=
from which we get the equations
+ 2y
w
x
2w
+ 4y
2x
= 1
+ 2z = 0
= O
+ 4z = 1
Subtracting twice the first equation from the third yields 0 = - 2, which is clearly
absurd. Thus, there is no solution. (Row reduction gives the same result but is not
really needed here.) We deduce that no such matrix B' exists; that is, B is not invert­
ible. (In fact, it does not even have an inverse that works on one side, let alone two!)
4
R e m a rks
Even though we have seen that matrix multiplication is not, in general, commutative, A' (if it exists) must satisfy A' A = AA' .
•
The examples above raise two questions, which we will answer in this section:
•
( 1) How can we know when a matrix has an inverse?
(2) If a matrix does have an inverse, how can we find it?
We have not ruled out the possibility that a matrix A might have more than
•
one inverse. The next theorem assures us that this cannot happen.
Theorem 3 . 6
If A is an invertible matrix, then its inverse is unique.
Proof
In mathematics, a standard way to show that there is just one of something is
to show that there cannot be more than one. So, suppose that A has two inverses-say,
A ' and A " . Then
Thus,
AA ' = I = A 'A and AA " = I = A " A
A ' = A 'I = A ' (AA " ) = (A 'A )A " = IA " = A "
Hence, A ' = A", and the inverse is unique.
Thanks to this theorem, we can now refer to the inverse of an invertible matrix.
From now on, when A is invertible, we will denote its (unique) inverse by A - 1 (pro­
nounced "A inverse").
Warning
Do not be tempted to write A - l =
1
A
! There is no such operation as
"division by a matrix." Even if there were, how on earth could we divide the scalar 1 by
Section
3. 3 The Inverse of a Matrix
165
the matrix A? If you ever feel tempted to "divide" by a matrix, what you really want to do
is multiply by its inverse.
We can now complete the analogy that we set up at the beginning of this section.
Theorem 3 . 1
If A is an invertible n X n matrix, then the system of linear equations given by
Ax = b has the unique solution x = A - l b for any b in !R n .
Theorem 3.7 essentially formalizes the observation we made at the beginning
of this section. We will go through it again, a little more carefully this time. We are
asked to prove two things: that Ax = b has a solution and that it has only one solution.
(In mathematics, such a proof is called an "existence and uniqueness" proof.)
To show that a solution exists, we need only verify that x = A - l b works. We check
that
Proof
So A - l b satisfies the equation Ax = b, and hence there is at least this solution.
To show that this solution is unique, suppose y is another solution. Then Ay = b,
and multiplying both sides of the equation by A - l on the left, we obtain the chain of
implications
A - 1 (Ay) = A - 1 b => (A- 1A) y = A - 1 b => Iy = A - 1 b => y = A - 1 b
Thus, y is the same solution as before, and therefore the solution is unique.
So, returning to the questions we raised in the Remarks before Theorem 3.6, how
can we tell if a matrix is invertible and how can we find its inverse when it is invert­
ible? We will give a general procedure shortly, but the situation for 2 X 2 matrices is
sufficiently simple to warrant being singled out.
Theorem 3 . 8
If A =
[ : �],
then A is invertible if ad - be * 0, in which case
A-1 -
[ ]
1
d
ad - be - e
If ad - be = 0 , then A is not invertible.
[ ]
-
b
a
The expression ad - be is called the determinant of A, denoted det A. The formula
a b
1
for the inverse of
(when it exists) is thus
times the matrix obtained by
e d
dct A
interchanging the entries on the main diagonal and changing the signs on the other
two entries. In addition to giving this formula, Theorem 3.8 says that a 2 X 2 matrix
A is invertible if and only if det A * 0. We will see in Chapter 4 that the determinant
can be defined for all square matrices and that this result remains true, although there
is no simple formula for the inverse of larger square matrices.
--
[ ae db ] [ -ed - ab ] [aedd -- bede -ebab daba] [ad - be ad - be]
Proof
Suppose that det A = ad - be * 0. Then
=
-
+
+
=
0
0
= det A
[ OJ
l
0 1
166
Chapter
3 Matrices
Similarly,
[ ] [ ] [ OJ
[ : �] ( � [ � - � ] ) [ � � ]
( � [ � - � ] ) [ : �] [ � � ]
[ ]
d -b a b
l
= det A
-e a e d
0 1
Since det A * 0, we can multiply both sides of each equation by 1 / det A to obtain
=
de A -
and
=
de A -
[Note that we have used property (d) of Theorem 3.3.] Thus, the matrix
1
d -b
det A - e
a
satisfies the definition of an inverse, so A is invertible. Since the inverse of A is unique,
by Theorem 3.6, we must have
A-1 -
[ ]
d -b
det A - e
a
1
--
Conversely, assume that ad - be = 0. We will consider separately the cases where
a * 0 and where a = 0. If a * 0, then d = be/a, so the matrix can be written as
where k = e/a. In other words, the second row of A is a multiple of the first. Referring
to Example 3.23(b), we see that if A has an inverse
[; ;] ,
then
and the corresponding system of linear equations
+ by
aw
ax
kaw
�
+ kby
kax
= 1
O
+ bz = 0
=
+ kbz = 1
has no solution. (Why?)
If a = 0, then ad - be = 0 implies that be = 0, and therefore either b or e is 0.
Thus, A is of the form
[ � �] [ � �]
[ � �] [; ;] [ � � ] [ � � ] .
or
In the first case,
�
*
have an inverse. (Verify this.)
Consequently, if ad - be = 0, then A is not invertible.
Similarly,
[ � �]
cannot
Section
Exa m p l e 3 . 2 4
Find the inverses o f A =
Solution
[ l 2 ] [ 12 --515 ]
1(4)1- 2 --2 2 - : �
_ _-_2 [ 1 ] [ - --]
12 ( - 5) - ( - 15)
3 4
We have <let A =
A-1
�
Exa m p l e 3 . 2 5
3. 3 The Inverse of a Matrix
and B =
(3) =
4
, if they exist.
4
* 0, so A is invertible, with
=
-3
(Check this.)
On the other hand, <let B =
2
2
(4) = 0, so B is not invertible.
2yy - 2
Use the inverse o f the coefficient matrix to solve the linear system
x+ =
3x + 4 =
Solution
-+-
3
The coefficient matrix is the matrix A =
[ _ � ];
161
[� ! l
whose inverse we com­
1
puted in Example 3.24. By Theorem 3.7, Ax = b has the unique solution x = A - b.
Here we have b =
thus, the solution to the given system is
x=
[ - � -tJ [ _ � ] [ -�]
22
1
1
Solving a linear system Ax = b via x = A - b would appear to be a good
method. Unfortunately, except for X coefficient matrices and matrices with cer­
tain special forms, it is almost always faster to use Gaussian or Gauss-Jordan elimi­
nation to find the solution directly. (See Exercise 3 .) Furthermore, the technique of
Example
works only when the coefficient matrix is square and invertible, while
elimination methods can always be applied.
Remark
3.25
Properties of Invertible Malrices
The following theorem records some of the most important properties of invertible
matrices.
Theorem 3 . 9
a. If A is an invertible matrix, then A - I is invertible and
b. If A is an invertible matrix and c is a nonzero scalar, then cA is an invertible
matrix and
_l_
(cA) - 1 = c A - 1
c. If A and B are invertible matrices of the same size, then AB is invertible and
168
Chapter
3 Matrices
d. If A is an invertible matrix, then A T is invertible and
e. If A is an invertible matrix, then A n is invertible for all nonnegative inte­
gers n and
(A " ) - 1 = (A - 1 ) "
Proof We will prove properties (a), (c), and (e), leaving properties (b) and (d) to be
proven in Exercises 1 4 and 1 5 .
(a) To show that A - i i s invertible, we must argue that there i s a matrix X such that
A - 1x = I = XA - 1
But A certainly satisfies these equations in place of X, so A - l is invertible and A is an
inverse of A - 1 . Since inverses are unique, this means that (A - 1 ) - 1 = A.
( c) Here we must show that there is a matrix X such that
(AB ) X = I = X (AB )
The claim is that substituting B - 1A - l for X works. We check that
�
where we have used associativity to shift the parentheses. Similarly, (B - 1A - l ) (AB ) = I
(check!), so AB is invertible and its inverse is B - 1A - 1 .
(e) The basic idea here is easy enough. For example, when n = 2, we have
A 2 (A - 1 ) 2 = AAA - IA - I = A L4. - I = AA - I = I
Similarly, (A - I ) 2A 2 = I. Thus, (A - I ) 2 is the inverse of A 2 . It is not difficult to see that
a similar argument works for any higher integer value of n. However, mathematical
induction is the way to carry out the proof.
The basis step is when n = 0, in which case we are being asked to prove that A 0 is
invertible and that
�
This is the same as showing that I is invertible and that r 1 = I, which is clearly true.
(Why? See Exercise 1 6.)
Now we assume that the result is true when n = k, where k is a specific nonnega­
tive integer. That is, the induction hypothesis is to assume that A k is invertible and that
The induction step requires that we prove that Ak+I is invertible and that
k
k1
k
1 1
1k1
(A + ) - = (A - ) + . Now we know from (c) that A + = A A is invertible, since A
and (by hypothesis) A k are both invertible. Moreover,
(A - 1 ) k+ 1 = (A - 1 ) kA - 1
by the induction hypothesis
= (A k ) - 1A - 1
= (AA k ) - 1
by property (c)
= (A k+ 1 ) - 1
Section
3. 3 The Inverse of a Matrix
169
Therefore, A n is invertible for all nonnegative integers n , and (A n ) - 1 = (A - l ) n by the
principle of mathematical induction.
R e m a rks
While all of the properties of Theorem 3.9 are useful, ( c) is the one you should
highlight. It is perhaps the most important algebraic property of matrix inverses. It
is also the one that is easiest to get wrong. In Exercise 17, you are asked to give a
counterexample to show that, contrary to what we might like, (AB) - 1 * A - l B - 1 in
general. The correct property, (AB) - 1 = B - 1A - 1 , is sometimes called the socks-and­
shoes rule, because, although we put our socks on before our shoes, we take them off
in the reverse order.
•
Property (c) generalizes to products of finitely many invertible matrices: If A 1 ,
A 2 , . . . , A n are invertible matrices of the same size, then A 1 A 2 • • • A n is invertible and
•
(A 1A 2 · · · A n ) - 1 = A ;;- 1 · · · A2 1A� 1
(See Exercise 1 8.) Thus, we can state:
The inverse of a product of invertible matrices is the product of their inverses in
the reverse order.
1
1 1
a
b
a
+
b
square matrices, (A + B) - 1 = A - 1 + B - 1 (and, indeed, this is not true in general; see
Exercise 1 9 ) . In fact, except for special matrices, there is no formula for (A + B) - 1 .
•
Since, for real numbers, -- * - + -, we should not expect that, for
Property (e) allows us to define negative integer powers of an invertible
matrix:
•
Definition
defined by
If A is an invertible matrix and n is a positive integer, then A - n is
With this definition, it can be shown that the rules for exponentiation, A rA s = A'· + s
and (AT = A'5, hold for all integers r and s, provided A is invertible.
One use of the algebraic properties of matrices is to help solve equations involving
matrices. The next example illustrates the process. Note that we must pay particular
attention to the order of the matrices in the product.
Exa m p l e 3 . 2 6
Solve the following matrix equation for X (assuming that the matrices involved are
such that all of the indicated operations are defined):
A - 1 (BX) - 1 = (A - 1 B 3 ) 2
110
Chapter
3 Matrices
Solulion
There are many ways to proceed here. One solution is
A - 1 (BX)- 1 = (A - 1 B 3 ) 2 ==? ((BX)A) - 1 = (A - 1 B 3 ) 2
=} [ (( BX)A ) - 1 ] - 1 = [ (A - 1 B 3 ) 2 ] - 1
==? (BX)A = [ (A - 1 B 3 )(A- 1 B 3 ) J - 1
==? ( BX)A = B - 3 (A- 1 ) - 1 B - 3 (A- 1 ) - 1
==? BXA = B - 3AB - 3A
==? B - 1 BXAA - 1 = B - 1 B - 3AB - 3AA - 1
==? IXI = B - 4AB - 3I
==? X = B - 4AB - 3
�
(Can you justify each step?) Note the careful use of Theorem 3.9(c) and the expansion
of (A - 1 B 3 ) 2 • We have also made liberal use of the associativity of matrix multiplica­
tion to simplify the placement (or elimination) of parentheses.
Elemen1arv Malrices
We are going to use matrix multiplication to take a different perspective on the row
reduction of matrices. In the process, you will discover many new and important
insights into the nature of invertible matrices.
If
we find that
In other words, multiplying A by E (on the left) has the same effect as interchanging
rows 2 and 3 of A. What is significant about E? It is simply the matrix we obtain by
applying the same elementary row operation, R 2 � R 3 , to the identity matrix J3 . It
turns out that this always works.
An elementary matrix is any matrix that can be obtained by per­
forming an elementary row operation on an identity matrix.
DefiniliOD
Since there are three types of elementary row operations, there are three cor­
responding types of elementary matrices. Here are some more elementary matrices.
Exa m p l e 3 . 2 1
Let
Section
3. 3 The Inverse of a Matrix
111
Each of these matrices has been obtained from the identity matrix I4 by applying a
single elementary row operation. The matrix E 1 corresponds to 3 R2 , E2 to R 1 � R 3 ,
and E3 to R4 - 2R 2 . Observe that when we left-multiply a 4 X n matrix by one of these
elementary matrices, the corresponding elementary row operation is performed on
the matrix. For example, if
A=
then
and
[
"[ "
a12
a 2 1 a 22 a 2 3
a 31 a 3 2 a 33
a41 a42 a43
""
]
"
"
[
au a12
3a 2 1 3a 22 3a 2 3 ' E A =
E1A =
2
a 31 a 3 2 a 33
a41 a42 a43
a12
au
a 22
a ,,
E 3A =
a32
a 31
a
a41 - 2a 2 1 42 - 2a 22
] ""
[ :
a 3 2 a ,,
a z1 a 22 a z3 '
a 11 a 1 2 a 13
a 41 a42 a 43
a 13
a,,
a 33
a43 - 2a 2 3
l
Example 3.27 and Exercises 24-30 should convince you that any elementary
row operation on any matrix can be accomplished by left-multiplying by a suitable
elementary matrix. We record this fact as a theorem, the proof of which is omitted.
Theor em 3 . 1 0
Let E be the elementary matrix obtaine d by performing an elementary row opera­
A,
tion on In - If the same elementary row operation is performed on an n X r matrix
the result is the same as the matrix EA.
Remark
From a computational point of view, it is not a good idea to use el­
ementary matrices to perform elementary row operations-just do them directly.
However, elementary matrices can provide some valuable insights into invertible
matrices and the solution of systems of linear equations.
We have already observed that every elementary row operation can be "undone:'
or "reversed:' This same observation applied to elementary matrices shows us that
they are invertible.
Exa m p l e 3 . 2 8
Let
0
0
�
0
4
0
�]
1
, and E3 =
[ � � �]
-2 0 1
Then E 1 corresponds to R 2 � R3 , which is undone by doing R2 � R3 again. Thus,
E 1 - l = E 1 . (Check by showing that E l = E 1 E 1 = I.) The matrix E2 comes from 4R2 ,
112
Chapter
3 Matrices
which is undone by performing iR 2 • Thus,
which can be easily checked. Finally, E3 corresponds to the elementary row opera­
tion R 3 - 2R 1 , which can be undone by the elementary row operation R 3 + 2R 1 . So,
in this case,
(Again, it is easy to check this by confirming that the product of this matrix and E3 ,
in both orders, is I.)
Notice that not only is each elementary matrix invertible, but its inverse is another
elementary matrix of the same type. We record this finding as the next theorem.
Theorem 3 . 1 1
Each elementary matrix is invertible, and its inverse is an elementary matrix of the
same type.
The Fundam en1a1 Theorem ol lnvenible Malrices
We are now in a position to prove one of the main results in this book-a set of
equivalent characterizations of what it means for a matrix to be invertible. In a sense,
much of linear algebra is connected to this theorem, either in the development of
these characterizations or in their application. As you might expect, given this intro­
duction, we will use this theorem a great deal. Make it your friend!
We refer to Theorem 3 . 1 2 as the first version of the Fundamental Theorem, since
we will add to it in subsequent chapters. You are reminded that, when we say that a set
of statements about a matrix A are equivalent, we mean that, for a given A, the state­
ments are either all true or all false.
Theorem 3 . 1 2
The Fundamental Theorem of Invertible Matrices: Version 1
Let A be an n X n matrix. The following statements are equivalent:
a. A is invertible.
b. Ax = b has a unique solution for every b in !R n .
c. Ax = 0 has only the trivial solution.
d. The reduced row echelon form of A is Iw
e. A is a product of elementary matrices.
Section
3. 3 The Inverse of a Matrix
113
We will establish the theorem by proving the circular chain of implications
Proof
( a ) =} ( b ) =} ( c ) =} ( d ) =} ( e ) =} ( a )
(a) =} (b) We have already shown that if A is invertible, then Ax = b has the
unique solution x = A - 1 b for any b in !R n (Theorem 3.7).
(b) =} (c) Assume that Ax = b has a unique solution for any b in !R n . This implies,
in particular, that A x = has a unique solution. But a homogeneous system A x =
always has x = as one solution. So in this case, x = must be the solution.
(c) =} (d) Suppose that Ax = has only the trivial solution. The corresponding
system of equations is
0
0
0
0
0
a 11 x 1 + a 1 2x2 + · · · + a 1nxn = 0
a 11 X 1 + a 2 1X2 + . . . + a 1nXn = 0
and we are assuming that its solution is
= O
= O
In other words, Gauss-Jordan elimination applied to the augmented matrix of the
system gives
[A l o J =
[ ""
�
a12
a , a21
a ln
a2n
a n ] a n2
a nn
f l [!
0
�
0
0
0
1
!]
[In l o J
Thus, the reduced row echelon form of A is In ( d) =} (e) If we assume that the reduced row echelon form of A is In , then A can be
reduced to In using a finite sequence of elementary row operations. By Theorem 3. 10,
each one of these elementary row operations can be achieved by left-multiplying by an
appropriate elementary matrix. If the appropriate sequence of elementary matrices is
E 1 , E2 , , Ek (in that order), then we have
•
•
.
Ek · · · E2E 1A = In
According to Theorem 3. 1 1 , these elementary matrices are all invertible. Therefore,
so is their product, and we have
Again, each E; - 1 is another elementary matrix, by Theorem 3. 1 1 , so we have written
A as a product of elementary matrices, as required.
(e) =} (a) If A is a product of elementary matrices, then A is invertible, since
elementary matrices are invertible and products of invertible matrices are invertible.
114
Chapter
3 Matrices
Exa m p l e 3 . 2 9
If possible, express A =
Solution
[� �J
as a product of elementary matrices.
[2 J
[ 2l J [ l
[ l J � [ l OJ
We row reduce A as follows:
A=
1
3 �
3
�
0
3 �·
3
0
3
-3
0
= I2
0
-3
1
J
Thus, the reduced row echelon form of A is the identity matrix, so the Fundamental
Theorem assures us that A is invertible and can be written as a product of elementary
matrices. We have E4E3 E2 E 1A = I, where
are the elementary matrices corresponding to the four elementary row operations
used to reduce A to I. As in the proof of the theorem, we have
[ O l J [ 2l O J [ l - l J [ l O J
1 0
as required.
1
0
1
0
-3
Remark
Because the sequence of elementary row operations that transforms A
into I is not unique, neither is the representation of A as a product of elementary
matrices. (Find a different way to express A as a product of elementary matrices.)
"Never bring a cannon o n stage in
Act I unless you intend to fire it by
the last act:' - Anton Chekhov
Theorem 3 . 1 3
The Fundamental Theorem is surprisingly powerful. To illustrate its power, we
consider two of its consequences. The first is that, although the definition of an in vertible matrix states that a matrix A is invertible if there is a matrix B such that both
AB = I and BA = I are satisfied, we need only check one of these equations. Thus, we
can cut our work in half!
Let A be a square matrix. If B is a square matrix such that either AB = I or BA = I,
then A is invertible and B = A - 1 •
0.
0.
0
Proof Suppose BA = I. Consider the equation Ax =
Left-multiplying by B, we have
BAx = BO. This implies that x = Ix = Thus, the system represented by Ax = has the
unique solution x = From the equivalence of (c) and (a) in the Fundamental Theo­
rem, we know that A is invertible. (That is, A - 1 exists and satisfies AA i = I = A iA.)
If we now right-multiply both sides of BA = I by A - i , we obtain
BAA - i = IA - i =} BI = A - i =} B = A i
0.
-
-
-
(The proof in the case of AB = I is left as Exercise 4 1 .)
The next consequence of the Fundamental Theorem is the basis for an efficient
method of computing the inverse of a matrix.
Section
Theorem 3 . 1 4
3. 3 The Inverse of a Matrix
115
Let A be a square matrix. If a sequence of elementary row operations reduces A
to I, then the same sequence of elementary row operations transforms I into A - i .
Proof If A is row equivalent to I, then we can achieve the reduction by left­
multiplying by a sequence E 1 , E2 ,
, Ek of elementary matrices. Therefore, we have
Ek · · · E2 E 1A = I. Setting B = Ek · · · E2 E 1 gives BA = I. By Theorem 3 . 1 3, A is invert­
ible and A - l = B. Now applying the same sequence of elementary row operations to
I is equivalent to left-multiplying I by Ek · · · E2 E 1 = B. The result is
Ek · · · E2E 1 I = BI = B = A - 1
•
•
•
Thus, I is transformed into A - i by the same sequence of elementary row operations.
The Gauss-Jordan Melhod for Compuling lhe Inverse
We can perform row operations on A and I simultaneously by constructing a "super­
augmented matrix" [A I I ] . Theorem 3 . 1 4 shows that if A is row equivalent to I [which,
by the Fundamental Theorem (d) � (a), means that A is invertible] , then elementary
row operations will yield
If A cannot be reduced to I, then the Fundamental Theorem guarantees us that A is
not invertible.
The procedure just described is simply Gauss-Jordan elimination performed on an
n X 2n, instead of an n X (n + 1), augmented matrix. Another way to view this pro­
cedure is to look at the problem of finding A -I as solving the matrix equation AX = In
for an n X n matrix X. (This is sufficient, by the Fundamental Theorem, since a right
inverse of A must be a two-sided inverse.) If we denote the columns of X by x1 , . . . , Xn'
then this matrix equation is equivalent to solving for the columns of X, one at a time.
Since the columns of In are the standard unit vectors e 1 , . . . , en , we thus have n systems
of linear equations, all with coefficient matrix A:
Since the same sequence of row operations is needed to bring A to reduced row
echelon form in each case, the augmented matrices for these systems, [A I e 1 ] , . . . ,
[A I e n ] , can be combined as
We now apply row operations to try to reduce A to In , which, if successful, will simul­
taneously solve for the columns of A - i , transforming In into A - i .
We illustrate this use of Gauss-Jordan elimination with three examples.
Exa m p l e 3 . 3 0
Find the inverse of
A=
if it exists.
[� � �i
1 3
-3
-
116
Chapter
3 Matrices
Solulion
Gauss-Jordan elimination produces
[A I I J
R2 - 2R,
R3 - R1
�
H:J R,
R3-R2
�
R, + R3
R2 + 3R3
�
R, - 2R2
�
Therefore,
[:
[:
[:
[:
[:
[:
A-1 =
[
2
2
3
0
-1
4 0
-3 0 0
2
-2
1
-1
2
-1
-3
-2 - 1
1
2
1
0
�:
0
6 -2 1
-2 - 1 0
-1
-3
0
I
-2
0
0
1
-2
2 0 -1
1 0 -5
-2
0
0 0 9
1 0 -5
0 1 -2
9 -t - 5
3
-5
-2
t 1
I
2
I
-2
I
2
1
2
!
3
-2
1
]
2
!
1]
�]
�]
�:
- :i
(You should always check that AA - I = I by direct multiplication. By Theorem 3 . 1 3,
we do not need to check that A - iA = I too.)
Remark
Notice that we have used the variant of Gauss-Jordan elimination that
first introduces all of the zeros below the leading ls, from left to right and top to
bottom, and then creates zeros above the leading l s, from right to left and bottom to
top. This approach saves on calculations, as we noted in Chapter 2, but you may find
it easier, when working by hand, to create all of the zeros in each column as you go.
The answer, of course, will be the same.
Exa m p l e 3 . 3 1
Find the inverse of
A=
if it exists.
[ ! - :i
-2
-1
2
-2
Section
3. 3 The Inverse of a Matrix
111
Solution
We proceed as in Example 3.30, adjoining the identity matrix to A and
then trying to manipulate [A I I] into [I I A - 1 ] .
[ -:
[:
,-,., [ I
[A I IJ
1
-1
2
-2
R2 + 2R1
R3 + R1
�
�
1
1
3
�]
-4 1 0
6 0 1
-2 0 0
0
-4
-2 2 1
-6 1 0
2 -1
0
-3 2
0 0
0 -5
�]
0
1
-3
�]
At this point, we see that it is not possible to reduce A to I, since there is a row of zeros
on the left-hand side of the augmented matrix. Consequently, A is not invertible.
4
As the next example illustrates, everything works the same way over "11.P. , where
p is prime.
Exa m p l e 3 . 3 2
Find the inverse of
A =
if it exists, over "11.. 3 .
Solution 1
in "11.. 3 .
[� �]
We use the Gauss-Jordan method, remembering that all calculations are
[A I I] =
�
2R1
�
R1 + R1
�
R1 + 2R2
[ �] ,
0
Thus , A - 1 =
2
[� 1 �]
[� 1 �]
[� 1 �]
[� 1 �]
2 1
00
1 2
0 0
1 2
1 2
0 0
1 2
and it is easy to check that, over "11.. 3 , AA - l = I.
Since A is a 2 X 2 matrix, we can also compute A - l using the formula
given in Theorem 3.8. The determinant of A is
Solution 2
det A = 2 ( 0 ) - 2 ( 2 ) = - 1 = 2
in "11.. 3 (since 2 + 1 = O ) . Thus, A - l exists and is given by the formula in Theorem 3.8.
We must be careful here, though, since the formula introduces the "fraction" l /det A
Chapter
118
3 Matrices
and there are no fractions in Z 3 . We must use multiplicative inverses rather than
division.
Instead of l /det A = 1 /2, we use T 1 ; that is, we find the number x that satisfies
the equation 2x = 1 in Z 3 • It is easy to see that x = 2 is the solution we want: In Z 3 ,
T 1 = 2, since 2(2) = 1 . The formula for A - 1 now becomes
A-1 = T1
[ _ � -� ] [ � � ] [ � � ]
=2
=
which agrees with our previous solution.
..
I
Exercises 3 . 3
In Exercises 1 - 1 0, find the inverse of the given matrix (if it
exists) using Theorem 3.8.
[� �]
[! !]
[[i n ]
[IO. � -�]
[ ]
1.
2.
3.
4.
5.
7.
- 1 .5
0.5
[ �_� - �� ]
[ ]
[[ ] ]
l / V2 l / V2
- 1 ; V2 1 / \/2
3.55 0.25
8.
8.52 0.60
6.
- 4.2
2.4
9.
•
l /a l /b
, where neither a, b, c, nor d is O
1 /c l /d
•
In Exercises 1 1 and 12, solve the given system using the
method of Example 3.25.
12. X 1 - X2 = 1
2x 1 + x2 = 2
1 1 . 2X + y = - 1
5x + 3y = 2
13. Let A =
For larger systems, the difference is even more
pronounced, and this explains why computer
systems do not use one of these methods to solve
linear systems.
14. Prove Theorem 3.9(b).
15. Prove Theorem 3.9(d).
16. Prove that the n X n identity matrix In is invertible and
that ln- l = Iw
17. (a) Give a counterexample to show that (AB) - 1 *
A - 1 B - 1 in general.
(h) Under what conditions on A and B is (AB) - 1 =
A - 1 B - 1 ? Prove your assertion.
18. By induction, prove that if A 1 , A 2 , , A n are invertible
matrices of the same size, then the product A 1A 2 · · · A n
A n ) - 1 = A ;;- 1 · · · A2 1 A � 1 •
is invertible and (A 1A 2
19. Give a counterexample to show that (A + B) - 1 *
A - 1 + B - 1 in general.
[ � !l [ ;] [ �J
h1 =
, h2 =
-
, and b 3 =
(a) Find A - 1 and use it to solve the three systems
Ax = h 1 , Ax = h2 , and Ax = h 3 .
[ �] .
(h) Solve all three systems at the same time by row re­
ducing the augmented matrix [A I h 1 h 2 h 3 ] using
Gauss-Jordan elimination.
(c) Carefully count the total number of individual
multiplications that you performed in (a) and in
(b) . You should discover that, even for this 2 X 2
example, one method uses fewer operations.
•
•
•
•
In Exercises 20 -23, solve the given matrix equation for X.
Simplify your answers as much as possible. (In the words of
Albert Einstein, "Everything should be made as simple as pos­
sible, but not simpler.") Assume that all matrices are invertible.
21. AXB = ( BA ) 2
20. XA 2 = A - 1
1
1
1
22. (A - X) - = A ( B - 2A ) - 23. ABXA - 1 B - 1 = I + A
In Exercises 24-30, let
A=
c�
[:
[:
] [ � �l
] H -:]
� , B=
2
1
-1
-1
2
-1
1 ' D=
-1
1
-1
1
2
2
-1
-1
-1
Section
In each case, find an elementary matrix E that satisfies the
given equation.
24. EA = B
25. EB = A
26. EA = C
28. EC = D
27. EC = A
29. ED = C
30. Is there an elementary matrix E such that EA = D?
Why or why not?
In Exercises 48-63, use the Gauss-Jordan method to find the
inverse of the given matrix (if it exists).
48.
31.
50.
33.
35.
37.
0
1
0
0
c
0
32.
34.
36.
38.
[� �]
[ _� � ]
[: :J
[i l H
0
1
0
0
.
39. A =
•
.
[ _ � -�J
52.
54.
0
In Exercises 39 and 40, find a sequence of elementary
matrices E1, E2 , , Ek such that Ek · · · E2 E1A = I. Use this
sequence to write both A and A i as products of elementary
matrices.
56.
-
40. A =
[ � �]
41. Prove Theorem 3 . 1 3 fo r the case o f AB = I.
42. (a) Prove that if A is invertible and AB = 0, then
[ � :J
[� - �]
[: -� -� ]
:[ � : J
[: � �]
0
49.
Thus, something that is idempotent has the "same
power" when squared.)
( a) Find three idempotent X matrices.
(b) Prove that the only invertible idempotent n X n
matrix is the identity matrix.
45. Show that if A is a square matrix that1 satisfies the
equation A 2 - 2A + I = 0, then A - = 2I - A.
46. Prove that if a symmetric matrix is invertible, then its
inverse is symmetric also.
22
[ -2 4]
3
-1
-1
[
0 2 V2
V2
0
- 4 V2 V2
58.
1
0
0
3
0
0
B = 0.
(b) Give a counterexample to show that the result in
part (a) may fail if A is not invertible.
43. (a) Prove that if A is invertible and BA = CA, then
B = C.
(b) Give a counterexample to show that the result in
part (a) may fail if A is not invertible.
44. A square matrix A is called idempotent if A 2 = A.
(The word idempotent comes from the Latin idem,
meaning "same;' and potere, meaning "to have power:'
119
47. Prove that if A and B are square matrices and AB is
invertible, then both A and B are invertible.
In Exercises 3 1 -38, find the inverse of the given elementary
matrix.
[� �]
[� �]
[ i - �]
[ i n' H
3. 3 The Inverse of a Matrix
60.
61.
[! !]
[: 2 OJ
over Z 5
5
63.
62.
[
0
1
[:
4 over Z 7
6 1
Partitioning large square matrices can sometimes make their
inverses easier to compute, particularly if the blocks have
a nice form. In Exercises 64-68, verify by block multip lica­
tion that the inverse of a matrix, ifpartitioned as shown, is
as claimed. (Assume that all inverses exist as needed.)
Chapter
180
65.
66.
67.
3 Matrices
[ � �r [
[� �r [
[� �r
l
l
l
]
(BC) - 1 B
- (BC) - 1
1
C(BC) - I - C(BC) - 1 B
(I - BC) - 1
- (I - BC) - 1 B
- C(I - BC) - 1 I + C(I - BC) - 1 B
[
]
(BD- 1 C) - 1 BD - 1
- (BD- 1 C) - 1
- D - 1 c(BD- 1 c) - l D - 1 - D - 1 c(BD- 1 c) - 1 BD - l
A B -1 - p Q
1 1
68.
- R S , where P = (A - BD - C) - ,
C D
- 1 , R = -D - 1 CP, and S = D - 1
Q = -PBD
1
+ D - CPBD - 1
[ ] [ ]
•
In Exercises 69-72, partition the given matrix so that you
can apply one of the formulas from Exercises 64-68, and
then calculate the inverse using that formula.
]
�]
0 0
1 0
3 1
2 0
70. The matrix in Exercise 58
71.
[� �:
0
0 1
-1 1
1 0
72.
T h e LU Factorizatio n
Just as it is natural (and illuminating) to factor a natural number into a product of
other natural numbers-for example, 30 = 2 3 5-it is also frequently helpful to fac­
tor matrices as products of other matrices. Any representation of a matrix as a product
of two or more other matrices is called a matrix factorization. For example,
·
·
[ ] [ ][ ]
3
9
-1
-5
=
l 0 3
3 1 0
-1
-2
is a matrix factorization.
Needless to say, some factorizations are more useful than others. In this section,
we introduce a matrix factorization that arises in the solution of systems of linear
equations by Gaussian elimination and is particularly well suited to computer imple­
mentation. In subsequent chapters, we will encounter other equally useful matrix
factorizations. Indeed, the topic is a rich one, and entire books and courses have been
devoted to it.
Consider a system of linear equations of the form Ax = b, where A is an n X n
matrix. Our goal is to show that Gaussian elimination implicitly factors A into a prod­
uct of matrices that then enable us to solve the given system (and any other system
with the same coefficient matrix) easily.
The following example illustrates the basic idea.
Exa m p l e 3 . 3 3
Let
1
-1
5
Section
3.4 The Factorization
LU
Row reduction of A proceeds as follows:
181
u
-1
5
(1)
The three elementary matrices £ 1 , E2 , E3 that accomplish this reduction of A to
echelon form U are (in order) :
0
1
0
0
1
0
Hence,
Solving for A, we get
A = E 1- 1 E2- 1 E3- 1 U =
[� �J U �m �J u
LU
u
�
J
�
u
0
1
0
0
1
0
0
1
-2
0
-2
Thus, A can be factored as
A = LU
where U is an upper triangular matrix (see the exercises for Section 3.2), and L is unit
lower triangular. That is, L has the form
1948
(1912-1954)
The LU factorization was introduced
in
by the great English
mathematician Alan M. Turing
in a paper entitled
"Rounding-off Errors in Matrix
Processes" ( Quarterly Journal of
(1948), 287-308).
Mechanics and Applied Mathematics,
1
pp.
During
World War II, Turing was
instrumental in cracking the
German "Enigma'' code. However,
he is best known for his work in
mathematical logic that laid the
theoretical groundwork for the
development of the digital computer
and the modern field of artificial
intelligence. The "Turing test"
that he proposed in
is still
used as one of the benchmarks in
addressing the question of whether
a computer can be considered
"intelligent:'
1950
r � [: :
with zeros above and l s on the main diagonal.
!]
The preceding example motivates the following definition.
D e f i n i t i o n Let A be a square matrix. A factorization of A as A = LU, where
L is unit lower triangular and U is upper triangular, is called an LU factorization
of A.
R e m a rks
Observe that the matrix A in Example 3.33 had an LU factorization because no
row interchanges were needed in the row reduction of A. Hence, all of the elementary
•
matrices that arose were unit lower triangular. Thus, L was guaranteed to be unit
182
Chapter
3 Matrices
�
lower triangular because inverses and products of unit lower triangular matrices are
also unit lower triangular. (See Exercises 29 and 30.)
If a zero had appeared in a pivot position at any step, we would have had to swap
rows to get a nonzero pivot. This would have resulted in L no longer being unit lower
triangular. We will comment further on this observation below. (Can you find a ma­
trix for which row interchanges will be necessary?)
•
The notion of an LU factorization can be generalized to nonsquare matrices
by simply requiring U to be a matrix in row echelon form. (See Exercises 1 3 and 1 4 . )
•
Some books define an LU factorization of a square matrix A to be any factor­
ization A = LU, where L is lower triangular and U is upper triangular.
The first remark above is essentially a proof of the following theorem.
Theorem 3 . 1 5
If A is a square matrix that can be reduced to row echelon form without using any
row interchanges, then A has an LU factorization.
To see why the LU factorization is useful, consider a linear system Ax = b, where
the coefficient matrix has an LU factorization A = LU. We can rewrite the system
Ax = b as LUx = b or L( Ux) = b. If we now define y = Ux, then we can solve for x in
two stages:
1. Solve Ly = b for y byforward substitution (see Exercises 25 and 26 in Section 2 . 1 ) .
2. Solve Ux = y fo r x by back substitution.
Each of these linear systems is straightforward to solve because the coefficient matri­
ces L and U are both triangular. The next example illustrates the method.
Exa m p l e 3 . 3 4
Use an LU factorization of A =
[ ! !]
to ,olve Ax � b, whece b �
-2
0
1
-1
5
-2
Solution
In Example 3.33, we found that
A =
[f:J
[ � � i [� - � -� ]
2
-1
1
0
2
[-: l
= LU
As outlined above, to solve Ax = b (which is the same as L( Ux) = b ), we first solve
Ly � b fm y �
Thi' ' ' ju,t the Hnem- 'l"tem
Y1
-4
2y 1 + Y2
-y 1 - 2y2 + y3 = 9
-
Forward substitution (that is, working from top to bottom) yields
Y i = l , y2 = - 4
2y i = - 6 , y = 9 + Y i + 2y2 = - 2
3
Section
2x 1 +
3.4 The Factorization
LU
183
x2 + 3x3 =
- 3x2 - 3x3 = - 6
2x3 = - 2
and back substitution quickly produces
X3 = - 1 ,
- 3x2 = - 6 + 3x3 = - 9 so that x2 = 3, and
2x 1 = 1 - x2 - 3x3 = 1 so that x1 = t
Thmfoce, the rnlution to the given 'Y'tem Ax � b ; , x �
[ _!l
·
A n Easv wav to Find L U Facto rizations
ii>-"'--
In Example 3.33, we computed the matrix L as a product of elementary matrices.
Fortunately, L can be computed directly from the row reduction process without our
needing to compute elementary matrices at all. Remember that we are assuming that
A can be reduced to row echelon form without using any row interchanges. If this is
the case, then the entire row reduction process can be done using only elementary
row operations of the form R; - kR1 . (Why do we not need to use the remaining
elementary row operation, multiplying a row by a nonzero scalar?) In the operation
R; - kRj, we will refer to the scalar k as the multiplier.
In Example 3.33, the elementary row operations that were used were, in order,
2)
-1)
-2)
R2 - 2R 1
(multiplier
R3 + R 1 = R 3 - ( - l)R 1
(multiplier
R3 + 2R2 = R 3 - ( - 2)R2
(multiplier
The multipliers are precisely the entries of L that are below its diagonal! Indeed,
=
=
=
and L 2 1 = 2, L 3 1 = - 1 , and L 3 2 = - 2. Notice that the elementary row operation
R; - kR1 has its multiplier k placed in the (i, j) entry of L.
Exa m p l e 3 . 3 5
Find an LU factorization of
A=
r ! 4 ! --�4�1
3 2
-9 5
5
-2
-1
184
Chapter
3 Matrices
Solulion
] ['
[;
. -HJ•, [ :
Reducing A to row echelon form, we have
A=
[j
4
2
5
3
8
5
-2
-4
- 10
-1
-4
]
!
]
]
R2 - 2R1
R3-R1
R4- ( - 3)R1
1 3 -4
Q 2 2 -2
2
0
0 8 7 -1
R, - !R2
R4-4R1
3 -4
2 -2
4
- 1 -8
-----+
-----+
-----+
2
0
0
3 -4
2 2 -2
4
0 0 1
0 0 0 -4
= U
The first three multipliers are 2, 1 , and - 3, and these go into the subdiagonal entries
of the first column of L. So, thus far,
The next two multipliers are t and 4, so we continue to fill out L:
The final multiplier, - 1 , replaces the last * in L to give
Thus, an LU factorization of A is
![ ! -��i [ � � � i [ � � =�i
0
A=
4
3 2
-9 5
as is easily checked.
5
-2
-1
-4
1 t
-3 4
-1
0
1
2
0 0 1
0 0 0
4
-4
= LU
Section
3.4 The LU Factorization
185
R e m a rks
In applying this method, it is important to note that the elementary row opera­
tions R; - kRj must be performed from top to bottom within each column (using the
diagonal entry as the pivot), and column by column from left to right. To illustrate
what can go wrong if we do not obey these rules, consider the following row reduction:
•
A = [ � � �i [ � � �i [� � �] U
L L32-= L2- 1 =
L � [i : �]
A LU.
LUL
A.)
2 2
1
�'
0 0
-1
�
0
0
This time the multipliers would be placed in as follows:
get
.,._..
but
-1
2,
1 . We would
(Check this! Find a correct factorization of
An alternative way to construct is to observe that the multipliers can be
obtained directly from the matrices obtained at the intermediate steps of the row
reduction process. In Example 3.33, examine the pivots and the corresponding col­
umns of the matrices that arise in the row reduction
•
*
1
-1
5
u
A.
The first pivot is 2, which occurs in the first column of Dividing the entries of
this column vector that are on or below the diagonal by the pivot produces
[± J [ _ � -
A1
The next pivot is - 3, which occurs in the second column of . Dividing the entries
of this column vector that are on or below the diagonal by the pivot, we obtain
U.
The final pivot (which we did not need to use) is 2, in the third column of Divid­
ing the entries of this column vector that are on or below the diagonal by the pivot,
we obtain
±[J [J
If we place the resulting three column vectors side by side in a matrix, we have
L
u -� J
which is exactly once the above-diagonal entries are filled with zeros.
186
Chapter
3 Matrices
In Chapter 2, we remarked that the row echelon form of a matrix is not unique.
However, if an invertible matrix A has an LU factorization A = LU, then this factoriza­
tion is unique.
Theorem 3 . 1 6
�
If A is an invertible matrix that has an LU factorization, then L and U are unique.
Proof Suppose A = LU and A = L 1 U1 are two LU factorizations of A. Then LU =
L 1 U1 , where L and L 1 are unit lower triangular and U and U1 are upper triangular. In
fact, U and U1 are two (possibly different) row echelon forms of A.
By Exercise 30, L 1 is invertible. Because A is invertible, its reduced row echelon
form is an identity matrix I by the Fundamental Theorem of Invertible Matrices.
Hence U also row reduces to I (why?) and so U is invertible also. Therefore,
Lj 1 ( L U) u - 1 = Lj 1 ( L 1 U1 ) u - 1 so ( Ll 1 L ) ( uu - 1 ) = ( Ll 1 L 1 ) ( UI u - 1 )
Hence,
�
( Ll 1 L ) I = I( UI u - 1 ) so Li 1 L = U1 u - I
But Lj 1 L is unit lower triangular by Exercise 29, and U1 u - 1 is upper triangular.
(Why?) It follows that Lj 1 L = U1 u- 1 is both unit lower triangular and upper tri­
angular. The only such matrix is the identity matrix, so Lj 1 L = I and U1 u - 1 = I. It
follows that L = L 1 and U = U1 , so the LU factorization of A is unique.
The pr LU Facto rization
We now explore the problem of adapting the LU factorization to handle cases where
row interchanges are necessary during Gaussian elimination. Consider the matrix
A straightforward row reduction produces
which is not an upper triangular matrix. However, we can easily convert this into
upper triangular form by swapping rows 2 and 3 of B to get
Alternatively, we can swap rows 2 and 3 of A first. To this end, let P be the elementary
matrix
Section
3.4 The LU Factorization
181
corresponding to interchanging rows 2 and 3, and let E be the product of the
elementary matrices that then reduce PA to U (so that E - 1 = L is unit lower triangu­
lar). Thus EPA = U, so A = (EP) - 1 U = P - 1 E - 1 U = P - 1 LU.
Now this handles only the case of a single row interchange. In general, P will be
the product P = Pk · · ·P2 P 1 of all the row interchange matrices P 1 ,P2 , , Pk (where
P 1 is performed first, and so on) . Such a matrix P is called a permutation matrix. Ob­
serve that a permutation matrix arises from permuting the rows of an identity matrix
in some order. For example, the following are all permutation matrices:
•
[� �] . [ :
•
•
0
0
Fortunately, the inverse of a permutation matrix is easy to compute; in fact, no calcu­
lations are needed at all!
Theorem 3 . 1 1
If P is a permutation matrix, then p - l = P T.
We must show that p Tp = I. But the ith row of p T is the same as the ith
column of P, and these are both equal to the same standard unit vector e, because P
is a permutation matrix. So
Proof
( prp) ;; = ( ith row of p r ) ( ith column of P) = e re = e · e = 1
This shows that diagonal entries of pTp are all ls. On the other hand, if j i= i, then
the jth column of P is a different standard unit vector from e-say e ' . Thus, a typical
off-diagonal entry of prp is given by
( p rp) iJ = ( ith row of p r ) (jth column of P) = e re ' = e · e ' = 0
Hence pTP is an identity matrix, as we wished to show.
Thus, in general, we can factor a square matrix A as A = P - 1 LU = P rLU.
Let A be a square matrix. A factorization of A as A = PTLU, where
P is a permutation matrix, L is unit lower triangular, and U is upper triangular, is
called a PrLUfactorization of A.
Definition
Exa m p l e 3 . 3 6
Find o pTLU fadmizotion of A �
[: � n
First we reduce A to row echelon form. Clearly, we need at least one row
interchange.
2
2
0
R1 +-* Rz
R, - 2R1
�
�
0
0
2
A =
1
-3
Solution
[ : ! ] [ i ! ] [ i - �l
[: -!]
�
Rz +-* R3
2
-3
0
188
Chapter
3 Matrices
We have used two row interchanges (R 1 � R 2 and then R 2 � R 3 ), so the required
permutation matrix is
p = P2 P 1 =
[i �m :J [: �]
1
0
0
1
0
0
0
0
We now find an LU factorization of PA.
PA =
[: �m !] [� !] [: -!]
[ � : rn � � : J [ i -� - ! :
1
0
0
2
1
0
0
2
1
2
-3
0
R2 - 2R1
u
Hence L 2 1 = 2, and so
A � P'L U �
The discussion above justifies the following theorem.
Theorem 3 . 1 8
Every square matrix has a PTLU factorization.
Even for an invertible matrix, the P TLU factorization is not unique. In
Example 3.36, a single row interchange R 1 � R 3 also would have worked, leading to
a different P. However, once P has been determined, L and U are unique.
Remark
Comp utational Considerations
IfA is n X n, then the total number of operations (multiplications and divisions) required
to solve a linear system Ax = b using an LU factorization of A) is T(n) n 3 /3, the same
as is required for Gaussian elimination. (See the Exploration "Counting Operations;'
in Chapter 2.) This is hardly surprising since the forward elimination phase produces
the LU factorization in n 3 /3 steps, whereas both forward and backward substitution
require n 2 /2 steps. Therefore, for large values of n, the n 3 /3 term is dominant. From
this point of view, then, Gaussian elimination and the LU factorization are equivalent.
However, the LU factorization has other advantages:
=
=
=
From a storage point of view, the L U factorization is very compact because
we can overwrite the entries of A with the entries of L and U as they are computed. In
Example 3.33, we found that
•
A=
�[ - � � i [ � � � i [ � - � � ]
-
-2
This can be stored as
5 5
-1
-2
1
[ � =� �]
-1
-2
-
2
0
0
2
= LU
Section
3.4 The LU Factorization
189
with the entries placed in the order ( 1 , 1 ) , ( 1 ,2), ( 1 ,3), (2, 1 ) , (3, 1 ) , (2,2), (2,3), (3,2),
(3,3). In other words, the subdiagonal entries of A are replaced by the corresponding
multipliers. (Check that this works! )
•
Once an LU factorization of A has been computed, it can be used to solve as
many linear systems of the form Ax
as we like. We just need to apply the method
of Example 3.34, varying the vector each time.
•
For matrices with certain special forms, especially those with a large number
of zeros (so-called "sparse" matrices) concentrated off the diagonal, there are methods
that will simplify the computation of an LU factorization. In these cases, this method
is faster than Gaussian elimination in solving Ax
•
For an invertible matrix A, an LU factorization of A can be used to find A - 1 ,
if necessary. Moreover, this can be done in such a way that it simultaneously yields a
factorization of A - 1 . (See Exercises 1 5 - 1 8.)
b= b
= b.
Remark If you have a CAS (such as MATLAB) that has the LU factorization
built in, you may notice some differences between your hand calculations and the
computer output. This is because most CAS's will automatically try to perform partial
pivoting to reduce roundoff errors. (See the Exploration "Partial Pivoting;' in Chapter
2.) Turing's paper is an extended discussion of such errors in the context of matrix
factorizations.
I
This section has served to introduce one of the most useful matrix factorizations.
In subsequent chapters, we will encounter other equally useful factorizations.
Exercises 3 . 4
=b
1. A = [ l ] [ J [ l ] ' b = [ S J
= [� -� ] = [t � ][� - ! l b = [ � ]
= [ - ! : =�J H ; � l
�[ = !], b [ - � 1
= [ � = � � 1 [ -�; � 1
[� -� H b � U_
In Exercises 1 -6, solve the system Ax
LU factorization of A.
1 O
-1 1
-2
2 5
5. A
=
6. A
=
using the given
-2
0 6
1
2. A
3. A
_
_
x
4. A
4
0 0
-1
x
-�
2 2
=
0
0
0 1
]
[ ]
In Exercises 7-12,find an LUfactorization of the given matrix.
7·
[
1
-3
2
-1
s.
2
3
-4
1
Chapter
190
9.
11.
[: ; :l
[ � : _; -:i
[ -� ! -� � i
-1
12.
3 Matrices
-2
4 4
6 9
-9
In Exercises 1 9-22, write the given permutation matrix as a
product of elementary (row interchange) matrices.
19. [ : � ]
0
0
0
21.
7 3
5 8
Generalize the definition of LUfactorization to nonsquare
matrices by simply requiring U to be a matrix in row ech­
elon form. With this modification, find an LUfactorization
of the matrices in Exercises 13 and 14.
[
1 c
13. 0 3 3
0 0 0
14.
[�
1
-2
2
-7
1
3
-�
0
3
3
-3
]
-1
8
5
-6
-�]
Fo r a n invertible matrix with a n L Ufactorization1 A1 = LU,
both L and U will be invertible
and1 A - l = u1 - L - . In
1
Exercises 15 and 1 6, find L , u - , and A - for the given
matrix.
15. A in Exercise 1
16. A in Exercise 4
The inverse of a matrix can also be computed by solving sev­
eral systems of equations using the method ofExample 3.34.
For an n X n matrix A, to find its inverse we need to solve
AX = In for the n X n matrix X. Writing this equation as
A [ x1 x 2 · · x n ] = [ e 1 e2 · · · e n ] , using the matrix-column
form ofAX, we see that we need to solve n systems of linear
equations: Ax1 = e 1 , Ax2 = e2 , . . . , AXn = en · Moreover, we
can use the factorization A = LU to solve each one of these
systems.
0
1
22. 0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
[ -; �]
In Exercises 23-25, find a PTLUfactorization of the given
matrix A.
23. A =
25. A =
H !]
2
3
[ -;
-1
1
1
0
-1
1
0
24. A =
il
2
1
3
-1
26. Prove that there are exactly n! n X n permutation
matrices.
In Exercises 27-28, solve the system Ax = b using the given
factorization A = PTLU. Because ppT = I, PTLUx = b can
be rewritten as LUx = Pb. This system can then be solved
using the method of Example 3.34.
27. A =
•
In Exercises 1 7 and 18, use the approach just outlined to
find A - l for the given matrix. Compare with the method of
Exercises 15 and 1 6.
18. A in Exercise 4
17. A in Exercise 1
20.
[ ;]
1
0
0
0
[� � ]
0 0
0 1
1 0
0 0
[� � �l [� � �l [� C :l
m
[ : �l l
[ ! �- [ : : J [ : : J
[ �:i
[� r
-
1
-1
3
1
0
x
0 0 1
� P'L U, b �
3
28. A =
0
0
0
x
1
-1
0
= P TL U, b =
2
-t 1
0
1
-1
Section
29. Prove that a product of unit lower triangular matrices
is unit lower triangular.
30. Prove that every unit lower triangular matrix is
invertible and that its inverse is also unit lower
triangular.
An LDUfactorization of a square matrix A is a factoriza­
tion A = LDU, where L is a unit lower triangular matrix,
D is a diagonal matrix, and U is a unit upper triangular matrix (upper triangular with ls on its diagonal). In
Exercises 31 and 32, find an LDUfactorization of A.
3. 5 Subspaces, Basis, Dimension, and Rank
191
31. A in Exercise 1
32. A in Exercise 4
33. If A is symmetric and invertible and has an LDU
factorization, show that U = L r.
34. If A is symmetric and invertible and A = LDL T (with L
unit lower triangular and D diagonal), prove that this
factorization is unique. That is, prove that if we also
have A = L1D1L[ (with L1 unit lower triangular and D1
diagonal), then L = L1 and D = D1•
S u b s p aces , Basis , D i m e n s io n , and R a n k
z
2u + v
x
y
Figure 3 . 2
This section introduces perhaps the most important ideas in the entire book. We have
already seen that there is an interplay between geometry and algebra: We can often
use geometric intuition and reasoning to obtain algebraic results, and the power of
algebra will often allow us to extend our findings well beyond the geometric settings
in which they first arose.
In our study of vectors, we have already encountered all of the concepts in this
section informally. Here, we will start to become more formal by giving definitions
for the key ideas. As you'll see, the notion of a subspace is simply an algebraic
generalization of the geometric examples of lines and planes through the origin. The
fundamental concept of a basis for a subspace is then derived from the idea of direc­
tion vectors for such lines and planes. The concept of a basis will allow us to give a
precise definition of dimension that agrees with an intuitive, geometric idea of the
term, yet is flexible enough to allow generalization to other settings.
You will also begin to see that these ideas shed more light on what you already
know about matrices and the solution of systems of linear equations. In Chapter 6,
we will encounter all of these fundamental ideas again, in more detail. Consider this
section a "getting to know you" session.
A plane through the origin in IR 3 "looks like" a copy of IR 2 • Intuitively, we would
agree that they are both "two-dimensional:' Pressed further, we might also say that
any calculation that can be done with vectors in IR 2 can also be done in a plane through
the origin. In particular, we can add and take scalar multiples (and, more generally,
form linear combinations) of vectors in such a plane, and the results are other vec­
tors in the same plane. We say that, like IR 2 , a plane through the origin is closed with
respect to the operations of addition and scalar multiplication. (See Figure 3.2.)
But are the vectors in this plane two- or three-dimensional objects? We might
argue that they are three-dimensional because they live in IR 3 and therefore have three
components. On the other hand, they can be described as a linear combination of just
two vectors-direction vectors for the plane-and so are two-dimensional objects liv­
ing in a two-dimensional plane. The notion of a subspace is the key to resolving this
conundrum.
192
Chapter
3 Matrices
Definition
A subspace of !R n is any collection S of vectors in !R n such that:
1 . The zero vector 0 is in S.
2. If u and v are in S, then u + v is in S. (S is closed under addition.)
3. If u is in S and c is a scalar, then cu is in S. (S is closed under scalar
multiplication.)
We could have combined properties (2) and (3) and required, equivalently, that S be
closed under linear combinations:
If u 1 , u2 , • . • , uk are in S and c 1 , c2 , . • . , ck are scalars,
then c 1 u 1 + c2 u2 + · · · + ckuk is in S.
Exa m p l e 3 . 3 1
Every line and plane through the origin in IR 3 is a subspace of IR 3 . It should be clear
geometrically that properties ( 1 ) through (3) are satisfied. Here is an algebraic proof
in the case of a plane through the origin. You are asked to give the corresponding
proof for a line in Exercise 9.
Let <JP be a plane through the origin with direction vectors v1 and v2 . Hence, <JP =
span (v1 , v2 ) . The zero vector 0 is in <JP, since 0 = Ov1 + Ov2 • Now let
be two vectors in <JP. Then
Thus, u + v is a linear combination of v1 and v2 and so is in <JP.
Now let c be a scalar. Then
4
which shows that cu is also a linear combination of v1 and v2 and is therefore in <JP. We
have shown that <JP satisfies properties ( 1 ) through ( 3) and hence is a subspace of IR 3
If you look carefully at the details of Example 3.3 7, you will notice that the fact
that v1 and v2 were vectors in IR 3 played no role at all in the verification of the prop­
erties. Thus, the algebraic method we used should generalize beyond IR 3 and apply
in situations where we can no longer visualize the geometry. It does. Moreover, the
method of Example 3.37 can serve as a "template" in more general settings. When we
generalize Example 3.3 7 to the span of an arbitrary set of vectors in any !R n , the result
is important enough to be called a theorem.
Theorem 3 . 1 9
Let v1 , v2 , . . . , vk be vectors in !R n . Then span(v1 , v2 , . . . , vk) is a subspace of !R n .
Let S = span (v1 , v2 , . . . , vk) · To check property ( 1 ) of the definition, we simply
observe that the zero vector 0 is in S, since 0 = Ov1 + Ov2 + · · · + Ovk.
Proof
3. 5 Subspaces, Basis, Dimension, and Rank
Section
193
Now let
be two vectors in S. Then
u + v = ( c 1 v1 + c2v2 + · · · + ckvk ) + ( d 1 v1 + d2v2 + · · · + dkvk )
= ( c 1 + d 1 ) v1 + ( cz + dz ) V2 + · · · + ( ck + dk ) vk
Thus, u + v is a linear combination of v1 , v2 , , vk and so is in S. This verifies prop­
erty ( 2).
To show property (3), let c be a scalar. Then
•
.
.
cu = c ( c 1 v1 + c2v2 + · · · + c kvk )
= ( cc 1 ) v1 + ( cc2 ) v2 + · · · + ( cck ) vk
which shows that cu is also a linear combination of v1 , v2 , . . . , vk and is therefore
in S. We have shown that S satisfies properties ( 1 ) through (3) and hence is a subspace
of ll�r.
We will refer to span (v1 , v2 ,
, vk) as the subspace spanned by v1 , v2 ,
, vk.
We will often be able to save a lot of work by recognizing when Theorem 3 . 1 9 can be
applied.
.
Exa m p l e 3 . 3 8
Show that the set of all vectors
forms a subspace of IR\ 3 .
Solullon
•
.
[;]
.
.
•
that satisfy the conditions x = 3y and z = - 2y
z
Substitufog the two conditions into
[�]
Since y is arbitrary, the given set of vectors is span
of lR\ 3 , by Theorem 3 . 1 9.
[ _n
yields
( [ �] )
-2
and is thus a subspace
4
Geometrically, the set of vectors in Example 3.38 represents the line through the
migin in GI' with dimtion vectm
194
Chapter
3 Matrices
- 2y
Exa m p l e 3 . 3 9
[; ]
Determine whether the set of all vectors
and z =
is a subspace of IR 3 .
Solulion
....,..
3y
that satisfy the conditions x =
z
3[ y
-:2y l l
+ 1
This time, we have all vectors of the form
The zero vector is not of this form. (Why not? Try solving
3[ y [ )
-:2y l H H'""'
1
property ( 1 ) does not hold, so this set cannot be a subspace of IR 3 .
Exa m p l e 3 . 4 0
[;], y
[ :2 ]
where
Determine whether the set o f all vectors
S o lulion
These are the vectors of the form
= :x2-, i s a subspace o f IR 2 .
- call this set S. This time 0 =
belongs to S (take x = O), so property ( 1 ) holds. Let u =
Then
u + v =
2.
[ J
x� + x�
[ :� ]
and v =
[:� ]
[�]
be in S.
X 1 + Xz
which, in general, is not in S, since it does not have the correct form; that is,
x � + x? * (x1 + x2 ) To be specific, we look for a counterexample. If
u =
(2)
[�]
[!]
2IR . [ ! ]
and
v =
then both u and v are in S, but their sum u + v =
property fails and S is not a subspace of
(3)
is not in S since 5 *
32
. Thus,
.+
In order for a set S to be a subspace of some !R n , we must prove that
properties ( 1 ) through hold in general. However, for S to fail to be a subspace of !R n ,
it is enough to show that one of the three properties fails to hold. The easiest course is
usually to find a single, specific counterexample to illustrate the failure of the property.
Once you have done so, there is no need to consider the other properties.
Remark
Subspaces Associaled wilh Malrices
2;
A great many examples of subspaces arise in the context of matrices. We have already
encountered the most important of these in Chapter we now revisit them with the
notion of a subspace in mind.
Section
Definition
3. 5 Subspaces, Basis, Dimension, and Rank
195
Let A be an m x n matrix.
1 . The row space of A is the subspace row(A) of !R n spanned by the rows of A.
2. The column space of A is the subspace col (A) of !R m spanned by the columns
of A.
Remark
Observe that, by Example 3.9 and the Remark that follows it, col(A)
consists precisely of all vectors of the form Ax where x is in !R n .
Exa m p l e 3 . 4 1
Consider the matrix
( ' ) Dctmnine wheth" b �
[�]
[
1
A= 0
3
;,
-1
1
-3
i
in the column 'P"' of A.
(b) Determine whether w = [ 4 5 ] is in the row space of A.
(c) Describe row(A) and col (A) .
Solution
(a) By Theorem 2.4 and the discussion preceding it, b is a linear combination of the
columns of A if and only if the linear system Ax = b is consistent. We row reduce
the augmented matrix as follows:
[ � �: � ] [ � � � ]
�
Thus, the system is consistent (and, in fact, has a unique solution) . Therefore, b is
in col (A) . (This example is just Example 2 . 1 8, phrased in the terminology of this
section.)
(b) As we also saw in Section 2.3, elementary row operations simply create linear
combinations of the rows of a matrix. That is, they produce vectors only in the row
space of the matrix. If the vector w is in row(A), then w is a linear combination of the
rows of A, so if we augment A by w as
___...
[�],
[�' ]
it will be possible to apply elementary row
operations to this augmented matrix to reduce it to form
using only elementary
row operations of the form R; + kRj , where i > j -in other words, workingfrom top
to bottom in each column. (Why?)
In this example, we have
[�] [ � - � i
3 -3
4 5
��
�:=
!�:
�
[� - � i [� -� i
0
0
0
9
��
�
R 4 - 9R2
0
0
0
0
��
196
Chapter
3 Matrices
�
Therefore, w is a linear combination of the rows of A (in fact, these calculations show
that w = 4 [ l - 1 ] + 9 [ O l ] -how?), and thus w is in row(A) .
( c) It is easy to check that, fo r any vector w = [ x y ] , the augmented matrix
reduces to
[�]
in a similar fashion. Therefore, every vector in IR 2 is in row(A), and so row(A) = IR 2 .
Finding col (A) is identical to solving Example 2.2 1 , wherein we determined that
it coincides with the plane (through the origin) in IR 3 with equation 3x - z = 0. (We
will discover other ways to answer this type of question shortly.)
Remark We could also have answered part (b) and the first part of part (c) by
observing that any question about the rows of A is the corresponding question about
the columns of A T. So, for example, w is in row(A) if and only if wT is in col (A T ) . This
is true if and only if the system A Tx = wT is consistent. We can now proceed as in
part (a) . (See Exercises 2 1 -24.)
The observations we have made about the relationship between elementary row
operations and the row space are summarized in the following theorem.
Theorem 3 . 2 0
Let B be any matrix that is row equivalent to a matrix A. Then row(B) = row(A).
The matrix A can be transformed into B by a sequence of row operations.
Consequently, the rows of B are linear combinations of the rows of A; hence, linear
combinations of the rows of B are linear combinations of the rows of A. (See Exer­
cise 2 1 in Section 2.3.) It follows that row(B) � row(A) .
O n the other hand, reversing these row operations transforms B into A. There­
fore, the above argument shows that row(A) � row(B) . Combining these results, we
have row(A) = row(B) .
Proof
There is another important subspace that we have already encountered: the set
of solutions of a homogeneous system of linear equations. It is easy to prove that this
subspace satisfies the three subspace properties.
Theorem 3 . 2 1
Let A be an m X n matrix and let N be the set of solutions of the homogeneous
linear system Ax = 0. Then N is a subspace of !R n .
[Note that x must be a (column) vector in !R n in order for Ax to be defined and
that 0 = O m is the zero vector in !R m .] Since AOn = Om , O n is in N. Now let u and v be
in N . Therefore, Au = 0 and Av = 0. It follows that
A ( u + v) = Au + Av = 0 + 0 = 0
Proof
Section
3. 5 Subspaces, Basis, Dimension, and Rank
191
Hence, u + v is in N . Finally, for any scalar c,
A ( c u ) = c (Au ) = c O = 0
and therefore cu is also in N. It follows that N is a subspace of !R n .
Definition
Let A be an m X n matrix. The null space of A is the subspace of
!R n consisting of solutions of the homogeneous linear system Ax = 0. It is denoted
by null(A).
The fact that the null space of a matrix is a subspace allows us to prove what intuition and examples have led us to understand about the solutions of linear systems:
They have either no solution, a unique solution, or infinitely many solutions.
Theorem 3 . 2 2
Let A be a matrix whose entries are real numbers. For any system of linear
equations Ax = b, exactly one of the following is true:
a. There is no solution.
b. There is a unique solution.
c. There are infinitely many solutions.
At first glance, it is not entirely clear how we should proceed to prove this theo­
rem. A little reflection should persuade you that what we are really being asked to
prove is that if (a) and (b) are not true, then ( c) is the only other possibility. That is, if
there is more than one solution, then there cannot be just two or even finitely many,
but there must be infinitely many.
If the system Ax = b has either no solutions or exactly one solution, we are
done. Assume, then, that there are at least two distinct solutions of Ax = b-say, x 1
and x2 . Thus,
Ax , = b and Ax2 = b
Proof
with x1 * x2 . It follows that
A ( x, - x2 ) = Ax 1 - Ax2 = b - b = 0
Set Xo = x 1 - x2 . Then Xo * 0 and AXo = 0. Hence, the null space of A is nontrivial,
and since null (A) is closed under scalar multiplication, CXo is in null (A) for every
scalar c. Consequently, the null space of A contains infinitely many vectors (since it
contains at least every vector of the form CXo and there are infinitely many of these) .
Now, consider the (infinitely many) vectors of the form x1 + CXo , as c varies through
the set of real numbers. We have
A ( x, + c x0 ) = Ax, + cAXo = b + c O = b
Therefore, there are infinitely many solutions of the equation Ax = b.
Basis
We can extract a bit more from the intuitive idea that subspaces are generalizations
of planes through the origin in IR 3 . A plane is spanned by any two vectors that are
198
Chapter
3 Matrices
parallel to the plane but are not parallel to each other. In algebraic parlance, two
such vectors span the plane and are linearly independent. Fewer than two vectors will
not work; more than two vectors is not necessary. This is the essence of a basis for a
subspace.
Definition
A basis for a subspace S of ll�r is a set of vectors in S that
1 . spans S and
2. is linearly independent.
Exa m p l e 3 . 4 2
Exa m p l e 3 . 4 3
In Section 2.3, we saw that the standard unit vectors e1, e2 , . . . e n in !R n are linearly
independent and span !R n . Therefore, they form a basis for !R n , called the standard
basis.
In Example 2 . 1 9, we showed that IR 2 = span
( [ � ] [� ] )
4
[ � ] [� ]
. Since _ and
_ ,
also linearly independent (as they are not multiples), they form a basis for IR 2 .
{ [�l [�] }
{ [ �l [�] }
are
4
A subspace can (and will) have more than one basis. For example, we have just
seen that IR 2 has the standard basis
. How_
ever, we will prove shortly that the number of vectors in a basis for a given subspace will
always be the same.
Exa m p l e 3 . 4 4
and the basis
Find a basis for S = span (u, v, w) , where
The vectors u, v, and w already span S, so they will be a basis for S if they
are also linearly independent. It is easy to determine that they are not; indeed, w =
2u - 3v. Therefore, we can ignore w, since any linear combinations involving u, v,
and w can be rewritten to involve u and v alone. (Also see Exercise 47 in Section 2.3.)
This implies that S = span (u, v , w) = span ( u, v) , and since u and v are certainly
linearly independent (why?), they form a basis for S. (Geometrically, this means that
u, v, and w all lie in the same plane and u and v can serve as a set of direction vectors
for this plane.)
Solution
�
4
Section
Exa m p l e 3 . 4 5
3. 5 Subspaces, Basis, Dimension, and Rank
199
Find a basis for the row space of
A=
Solution
[ -!
3
-1 0
2 1 -2
1 6
The reduced row echelon form of A is
0
1
0
0
1
2
0
0
0
0
1
0
By Theorem 3.20, row(A) = row(R), so it is enough to find a basis for the row space
of R. But row(R) is clearly spanned by its nonzero rows, and it is easy to check that
the staircase pattern forces the first three rows of R to be linearly independent. (This
is a general fact, one that you will need to establish to prove Exercise 33.) Therefore,
a basis for the row space of A is
{[1 0
0
- 1 ] , [0
1 2 0 3 ] , [0 0 0
4]}
We can use the method of Example 3.45 to find a basis for the subspace spanned
by a given set of vectors.
Exa m p l e 3 . 4 6
Rework Example 3.44 using the method from Example 3.45.
We transpose u, v, and w to get row vectors and then form a matrix with
these vectors as its rows:
Solution
[
!> ]
� � -!
Proceeding as in Example 3.45, we reduce B to its reduced row echelon form
1 0
and use the nonzero row vectors as a basis for the row space. Since we started with
column vectors, we must transpose again. Thus, a basis for span (u, v, w) is
{ [ H Ul l
R e m arks
In fact, we do not need to go all the way to reduced row echelon form-row ech­
elon form is far enough. If U is a row echelon form of A, then the nonzero row vectors
•
200
Chapter
3 Matrices
of U will form a basis for row(A) (see Exercise 33). This approach has the advantage of
(often) allowing us to avoid fractions. In Example 3.46, B can be reduced to
which gives us the basis
\HH - rn
for span (u, v, w) .
•
Observe that the methods used in Example 3.44, Example 3.46, and the Remark
above will generally produce different bases.
We now turn to the problem of finding a basis for the column space of a matrix A.
One method is simply to transpose the matrix. The column vectors of A become the
row vectors of A T' and we can apply the method of Example 3.45 to find a basis for
row(A T ) . Transposing these vectors then gives us a basis for col (A) . (You are asked to
do this in Exercises 2 1 -24.) This approach, however, requires performing a new set
of row operations on A T.
Instead, we prefer to take an approach that allows us to use the row reduced form
of A that we have already computed. Recall that a product Ax of a matrix and a vec­
tor corresponds to a linear combination of the columns of A with the entries of x as
coefficients. Thus, a nontrivial solution to Ax = 0 represents a dependence relation
among the columns of A. Since elementary row operations do not affect the solution
set, if A is row equivalent to R, the columns of A have the same dependence relation­
ships as the columns of R. This important observation is the basis (no pun intended!)
for the technique we now use to find a basis for col (A).
Exa m p l e 3 . 4 1
Find a basis for the column space of the matrix from Example 3.45,
A=
[-�
[�
1 3
1
1
-1 0
2
-2
6
1
-:i
Let a; denote a column vector of A and let r; denote a column vector of the
reduced echelon form
Solution
R�
0 1 0
1 2 0
0 0 1
0 0 0
-il
We can quickly see by inspection that r3 = r 1 + 2r2 and rs = - r 1 + 3r2 + 4r4 • (Check
that, as predicted, the corresponding column vectors of A satisfy the same depen­
dence relations.) Thus, r3 and rs contribute nothing to col (R) . The remaining column
Section
3. 5 Subspaces, Basis, Dimension, and Rank
201
vectors, r 1 , r2 , and r4 , are linearly independent, since they are just standard unit vec­
tors. The corresponding statements are therefore true of the column vectors of A.
Thus, among the column vectors ofA, we eliminate the dependent ones ( a3 and a5 ) ,
and the remaining ones will be linearly independent and hence form a basis for col(A).
What is the fastest way to find this basis? Use the columns of A that correspond to the
columns of R containing the leading ls. A basis for col(A) is
r . . . ., . .,
} � { [ - ;H - �H - m
Warning
Elementary row operations change the column space! In our example,
col (A) * col (R), since every vector in col (R) has its fourth component equal to 0 but
this is certainly not true of col (A) . So we must go back to the original matrix A to get
the column vectors for a basis of col (A) . To be specific, in Example 3.47, r 1 , r2 , and r4
do not form a basis for the column space of A.
Exa m p l e 3 . 4 8
Find a basis for the null space of matrix A from Example 3.47.
There is really nothing new here except the terminology. We simply have
to find and describe the solutions of the homogeneous system Ax = 0. We have al­
ready computed the reduced row echelon form R of A, so all that remains to be done
in Gauss-Jordan elimination is to solve for the leading variables in terms of the free
variables. The final augmented matrix is
0 1 0
2 0
0 0 1
0 0 0
Solution
If
then the leading l s are in columns 1 , 2, and 4, so we solve for x 1 , x2 , and x4 in terms of
the free variables x3 and x5 • We get x 1 = - x3 + x5 , x2 = - 2x3 - 3x5, and x4 = - 4x5.
Setting x3 = s and x5 = t, we obtain
-s + t
1
-1
X1
-3
- 2s - 3t
-2
X2
= s 1 + t 0 = su + tv
s
x = X3
0
-4t
-4
X4
0
X5
Thus, u and v span null(A ), and since they are linearly independent, they form a basis
for null(A).
4
202
Chapter
3 Matrices
Following is a summary of the most effective procedure to use to find bases for
the row space, the column space, and the null space of a matrix A.
1. Find the reduced row echelon form R of A.
2. Use the nonzero row vectors of R (containing the leading l s) to form a basis for
row(A) .
3. Use the column vectors o fA that correspond t o the columns o f R containing the
leading ls (the pivot columns) to form a basis for col (A) .
4. Solve fo r the leading variables o f Rx = 0 i n terms o f the free variables, set the
free variables equal to parameters, substitute back into x, and write the result as
a linear combination off vectors (where f is the number of free variables) . These
f vectors form a basis for null (A) .
I f we d o not need t o find the null space, then it i s faster t o simply reduce A t o row
echelon form to find bases for the row and column spaces. Steps 2 and 3 above remain
valid (with the substitution of the word "pivots" for "leading l s").
Dimension and Rank
We have observed that although a subspace will have different bases, each basis has
the same number of vectors. This fundamental fact will be of vital importance from
here on in this book.
Theorem 3 . 2 3
The Basis Theorem
Let S be a subspace of !R n . Then any two bases for S have the same number of
vectors.
Proof
Let B = {u 1 , u2 , . . . , u,.} and C = {v1 , v2 ,
, v, } be bases for S. We need to
prove that r = s. We do so by showing that neither of the other two possibilities, r < s
or r > s, can occur.
Suppose that r < s. We will show that this forces C to be a linearly dependent set
of vectors. To this end, let
.
Sherlock Holmes noted, "When
you have eliminated the impos­
sible, whatever remains, however
improbable, must be the truth"
(from The Sign of Four by Sir
Arthur Conan Doyle).
•
•
(1)
Since B i s a basis for S , we can write each V; a s a linear combination o f the
elements "f
V1 = a 1 1 u, + a 12 u2 + · · · + a , rur
Vz = a 1 1 u , + a z 2U2 + · · · + a zrUr
Substituting the Equations (2) into Equation ( 1 ) , we obtain
(2)
Section
3. 5 Subspaces, Basis, Dimension, and Rank
203
Regrouping, we have
( c 1 a ll + C2 a2 1 + . . . + c, as1 ) U1 + ( c 1 a 1 2 + C2 a22 + . . . + cs asz ) Uz
+ + ( c 1 a 1r + c2a2 r + + c, a ) u, = 0
Now, since B is a basis, the u/s are linearly independent. So each of the expressions in
·
·
·
·
·
·
parentheses must be zero:
c 1 a 11 + c2a2 1 + + c, a, 1 = 0
C 1 a 1 2 + C2 a22 + . . . + csas 2 = 0
·
·
·
This is a homogeneous system of r linear equations in the s variables c 1 c 2
, c5 (The
fact that the variables appear to the left of the coefficients makes no difference.) Since
r < s, we know from Theorem 2.3 that there are infinitely many solutions. In particu­
lar, there is a nontrivial solution, giving a nontrivial dependence relation in Equa­
tion ( 1 ) . Thus, C is a linearly dependent set of vectors. But this finding contradicts the
fact that C was given to be a basis and hence linearly independent. We conclude that
r < s is not possible. Similarly (interchanging the roles of B and C), we find that r > s
leads to a contradiction. Hence, we must have r = s, as desired.
,
, •
.
.
•
Since all bases for a given subspace must have the same number of vectors, we can
attach a name to this number.
If S is a subspace of !R n , then the number of vectors in a basis for S
is called the dimension of S, denoted dim S.
Definition
The zero vector 0 by itself is always a subspace of !R n . (Why?) Yet any set
containing the zero vector (and, in particular, { 0}) is linearly dependent, so { 0} cannot
have a basis. We define dim {O} to be 0.
Remark
Exa m p l e 3 . 4 9
Since the standard basis for !R n has n vectors, dim !R n = n . (Note that this result agrees
with our intuitive understanding of dimension for n :s 3.)
Exa m p l e 3 . 5 0
In Examples 3.45 through 3.48, we found that row(A) has a basis with three vectors,
col (A) has a basis with three vectors, and null (A) has a basis with two vectors. Hence,
dim(row(A)) = 3, dim(col (A)) = 3, and dim(null (A)) = 2.
A single example is not enough on which to speculate, but the fact that the row
and column spaces in Example 3.50 have the same dimension is no accident. Nor is
the fact that the sum of dim(col (A)) and dim(null (A)) is 5, the number of columns of
A. We now prove that these relationships are true in general.
204
Chapter
3 Matrices
Theorem 3 . 2 4
The row and column spaces of a matrix A have the same dimension.
Proof
Let R be the reduced row echelon form of A. By Theorem 3.20, row(A) =
row(R), so
dim(row (A)) = dim(row(R))
= number of nonzero rows of R
= number of leading l s of R
Let this number be called r.
Now col (A) * col (R), but the columns of A and R have the same dependence
relationships. Therefore, dim(col (A)) = dim(col (R)). Since there are r leading ls, R
has r columns that are standard unit vectors, e1, e2 , . . . , er. (These will be vectors in
!R m if A and R are m X n matrices.) These r vectors are linearly independent, and the
remaining columns of R are linear combinations of them. Thus, dim(col (R)) = r. It
follows that dim(row(A)) = r = dim(col (A)), as we wished to prove.
1878
(1849-1917),
The rank o f a matrix was first de­
fined in
by Georg Frobenius
although he defined
it using determinants and not as we
have done here. (See Chapter
Frobenius was a German
mathematician who received his
doctorate from and later taught
at the University of Berlin. Best
known for his contributions to
group theory, Frobenius used
matrices in his work on group
representations.
4.)
The rank of a matrix A is the dimension of its row and column
spaces and is denoted by rank(A) .
Definition
For Example 3.50, we can thus write rank(A) = 3.
R e m a rks
The preceding definition agrees with the more informal definition of rank that
was introduced in Chapter 2. The advantage of our new definition is that it is much
more flexible.
•
The rank of a matrix simultaneously gives us information about linear
dependence among the row vectors of the matrix and among its column vectors. In
particular, it tells us the number of rows and columns that are linearly independent
(and this number is the same in each case!).
•
Since the row vectors of A are the column vectors of A r, Theorem 3.24 has the
following immediate corollary.
Theorem 3 . 2 5
For any matrix A,
rank (A T) = rank (A )
Proof
We have
rank (A T) = dim ( col (A T))
= dim ( row (A ))
= rank (A )
The nullity of a matrix A is the dimension of its null space and is
denoted by nullity(A) .
Definition
Section
3. 5 Subspaces, Basis, Dimension, and Rank
205
In other words, nullity(A) is the dimension of the solution space of Ax = 0, which
is the same as the number of free variables in the solution. We can now revisit the
Rank Theorem (Theorem 2.2), rephrasing it in terms of our new definitions.
Theorem 3 . 2 6
The Rank Theorem
If A is an m X n matrix, then
rank (A ) + nullity (A ) = n
-
Proof
Let R be the reduced row echelon form of A, and suppose that rank(A) = r.
Then R has r leading ls, so there are r leading variables and n r free variables in the
solution to Ax = 0. Since dim(null (A)) = n r, we have
-
rank (A ) + nullity (A ) = r + (n
= n
-
r)
Often, when we need to know the nullity of a matrix, we do not need to know the
actual solution of Ax = 0. The Rank Theorem is extremely useful in such situations,
as the following example illustrates.
Exa m p l e 3 . 5 1
Find the nullity of each of the following matrices:
M=
N
[I �: -
� [�
and
1
4
7
-2
-3
:1
Since the two columns of M are clearly linearly independent, rank(M ) = 2.
Thus, by the Rank Theorem, nullity(M ) = 2 - rank(M ) = 2 - 2 = 0.
There is no obvious dependence among the rows or columns of N, so we apply
row operations to reduce it to
Solution
[:
2
0
-2
1
0
-� ]
We have reduced the matrix far enough (we do not need reduced row echelon form
here, since we are not looking for a basis for the null space). We see that there are only
two nonzero rows, so rank(N) = 2. Hence, nullity(N) = 4 rank(N) = 4 2 = 2.
-
-
4
The results of this section allow us to extend the Fundamental Theorem of
Invertible Matrices (Theorem 3 . 1 2 ) .
206
Chapter 3 Matrices
Theorem 3 . 2 1
The Fundamental Theorem of Invertible Matrices: Version 2
Let A be an n X n matrix. The following statements are equivalent:
The
of a matrix
wasSyldefined
in(1814-1887),
1884nullbyityJames
Joseph
vesterin
who
was
i
n
terested
-properties
of certain
matrices
that
do
not
change
under
types
of transformations.
Bornthe
insecond
England,
Syl
v
ester
became
presidentSociety.
of theInLondon
Mathematical
1878,
whi
l
e
teaching
at
Johns
Hopkins
University
imore, he
founded thein Balthetfirst
mathematical
journal in the United States.
invariants
American Journal of
Mathematics,
a. A is invertible.
b. Ax = b has a unique solution for every b in ll�r.
c. Ax = 0 has only the trivial solution.
d. The reduced row echelon form of A is In e. A is a product of elementary matrices.
f. rank(A) = n
g. nullity(A) = 0
h. The column vectors of A are linearly independent.
i. The column vectors of A span ll�r.
j. The column vectors of A form a basis for ll�r.
k. The row vectors of A are linearly independent.
1. The row vectors of A span !R n .
m. The row vectors of A form a basis for !R n .
We have already established the equivalence of (a) through (e). It remains to
be shown that statements (f) to (m) are equivalent to the first five statements.
(f) � (g) Since rank(A) + nullity(A) = n when A is an n X n matrix, it follows from
the Rank Theorem that rank(A) = n if and only if nullity(A) = 0.
(f) ==> (d) ==> ( c) ==> (h) If rank(A) = n , then the reduced row echelon form of A has
n leading ls and so is I - From (d) ==> (c) we know that Ax = 0 has only the trivial
n
solution, which implies that the column vectors of A are linearly independent, since
Ax is just a linear combination of the column vectors of A.
(h) ==> (i) If the column vectors of A are linearly independent, then Ax = 0 has only
the trivial solution. Thus, by (c) ==> (b), Ax = b has a unique solution for every b in
!R n . This means that every vector b in !R n can be written as a linear combination of the
column vectors of A, establishing (i).
(i) ==> (j) If the column vectors of A span W, then col (A) = !R n by definition,
so rank(A) = dim(col (A)) = n. This is (f), and we have already established that
(f) ==> (h) . We conclude that the column vectors of A are linearly independent and so
form a basis for !R n , since, by assumption, they also span !R n .
(j) ==> (f) If the column vectors of A form a basis for !R n , then, in particular, they are
linearly independent. It follows that the reduced row echelon form of A contains n
leading l s, and thus rank(A) = n .
The above discussion shows that (f) ==> (d) ==> (c) ==> (h) ==> (i) ==> (j) ==>
(f) � (g) . Now recall that, by Theorem 3.25, rank(A T ) = rank(A), so what we have
just proved gives us the corresponding results about the column vectors of A r_ These
are then results about the row vectors of A, bringing (k), (1), and (m) into the network of
equivalences and completing the proof.
Proof
Theorems such as the Fundamental Theorem are not merely of theoretical inter­
est. They are tremendous labor-saving devices as well. The Fundamental Theorem
has already allowed us to cut in half the work needed to check that two square matri­
ces are inverses. It also simplifies the task of showing that certain sets of vectors are
bases for !R n . Indeed, when we have a set of n vectors in !R n , that set will be a basis for
!R n if either of the necessary properties of linear independence or spanning set is true.
The next example shows how easy the calculations can be.
Section 3. 5 Subspaces, Basis, Dimension, and Rank
Exa m p l e 3 . 5 2
Show that the vectors
201
m
n
mr
and
form a basis for IR 3 .
Solution
According to the Fundamental Theorem, the vectors will form a basis for
IR 3 if and only if a matrix with these vectors as its columns (or rows) has rank 3. We
[� � � f l --7 [ � - � J
perform just enough row operations to determine this:
A
-
We see that A has rank 3, so the given vectors are a basis for IR 3 by the equivalence of
(f) and (j) .
4
The next theorem is an application of both the Rank Theorem and the Funda­
mental Theorem. We will require this result in Chapters 5 and 7.
Theorem 3 . 2 8
Let A be an m X n matrix. Then:
a. rank(A TA) = rank(A)
b. The n X n matrix A TA is invertible if and only if rank(A) = n .
Proof
(a) Since A TA is n X n , it has the same number of columns as A. The Rank Theorem
then tells us that
rank (A ) + nullity (A ) = n = rank (A TA ) + nullity (A TA )
Hence, to show that rank(A) = rank(A TA), it is enough to show that nullity(A) =
nullity(A TA). We will do so by establishing that the null spaces of A and A TA are the
same.
To this end, let x be in null (A) so that Ax = 0. Then A TAx = A To = 0, and thus
x is in null (A TA). Conversely, let x be in null (A TA). Then A TAx = 0, so xTA TAx =
xTO = 0. But then
and hence Ax = 0, by Theorem 1 .2 (d) . Therefore, x is in null (A), so null (A)
null (A TA), as required.
(b) By the Fundamental Theorem, the n X n matrix A TA is invertible if and only if
rank(A TA) = n. But, by (a) this is so if and only if rank(A) = n .
Coordin ates
We now return to one of the questions posed at the very beginning of this section:
How should we view vectors in IR 3 that live in a plane through the origin? Are they
two-dimensional or three-dimensional? The notions of basis and dimension will help
clarify things.
208
Chapter 3 Matrices
A plane through the origin is a two-dimensional subspace o f IR 3 , with any set of
two direction vectors serving as a basis. Basis vectors locate coordinate axes in the
plane/subspace, in turn allowing us to view the plane as a "copy" of IR 2 • Before we
illustrate this approach, we prove a theorem guaranteeing that "coordinates" that arise
in this way are unique.
Theorem 3 . 2 9
Let S be a subspace of !R n and let B = {v1 , v2 , . . . , vk } be a basis for S. For every
vector v in S, there is exactly one way to write v as a linear combination of the basis
vectors in B:
Proof
Since B is a basis, it spans S, so v can be written in at least one way as a linear
combination of v1, v2 , . . . , vk . Let one of these linear combinations be
Our task is to show that this is the only way to write v as a linear combination of
v1, v2 , . . . , vk . To this end, suppose that we also have
v = d1 v1 + d2v2 + · · · + dkvk
Then
Rearranging (using properties of vector algebra), we obtain
(c 1 - d 1 )v1 + (c2 - d2 )v2 + · · · + (ck - dk ) vk = 0
Since B is a basis, v1, v2 , . . . , vk are linearly independent. Therefore,
(c 1 - d1 ) = (c2 - d2 ) = · · · = (ck - dk ) = 0
In other words, c1 = d1, c 2 = d2 ,
, ck = dk, and the two linear combinations are
actually the same. Thus, there is exactly one way to write v as a linear combination of
•
•
•
the basis vectors in B.
Let S be a subspace of !R n and let B = {v1 , v2 , . . . , vk } be a basis for
S. Let v be a vector in S, and write v = c1v1 + c2v2 + · · · + ckvk . Then c1, c2 , . . . , ck
are called the coordinates of v with respect to B, and the column vector
Definition
is called the coordinate vector of v with respect to B.
Exa m p l e 3 . 5 3
Let E = { e 1 , e2 , e 3 } be the standard basis for IR 3 . Find the coordinate vector of
with respect to E.
Section 3. 5 Subspaces, Basis, Dimension, and Rank
Solution
Since v =
209
2e1 7e 4e 2
[ 47 ]
+ 2 + 3,
[v] E =
It should be clear that the coordinate vector of every (column) vector in !R n with
respect to the standard basis is just the vector itself.
Exa m p l e 3 . 5 4
ID Exompk 3.44, we <aw iliat u �
2u
[ -} m
[ _�]
�
and w �
[ ;l
-
ace th"e v"
tors in the same subspace (plane through the origin) S of IR 3 and that B = { u, v} is a
basis for S. Since w =
- 3v, we have
[w] B =
See Figure 3.3.
z
- 3v
,w
=
2u
-
3v
x ---.
The
respectcoordinates
to a basisof a vector with
Figure 3 . 3
I
y
Exercises 3 . 5
In Exercises 1 -4, let S be the collection of vectors
[; ]
in IR 2
that satisfy the given property. In each case, either prove that
S forms a subspace of IR 2 or give a counterexample to show
that it does not.
1. x =
2. x O, y
3. y = 2x
4. xy
0
2 20
20
In Exm;,,, 5-8, let S be the collea;on of vato"
[;]
;n D;l'
7. x - y + z = 1
9. Prove that every line through the origin in IR 3 is a sub­
space of IR 3 .
10. Suppose S consists of all points in IR 2 that are on the
x-axis or the y-axis (or both). (S is called the union of
the two axes.) Is S a subspace of IR 2 ? Why or why not?
In Exercises 1 1 and 12, determine whether b is in col(A)
and whether w is in row(A), as in Example 3.4 1 .
11. A =
b=
,w = [-1
1]
that satisfy the given property. In each case, either prove that
S forms a subspace of IR 3 or give a counterexample to show
that it does not.
5. x = y = z
6. z = 2x, y =
0
8. I x - y I = I Y - z I
12. A �
[ � � � l [�]
[ � -� � } [i} 2 4
�
w� [
-5
Chapter 3 Matrices
210
13. In Exercise 1 1 , determine whether w is in row(A),
using the method described in the Remark following
Example 3.4 1 .
14. In Exercise 1 2 , determine whether w is in row(A),
using the method described in the Remark following
Example 3.4 1 .
1 5 . IfA ;, tho mateix in F.xmi" 1 1 , b �
1 6 . I fA i' 1he mateix in Exmi" 12, i< v �
[ �: }
[ -:}
n null (A) ?
n nu!l(A) ?
In Exercises 1 7-20, give bases for row(A), col(A), and null(A).
17. A =
19. A =
20. A =
[� -�]
[
[� J
!]
H
1
18. A = 0
1
0
1
1
1
1
2
-1
-3
1
-4
l
0
-1
-1
-4 0 2
2
2
4
-2
In Exercises 21-24, find bases for row(A) and col(A) in the
given exercises using A T.
30. [ 0 1
-2
1
1 ], [3
- 1 0], [2
1 5
1]
For Exercises 31 and 32, find bases for the spans of the
vectors in the given exercises from among the vectors
themselves.
3 1 . Exercise 29
32. Exercise 30
33. Prove that if R is a matrix in echelon form, then a basis
for row(R) consists of the nonzero rows of R.
34. Prove that if the columns of A are linearly indepen­
dent, then they must form a basis for col (A) .
For Exercises 35-38, give the rank and the nullity of the
matrices in the given exercises.
35. Exercise 1 7
36. Exercise 1 8
37. Exercise 1 9
38. Exercise 2 0
39. I f A i s a 3 X 5 matrix, explain why the columns o f A
must be linearly dependent.
40. If A is a 4 X 2 matrix, explain why the rows of A must
be linearly dependent.
41. If A is a 3 X 5 matrix, what are the possible values of
nullity(A) ?
42. I f A i s a 4 X 2 matrix, what are the possible values of
nullity(A) ?
l
In Exercises 43 and 44, find all possible values of rank(A) as
a varies.
2 a
[ � � =�l
2 1 . Exercise 1 7
22. Exercise 1 8
23. Exercise 1 9
24. Exercise 20
25. Explain carefully why your answers to Exercises 1 7
and 2 1 are both correct even though there appear to be
differences.
26. Explain carefully why your answers to Exercises 1 8
and 2 2 are both correct even though there appear to
be differences.
Answer Exercises 45-48 by considering the matrix with the
given vectors as its columns.
In Exercises 27-30, find a basis for the span of the given
vectors.
46. Do
27.
[ - il [ - � J [ J [ - :J m. [ : J m
29. [ 2
28.
-3
1], [1
- 1 O ] , [4 -4 1 ]
4a 2
-2 1
45. Do
47. Do
44. A =
[J [ H [ : J
[ -J [ - n [ -1 ]
-2
-1
form a b"i' foe � ' ?
rn
ril m m
focm a bM1' fm � ' ?
focm a bMi' foe � ' '
a
Section 3. 6 Introduction to Linear Transformations
49. Do
50. Do
m [:J m
m [:J [�l
form a b.,i, fo, Zl?
fmm a bn<i• fo, Zj?
In Exercises 51 and 52, show that w is in span(B) andfind
the coordinate vector [w] 8.
In Exercises 53-56, compute the rank and nullity of the
given matrices over the indicated "ll..P "
53.
55.
56.
[ � :J
[� �]
1
1
0
ov°' z,
3 1
3 0
0 4
[!
4 0 0
3 5 1
0 2 2
1
54.
om Z;
�]
[ � �]
1
1
0
om Z,
211
57. If A is m X n, prove that every vector in null (A) is
orthogonal to every vector in row(A) .
58. I f A and B are n X n matrices o f rank n, prove that AB
has rank n.
59. (a) Prove that rank(AB) :::::: rank(B) . [Hint: Review
Exercise 29 in Section 3 . 1 . ]
(b) Give an example in which rank(AB) < rank(B).
60. ( a) Prove that rank(AB) :::::: rank(A). [Hint: Review
Exercise 30 in Section 3 . 1 or use transposes and
Exercise 59(a).]
(b) Give an example in which rank(AB) < rank(A) .
61. ( a) Prove that i f U i s invertible, then rank( UA) =
rank(A) . [Hint: A = U- 1 ( UA).]
(b) Prove that if V is invertible, then rank(A V) =
rank(A) .
62. Prove that an m X n matrix A has rank 1 i f and only if
A can be written as the outer product uvT of a vector u
in !Rm and v in !R n .
63. If an m X n matrix A has rank r, prove that A can be
written as the sum of r matrices, each of which has
rank 1. [Hint: Find a way to use Exercise 62.]
64. Prove that, for m X n matrices A and B, rank (A + B) ::::::
rank(A) + rank(B).
65. Let A be an n X n matrix such that A 2 = 0. Prove that
rank(A) :::::: n/2. [Hint: Show that col(A) � null(A) and
use the Rank Theorem.]
66. Let A be a skew-symmetric n X n matrix.
(See page 1 6 2 ) .
( a) Prove that xT Ax = 0 for all x in !R n .
(b) Prove that I + A is invertible. [Hint: Show that
null(I + A) = {O}.]
om Z,
0
I n t ro d u ct i o n to l i n e a r Tra n s f o r m a t i o n s
In this section, we begin to explore one o f the themes from the introduction t o this
chapter. There we saw that matrices can be used to transform vectors, acting as a type
of "function'' of the form w = T (v) , where the independent variable v and the de­
pendent variable w are vectors. We will make this notion more precise now and look
at several examples of such matrix transformations, leading to the concept of a linear
transformation- a powerful idea that we will encounter repeatedly from here on.
212
Chapter 3 Matrices
We begin by recalling some of the basic concepts associated with functions. You will
be familiar with most of these ideas from other courses in which you encountered func­
tions of the form f: IR ---+ IR [such as f(x) = x 2 ] that transform real numbers into real num­
bers. What is new here is that vectors are involved and we are interested only in functions
that are "compatible" with the vector operations of addition and scalar multiplication.
Consider an example. Let
Then
Thi, ,hows that A tmmfmmn into w �
[ _ :J
We can describe this transformation more generally. The matrix equation
[ � - � l [; ] [ � l
=
3
4
2x y
3x + 4y
gives a formula that shows how A transforms an arbitrary vector
vectm
[ "_ ]
[; ]
in IR 2 into the
2x y in D;l'. We denote this tmmfmmotion by T, and w'ite
3x + 4y
x � 2x �
y
3x + 4y
Y
r,( [ ] ) [ l
(Although technically sloppy, omitting the parentheses in definitions such as this one
is a common convention that saves some writing. The description of TA becomes
with this convention.)
With this example in mind, we now consider some terminology. A transformation
(or mapping or function) T from !R n to !R m is a rule that assigns to each vector v in !R n
a unique vector T (v) in !R m . The domain of T is !R n , and the codomain of T is !R m . We
indicate this by writing T : !R n ---+ !R m . For a vector v in the domain of T, the vector T (v)
in the co domain is called the image of v under (the action of) T. The set of all possible
images T (v) (as v varies throughout the domain of T) is called the range of T.
In our example, the domain of TA is IR 2 and its codomain is IR 3 , so we write
T,, D;l' --. D;l'. The image of v �
[_:J
is w � T, ( v) �
[ _iJ
What is the cange of
Section 3. 6 Introduction to Linear Transformations
213
TA? It consists of all vectors in the codomain IR 3 that are of the form
TA
�
[ -�i
[ ] [ � ] [�] [ �]
x
y
2x y = x
+y 3x + 4y
3
4
=
whi<h d mib" the "t of •ll linm wmbimtiom of the wlumn ve<to"
and
[�]
of A. In othe; wo'd', the 'ange of T IB the wlumn 'P"'e of A ! (We
will have more to say about this later-for now we'll simply note it as an interesting
observation.) Geometrically, this shows that the range of TA is the plane through the
origin in IR 3 with direction vectors given by the column vectors of A. Notice that the
range of TA is strictly smaller than the co domain of TA.
linear Tra nsformations
The example TA above is a special case of a more general type of transformation called
a linear transformation. We will consider the general definition in Chapter 6, but the
essence of it is that these are the transformations that "preserve" the vector operations
of addition and scalar multiplication.
D e fi n it i o n
A transformation T : !R n ---+ !R m is called a linear transformation if
1 . T (u + v) = T (u) + T (v) for all u and v in !R n and
2. T(cv) = cT (v) for all v in !R n and all scalars c.
Exa m p l e 3 . 5 5
Consider once again the transformation T : IR 2 ---+ IR 3 defined by
r[;] � [ :: ; � l
[;: J [;: J
Let's check that T is a linear transformation. To verify ( 1 ) , we let
u=
Then
and v =
214
Chapter 3 Matrices
To show (2 ), we let v =
[; ]
T( cv) =
and let c b e a scalar. Then
r(c [;] ) r( [ :;] )
:x__
c�
[ 32((cx)cx) 4(cy)(cy) ] [ cc((3x2 4y)y) ]
=
=
+
+
=
cT(v)
Thus, T is a linear transformation.
Remark
The definition of a linear transformation can be streamlined by com­
bining ( 1 ) and (2 ) as shown below.
T : !R n � !R m is a linear transformation if
In Exercise 53 , you will be asked to show that the statement above is equivalent
to the original definition. In practice, this equivalent formulation can save some
writing-try it!
Although the linear transformation T in Example 3.55 originally arose as a matrix
transformation TA, it is a simple matter to recover the matrix A from the definition of
T given in the example. We observe that
rn
T � TA, whm A �
[: � J
-
(N otke that when the vmiahles x ond y "' J;ned
up, the matrix A is just their coefficient matrix.)
Recognizing that a transformation is a matrix transformation is important, since,
as the next theorem shows, all matrix transformations are linear transformations.
Theorem 3 . 3 0
Let A be an m X n matrix. Then the matrix transformation TA : !R n � !R m defined by
is a linear transformation.
Section 3. 6 Introduction to Linear Transformations
Proof
215
Let u and v be vectors in !R n and let c be a scalar. Then
and
TA (cv) = A(cv) = c (Av) = cTA (v)
Hence, TA is a linear transformation.
Exa m p l e 3 . 5 6
Let F : IR 2 ---+ IR 2 be the transformation that sends each point to its reflection in the
x-axis. Show that F is a linear transformation.
From Figure 3.4, it is clear that F sends the point (x, y) to the point (x, -y).
Thus, we may write
Solution
y
( 1 , 2)
T
(x, y)
T
I
I
I
�+----+1�--+-��
:I 1--• x
I
I
I
I
We could proceed to check that F is linear, as in Example 3.55 (this one is even easier
to check! ), but it is faster to observe that
I
I
•
•
(x, - y )
( 1 , - 2)
Therefore, F
Reflection in the x-axis
Figure 3 . 4
[; ] [; ]
= A
, where A =
[ � � ],
_
so F is a matrix transformation. It
now follows, by Theorem 3.30, that F is a linear transformation.
Exa m p l e 3 . 5 1
Let R : IR 2 ---+ IR 2 be the transformation that rotates each point 90° counterclockwise
about the origin. Show that R is a linear transformation.
Solution
we have
As Figure 3.5 shows, R sends the point (x, y) to the point ( - y, x). Thus,
Hence, R is a matrix transformation and is therefore linear.
-y
90° rotation
Figure 3 . 5
A
x
Observe that if we multiply a matrix by standard basis vectors, we obtain the col­
umns of the matrix. For example,
We can use this observation to show that every linear transformation from !R n to
!R m arises as a matrix transformation.
216
Chapter 3 Matrices
Theorem 3 . 3 1
Let T : !R n � !R m b e a linear transformation. Then T is a matrix transformation.
More specifically, T = TA > where A is the m X n matrix
Let e1, e2 ,
, en be the standard basis vectors in !R n and let x be a vector
n
in !R . We can write x = x1e1 + x 2 e2 +· · · + xn e n (where the x/s are the components
of x) . We also know that T(e1), T(e2 ),
, T(e n ) are (column) vectors in !R m . Let A =
[ T(e1) : T(e2 ) : · : T(en )l be the m X n matrix with these vectors as its columns.
Then
Proof
•
•
.
•
•
.
•
•
T ( x) = T(x1 e 1 + X 2 e 2 + . . . + xn en )
= x1 T(e 1 ) + x2 T(e2 ) + · · · + Xn T(en )
[ T(e , ) T(e, ) : · · · T(e,) ]
�
as required.
[}]
� Ax
The matrix A in Theorem 3.31 is called the standard matrix of the linear trans­
formation T.
Exa m p l e 3 . 5 8
,......
Show that a rotation about the origin through an angle e defines a linear transforma­
tion from IR 2 to IR 2 and find its standard matrix.
Solulion
Let Re be the rotation. We will give a geometric argument to establish
the fact that R0 is linear. Let u and v be vectors in IR 2 • If they are not parallel, then
Figure 3.6(a) shows the parallelogram rule that determines u + v. If we now apply R0 ,
the entire parallelogram is rotated through the angle e, as shown in Figure 3.6(b ). But the
diagonal of this parallelogram must be R0(u) + R0(v), again by the parallelogram rule.
Hence, R0(u + v) = R0(u) + R0(v) . (What happens if u and v are parallel?)
y
y
u+v
r
(a)
-
I
u
I
I
I
Re ( v)'
x
x
(b)
Figure 3 . 6
,......
Similarly, if we apply Re to v and cv, we obtain R0(v) and Re(cv) , as shown
in Figure 3.7. But since the rotation does not affect lengths, we must then have
R8 ( cv) = cR8 ( v) , as required. (Draw diagrams for the cases 0 < c < 1 , - 1 < c < 0,
and c < - 1 .)
Section 3. 6 Introduction to Linear Transformations
211
y
y
'--_,-'
cos 8
CV
Re (cv)
Figure 3 . 1
( 1 , 0)
R 0(e1)
Figure 3 . 8
Therefore, Re is a linear transformation. According to Theorem 3.3 1 , we can find
its matrix by determining its effect on the standard basis vectors e1 and e2 of IFR 2 • Now,
1
c s0
as Figure 3.8 shows, R 0
= � .
0
sm ()
We can find Re
[] [ ]
[�]
[]
similarly, but it is faster to observe that Re
[�]
[] [ ]
must be per-
1
0
- sin ()
=
pendicular (counterclockwise) to R0
and so, by Example 3.57, R8
.
1
cos ()
0
(F 1gure 3 . 9) .
Therefore, the standard matrix of Re is
[
cos ()
. ()
sm
]
- sin ()
.
cos ()
y
R e(ez)
Figure 3 . 9
The result of Example 3.58 can now be used to compute the effect of any rota­
tion. For example, suppose we wish to rotate the point (2, - 1 ) through 60° about the
origin. (The convention is that a positive angle corresponds to a counterclockwise
Chapter 3 Matrices
218
rotation, while a negative angle is clockwise.) Since cos 60° = 1 /2 and sin 60° =
'\/3/2, we compute
y
R60
[] [
cos 60°
2
=
-1
sin 60°
- sin 60°
cos 60°
(2, I )
-
60° rotation
1 /2
'\/3/2
- '\/3/2
1 /2
( 2 + '\/3) /2
( 2 '\/3 - 1 ) /2
]
][ ]
2
-1
=
Exa m p l e 3 . 5 9
(a) Show that the transformation P : IR 2 ---+ IR 2 that projects a point onto the x-axis is
a linear transformation and find its standard matrix.
(b) More generally, if e is a line through the origin in IR 2 , show that the transforma­
tion Pe : IR 2 ---+ IR 2 that projects a point onto e is a linear transformation and find its
standard matrix.
Solution
y
(a) As Figure 3. 1 1 shows, P sends the point (x, y) to the point (x, O). Thus,
It follows that P is a matrix transformation (and hence a linear transformation) with
(x, y )
T
-+------+---+ x
(x, 0)
projection
Figure 3 . 1 1
A
2
-1
Thus, the image of the point (2, - 1 ) under this rotation is the point ((2 + '\/3)/2,
(2'\/3 - 1 )/2) ( 1 .87, 1 .23), as shown in Figure 3.10.
Figure 3 . 1 0
A
][ ] [
[
standard matrix
[� �] .
(b) Let the line e have direction vector d and let v be an arbitrary vector. Then Pe is
given by projd(v), the projection of v onto d, which you'll recall from Section 1 .2 has
the formula
Thus, to show that Pe is linear, we proceed as follows:
Pe( u + v) =
=
=
=
( � )
( )
( )
( ) ( )
d · (u + v)
d
d·d
d·v
d·u
d
d d
d·u
-
d·d
+
d·v
-
d·d
d·u
d +
d·d
d
d•v
d = Pe(u) + Pe(v)
d·d
Similarly, Pe(cv) = cPe(v) for any scalar c (Exercise 52). Hence, Pe is a linear
transformation.
Section 3. 6 Introduction to Linear Transformations
To find the standard matrix of Pe, we apply Theorem 3 .3 1 . If we let d
=
219
[�: ] ,
then
and
Thus, the standard matrix of the projection is
d 1 d2
d�
U (d f + d D d 1 d2/(d i + d i ) ]
] [ ddd2/(d?
+ d i ) d U (dl + d i )
=
1
As a check, note that in part (a) we could take d
x-axis. Therefore, d1
=
1 and d2
=
=
0, and we obtain A
e1 as a direction vector for the
=
[� �] ,
4
as before.
New Linear Tra nsformations from Old
If T : !R m � !R n and S : !R n � [RP are linear transformations, then we may follow T by
S to form the composition of the two transformations, denoted S T. Notice that, in
order for S T to make sense, the codomain of T and the domain of S must match
(in this case, they are both !R n ) and the resulting composite transformation S T goes
from the domain of T to the codomain of S (in this case, S T : !R m � [RP ). Figure 3 . 1 2
shows schematically how this composition works. The formal definition o f composi­
tion of transformations is taken directly from this figure and is the same as the cor­
responding definition of composition of ordinary functions:
0
0
0
0
( S o T ) (v)
=
S ( T ( v))
Of course, we would like S T to be a linear transformation too, and happily we
find that it is. We can demonstrate this by showing that S T satisfies the definition of
a linear transformation (which we will do in Chapter 6), but, since for the time being
we are assuming that linear transformations and matrix transformations are the same
thing, it is enough to show that S T is a matrix transformation. We will use the nota­
tion [ T ] for the standard matrix of a linear transformation T.
0
0
0
S
T
v •�•�• S(T(v)) (S T)(v)
T(v)
�m
�n
�P
=
The composition of transformations
Figure 3 . 1 2
0
220
Chapter 3 Matrices
Theorem 3 . 3 2
Let T : !R m ---+ !R n and S : !R n ---+ !RP b e linear transformations. Then S T : !R m ---+ !RP
is a linear transformation. Moreover, their standard matrices are related by
0
[S TJ = [SJ [ TJ
0
Let [S J = A and [ T J = B. (Notice that A is p X n and B is n X m.) If v is a vector
in !R m , then we simply compute
Proof
( S T ) (v) = S ( T ( v)) = S ( Bv) = A ( Bv) = (AB ) v
0
.......
(Notice here that the dimensions of A and B guarantee that the product AB makes
sense.) Thus, we see that the effect of S T is to multiply vectors by AB, from which
it follows immediately that S T is a matrix (hence, linear) transformation with
[S T J = [S J [ T J .
0
0
0
Isn't this a great result? Say it in words: "The matrix of the composite is the prod­
uct of the matrices:' What a lovely formula!
Exa m p l e 3 . 6 0
Consider the linear transformation T : IR 2 ---+ IR 3 from Example 3.55, defined by
r [:: J [ 3x1 + ]
=
x
2x 1 � x2
4X2
and the linear transformation S : IR 3 ---+ IR 4 defined by
Solulion
We see that the standard matrices are
[SJ
so Theorem 3.32 gives
[S o TJ
_ -l
[! � �
( S a T)
ITI
[ � -: l
: -�
i
[� m� � [ ]
+
[:; ] +
[ :J
[ -1 -l l [ '" + �,l
[SJ [ TJ
It follows that
and
0
3
-1
-1
6
3x 1 - 7x2
- xi X2
6x 1 3x2
1
3
Section 3. 6 Introduction to Linear Transformations
221
(In Exercise 29, you will be asked to check this result by setting
[;:] r[::J [
=
=
y3
x
2x 1 � x2
3x, + 4x2
]
and substituting these values into the definition of S, thereby calculating ( S T)
dice<tly.)
0
Exa m p l e 3 . 6 1
x,
.:+
Find the standard matrix of the transformation that first rotates a point 90° counter­
clockwise about the origin and then reflects the result in the x-axis.
[ 1 -1O ] .
3.56, respectively, where we found their standard matrices to be [ R ] =
[F] =
[ � �]
The rotation R and the reflection F were discussed in Examples 3.57 and
Solution
0
-
and
It follows that the composition F R has for its matrix
0
[F a R ] = [ F ] [R ] =
ii--""--
[]
[ 1 -1o ] [ o1 1 ] [ -1o 1 ]
-
0
0
-
=
0
(Check that this result is correct by considering the effect of F R on the standard
basis vectors e1 and e 2 . Note the importance of the order of the transformations:
R is performed before F, but we write F R. In this case, R F also makes sense. Is
R o F = F o R?)
0
0
°
4
Inverses of linear Transformations
Consider the effect of a 90° counterclockwise rotation about the origin followed by
a 90° clockwise rotation about the origin. Clearly this leaves every point in IR 2 un­
changed. If we denote these transformations by R90 and R_90 (remember that a nega­
tive angle measure corresponds to clockwise direction), then we may express this
as ( R90 R_90) (v) = v for every v in IR 2 . Note that, in this case, if we perform the
transformations in the other order, we get the same end result: (R_90 R9 0) (v) = v
for every v in IR 2 •
Thus, R9 0 R_9 0 (and R_9 0 R9 0 too) is a linear transformation that leaves every
vector in IR 2 unchanged. Such a transformation is called an identity transformation.
Generally, we have one such transformation for every !R n -namely, I : !R n � !R n such
that I(v) = v for every v in !R n . (If it is important to keep track of the dimension of the
space, we might write In for clarity.)
So, with this notation, we have R90 R_90 = I = R_90 R90. A pair of transforma­
tions that are related to each other in this way are called inverse transformations.
°
°
°
°
°
°
Let S and T be linear transformations from !R n to !R n . Then S and T
are inverse transformations if S T = In and T S = In -
D e fi n i t i o n
0
0
222
Chapter 3 Matrices
Remark Since this definition is symmetric with respect to S and T, we will say
that, when this situation occurs, S is the inverse of T and T is the inverse of S. Further­
more, we will say that S and T are invertible.
�
In terms of matrices, we see immediately that if S and T are inverse transformations, then [S J [ T J = [S T J = [ JJ = I, where the last I is the identity matrix. (Why
is the standard matrix of the identity transformation the identity matrix?) We
must also have [ T ] [S J = [ T S J = [ JJ = I. This shows that [S J and [ T J are inverse
matrices. It shows something more: If a linear transformation T is invertible, then its
standard matrix [ T J must be invertible, and since matrix inverses are unique, this
means that the inverse of T is also unique. Therefore, we can unambiguously use the
notation r - 1 to refer to the inverse of T. Thus, we can rewrite the above equations as
1
1
1
[ T J [ T - J = I = [ T - J [ T J , showing that the matrix of r - is the inverse matrix of [ T J .
We have just proved the following theorem.
0
0
Theorem 3 . 3 3
Let T : !R n ---+ !R n b e an invertible linear transformation. Then its standard matrix
[ T J is an invertible matrix, and
Remark Say this one in words too: "The matrix of the inverse is the inverse of
the matrix:' Fabulous!
Exa m p l e 3 . 6 2
Find the standard matrix of a 60° clockwise rotation about the origin in IR 2 •
Earlier we computed the matrix of a 60° counterclockwise rotation about
the origin to be
- V3/2
1 /2
Solulion
]
Since a 60° clockwise rotation is the inverse of a 60° counterclockwise rotation, we can
apply Theorem 3.33 to obtain
)-1 1 /2 V3/2
1 /2 - V3/2 - 1
[ R - 6o J - [ ( R6o J 1 /2
1 /2
- V3/2
V3/2
[
lt--l'--
Exa m p l e 3 . 6 3
] [
]
(Check the calculation of the matrix inverse. The fastest way is to use the 2 X 2 short­
cut from Theorem 3.8. Also, check that the resulting matrix has the right effect on the
standard basis in IR 2 by drawing a diagram.)
4
Determine whether projection onto the x-axis is an invertible transformation, and if
it is, find its inverse.
Solulion
The standard matrix of this projection P is
[� �] ,
since its determinant is 0. Hence, P is not invertible either.
which is not invertible
4
Section 3. 6 Introduction to Linear Transformations
223
Remark Figure 3 . 1 3 gives some idea why P in Example 3.63 is not invertible. The
projection "collapses" IR 2 onto the x-axis. For P to be invertible, we would have to have
a way of "undoing" it, to recover the point ( a, b) we started with. However, there are
infinitely many candidates for the image of ( a, O) under such a hypothetical "inverse:'
Which one should we use? We cannot simply say that P - 1 must send (a, 0) to (a, b),
since this cannot be a definition when we have no way of knowing what b should be.
(See Exercise 42.)
T (a, b)
I
� (a, b ')
I
(a, 0)
I
• (a, b ' ')
Projections are not invertible
Figure 3 . 1 3
Associalivilv
�
Theorem 3.3 (a) in Section 3.2 stated the associativity property for matrix multiplication: A (BC) = (AB) C. (If you didn't try to prove it then, do so now. Even with all
matrices restricted 2 X 2, you will get some feeling for the notational complexity
involved in an "elementwise" proof, which should make you appreciate the proof we
are about to give.)
Our approach to the proof is via linear transformations. We have seen that every
m
n
m X n matrix A gives rise to a linear transformation TA : !R ---+ !R ; conversely, every
m
n
linear transformation T : !R ---+ !R has a corresponding m X n matrix [ T ] . The two
correspondences are inversely related; that is, given A, [ TA] = A, and given T, Tr T J = T.
Let R = TA, S = Tn, and T = Tc. Then, by Theorem 3.32,
A ( BC) = (AB ) C if and only if R ( S T ) = ( R S ) a T
We now prove the latter identity. Let x be in the domain of T [and hence in the do­
main of both R (S T) and (R S) T-why?] . To prove that R (S T) = (R S) T,
it is enough to prove that they have the same effect on x. By repeated application of the
definition of composition, we have
0
�
0
0
0
0
0
0
0
0
0
0
( R a ( S a T ) ) ( x) = R ( ( S T ) ( x))
= R ( S ( T ( x)))
= ( R S ) ( T ( x)) = ( ( R S ) T ) ( x)
0
0
�
I
0
0
as required. (Carefully check how the definition of composition has been used four
times.)
This section has served as an introduction to linear transformations. In Chap­
ter 6, we will take a more detailed and more general look at these transformations.
The exercises that follow also contain some additional explorations of this important
concept.
Exercises 3 . 6
1 . Let TA : IR 2
IR 2 be the matrix transformation corre­
[� -:] .
[ � ] [ - �l
---+
sponding to A =
where u =
and v =
Find TA (u) and TA (v),
2. Let TA : IR 2
---+
[ ]
� !
[�] [ _�J .
IR 3 be the matrix transformation corre-
sponding to A =
TA (v), where u =
3
-1
. Find TA ( u) and
and v =
224
Chapter 3 Matrices
In Exercises 3-6, prove th at th e given transformation is a
linear transformation, using th e definition (or th e Remark
following Example 3 . 55) .
In Exercises 7- 1 0, give a counterexample to sh ow th at th e
given transformation is not a linear transformation.
I I
7. r
8. r
:2
Y
+
9. r
10. r
x7
y- 1
[; J [ l x l ]
[; J [ x l ]
[; J [ J
[; J [ J
In Exercises 1 1 - 1 4, find th e standard matrix of th e linear
transformation in th e given exercise.
12. Exercise 4
14. Exercise 6
1 1 . Exercise 3
13. Exercise 5
In Exercises 215-18,2 sh ow th at th e given transformation from IR to IR is linear by sh owing th at it is a matrix
transformation.
15. F reflects a vector in the y-axis.
16. R rotates a vector 45° counterclockwise about the
origin.
17. D stretches a vector by a factor of 2 in the x-component
and a factor of 3 in the y-component.
18. P projects a vector onto the line y = x.
19. The three types of elementary matrices give rise to five
types of 2 X 2 matrices with one of the following forms:
[� �] [� �]
[� �]
[� �] [� �]
In Exercises 20-25, find th e standard
matrix of th e given
2
2
linear transformation from IR to IR .
20. Counterclockwise rotation through 1 20° about the
origin
2 1 . Clockwise rotation through 30° about the origin
22. Projection onto the line y = 2x
23. Projection onto the line y = - x
24. Reflection in the line y = x
25. Reflection in the line y = - x
26. Let e be a line through the origin in IR 2 , Pe the linear
transformation that projects a vector onto e, and Fe the
transformation that reflects a vector in e.
(a) Draw diagrams to show that Fe is linear.
(b) Figure 3.14 suggests a way to find the matrix of Fe,
using the fact that the diagonals of a parallelogram
bisect each other. Prove that Fe (x) = 2Pe (x) - x,
and use this result to show that the standard matrix
of Fe is
(where the direction vector of e is d =
[ �: ] ).
(c) If the angle between e and the positive x-axis is e ,
show that the matrix of Fe is
[
cos W
sin W
sin W
- cos W
]
y
or
or
Each of these elementary matrices corresponds to a linear
transformation from IR 2 to IR 2 • Draw pictures to illustrate
the effect of each one on the unit square with vertices at
(0, O), ( 1 , O), (0, 1), and ( 1 , 1 ) .
Figure 3 . 1 4
In Exercises 27 and 28, apply part (b) or (c) of Exercise 26
to find th e standard matrix of th e transformation.
27. Reflection in the line y = 2x
Section 3. 6 Introduction to Linear Transformations
\/3
28. Reflection in the line y =
x
29. Check the formula for S T in Example 3.60, by
performing the suggested direct substitution.
0
In Exercises 30-35, verify Th eorem 3.32 byfinding th e
matrix of S T (a) by direct substitution and (b) by matrix
multiplication of [S] [T] .
0
225
41. If () is the angle between lines e and m (through the
origin), then Fm Fe = R + w · (See Exercise 26.)
42. (a) If P is a projection, then P P = P.
°
0
(b) The matrix of a projection can never be invertible.
43. If e, m , and n are three lines through the origin, then
Fn Fm Fe is also a reflection in a line through the
origin.
44. Let T be a linear transformation from IR 2 to IR 2 (or
from IR 3 to IR 3 ). Prove that T maps a straight line to a
straight line or a point. [Hint: Use the vector form of
the equation of a line.]
45. Let T be a linear transformation from IR 2 to IR 2 (or
from IR 3 to IR 3 ). Prove that T maps parallel lines to
parallel lines, a single line, a pair of points, or a single
point.
°
°
In Exercises 46-51, let ABCD be th e square with vertices
( - 1 , 1 ) , ( 1 , 1 ) , ( 1 , - 1 ), and ( l, - 1 ) . Use th e results in
Exercises 44 and 45 to find and draw th e image ofABCD
under th e given transformation.
46. T in Exercise 3
-
47. D in Exercise 1 7
48. P in Exercise 1 8
49. The projection in Exercise 2 2
In Exercises 36-39, find th e standard matrix of th e compos­
ite transformation from IR 2 to IR 2 •
36. Counterclockwise rotation through 60°, followed by
reflection in the line y = x
37. Reflection in the y-axis, followed by clockwise rotation
through 30°
38. Clockwise rotation through 45°, followed by projec­
tion onto the y-axis, followed by clockwise rotation
through 45°
39. Reflection in the line y = x, followed by counterclock­
wise rotation through 30°, followed by reflection in the
line y = -x
In Exercises 40-43, use matrices to prove th e given state­
ments about transformations from IR 2 to IR 2 •
40. If Re denotes a rotation (about the origin) through the
angle (), then R" R13 = Ra + f3 ·
0
50. T in Exercise 3 1
5 1 . The transformation in Exercise 37
52. Prove that Pe ( cv) = cPe ( v) for any scalar c
[Example 3.59(b) ] .
53. Prove that T : !R n ---+ !R m is a linear transformation if and
only if
for all v1 , v2 in !R n and scalars c1 , c2 .
54. Prove that (as noted at the beginning of this section)
the range of a linear transformation T : !R n ---+ !R m is the
column space of its matrix [ T ] .
55. If A is an invertible 2 X 2 matrix, what does the
Fundamental Theorem of Invertible Matrices assert
about the corresponding linear transformation TA in
light of Exercise 19?
Vi gnette
Rob o t i c s
I n 1 98 1 , the U.S. Space Shuttle Columbia blasted o ff equipped with a device called the
Shuttle Remote Manipulator System (SRMS). This robotic arm, known as Canadarm,
has proved to be a vital tool in all subsequent space shuttle missions, providing strong,
yet precise and delicate handling of its payloads (see Figure 3 . 1 5).
Canadarm has been used to place satellites into their proper orbit and to retrieve
malfunctioning ones for repair, and it has also performed critical repairs to the shut­
tle itself. Notably, the robotic arm was instrumental in the successful repair of the
Hubble Space Telescope. Since 1 998, Canadarm has played an important role in the
assembly and operation of the International Space Station.
;;;
Canadarm
<(
z
35
<(
z
Figure 3 . 1 5
226
A robotic arm consists of a series of links of fixed length connected at joints where
they can rotate. Each link can therefore rotate in space, or (through the effect of the
other links) be translated parallel to itself, or move by a combination (composition) of
rotations and translations. Before we can design a mathematical model for a robotic
arm, we need to understand how rotations and translations work in composition. To
simplify matters, we will assume that our arm is in IR 2 .
]
[
In Section 3.6, we saw that the matrix of a rotation R about the origin through an
cos e - sin e
angle e is a linear transformation with matrix .
(Figure 3 . 1 6(a)). If
sm e
cos e
, then a translation along v is the transformation
v
=
[:]
T ( x)
=
x + v or, equivalently, T
(Figure 3 . 1 6(b)) .
y
[; ] [; : : ]
=
y
T(x) = x + v
x
R(x)
(b) Translation
(a) Rotation
Figure 3 . 1 6
[�]
Unfortunately, translation is not a linear transformation, because T(O) * 0. How­
ever, there is a trick that will get us around this problem. We can represent the vector
x�
[; ]
"'
th, mtoc
in D;l'. This i& <oll,d "P""'nting x in homogrn,om coor­
dinates. Then the matrix multiplication
[
o ] [x] [ : l
represents the translated vector T (x) in homogeneous coordinates.
We can treat rotations in homogeneous coordinates too. The matrix multiplication
cos e
sin e
0
[i l [
- sin e
cos e
0
� ;
=
x cos e - y sin e
x sin e y cos e
l� [ �
represents the rotated vector R(x) in homogeneous coordinates. The composition T R
that gives the rotation R followed by the translation T is now represented by the product
0 a
1 b
0 1
cos e
sin e
0
- sin e o
cos e
0
=
cos e
si e
- sin e
cos e
0
0
�]
[Note that R o T * T o R. ]
To model a robotic arm, we give each link its own coordinate system (called a
frame) and examine how one link moves in relation to those to which it is directly
connected. To be specific, we let the coordinate axes for the link A; be X; and y;, with
the X;-axis aligned with the link. The length of A; is denoted by a;, and the angle
221
between X; and X; - 1 is denoted by e;. The joint between A; and A; - 1 is at the point
(0, O) relative to A; and (a;_1, O) relative to A;_ 1 . Hence, relative to A;_ 1 , the coordinate
system for A; has been rotated through e i and then translated along
[ a� I ]
(Figure 3 . 1 7) . This transformation is represented in homogeneous coordinates by
the matrix
Figure 3 . 1 1
To give a specific example, consider Figure 3 . 1 8(a). It shows an arm with three
links in which A 1 is in its initial position and each of the other two links has been
rotated 45° from the previous link. We will take the length of each link to be 2 units.
Figure 3 . 1 8(b) shows A 3 in its initial frame. The transformation
T3
=
[
cos 45
sin 45
0
- sin 45
cos 45
0
- 1 / \/2
l / v2
0
causes a rotation of 45° and then a translation by 2 units. As shown in 3 . 1 8(c), this
places A 3 in its appropriate position relative to A 2 's frame. Next, the transformation
T2
228
=
[
cos 45
sin 45
0
]� [ :2
- sin 45 2
cos 45
0
=
l / v2
1/
- 1 / v2
l / v2
0
is applied to the previous result. This places both A 3 and A 2 in their correct posi­
tion relative to A 1 , as shown in Figure 3 . 1 8 (d) . Normally, a third transformation Ti
(a rotation) would be applied to the previous result, but in our case, Ti is the identity
transformation because A 1 stays in its initial position.
Typically, we want to know the coordinates of the end (the "hand") of the robotic
arm, given the length and angle parameters-this is known as forward kinematics.
Following the above sequence of calculations and referring to Figure 3 . 1 8, we see that
Y3
Y1
(b) A 3 in its initial frame
(a) A three-link chain
YI
Y2
(c) T3 puts A3 in Az's initial frame
Figure 3 . 1 8
(d)
T2 T3 puts A3 in A 1's initial frame
we need to determine where the point (2, O) ends up after T3 and T2 are applied. Thus,
the arm's hand is at
- 1 / Vl
1 / V2
0
which represents the point ( 2 + V2, 2 + Vl) in homogeneous coordinates. It is easily
checked from Figure 3 . l S(a) that this is correct.
The methods used in this example generalize to robotic arms in three dimen­
sions, although in IR 3 there are more degrees of freedom and hence more variables.
The method of homogeneous coordinates is also useful in other applications, notably
computer graphics.
229
230
Chapter 3 Matrices
A p p l icati o n s
Markov Chains
A market research team is conducting a controlled survey to determine people's pref­
erences in toothpaste. The sample consists of 200 people, each of whom is asked to
try two brands of toothpaste over a period of several months. Based on the responses
to the survey, the research team compiles the following statistics about toothpaste
preferences.
Of those using Brand A in any month, 70% continue to use it the following month,
while 30% switch to Brand B; of those using Brand B in any month, 80% continue to
use it the following month, while 20% switch to Brand A. These findings are summa­
rized in Figure 3 . 1 9, in which the percentages have been converted into decimals; we
will think of them as probabilities.
�
0.30
Andreia Russian
A. Markovmathematician who
was
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taught at theHe
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( 1 856- 1 922)
Exa m p l e 3 . 6 4
0.80
0.20
Figure 3 . 1 9
Figure 3.19 is a simple example of a (finite) Markov chain. It represents an evolv­
ing process consisting of a finite number of states. At each step or point in time, the
process may be in any one of the states; at the next step, the process can remain in its
present state or switch to one of the other states. The state to which the process moves
at the next step and the probability of its doing so depend only on the present state
and not on the past history of the process. These probabilities are called transition
probabilities and are assumed to be constants (that is, the probability of moving from
state i to state j is always the same).
In the toothpaste survey described above, there are just two states-using Brand
A and using Brand B-and the transition probabilities are those indicated in
Figure 3.19. Suppose that, when the survey begins, 1 20 people are using Brand A and
80 people are using Brand B. How many people will be using each brand 1 month
later? 2 months later?
The number of Brand A users after 1 month will be 70% of those initially
using Brand A (those who remain loyal to Brand A) plus 20% of the Brand B users
(those who switch from B to A):
Solution
0.70 ( 120 ) + 0.20 ( 80)
=
1 00
Similarly, the number of Brand B users after 1 month will be a combination of those
who switch to Brand B and those who continue to use it:
0.30 ( 1 20 ) + 0.80 ( 80)
=
1 00
Section 3. 7 Applications
231
We can summarize these two equations in a single matrix equation:
[ �:�� �:!� ] [l !� ] [ � �� ]
[ ]
=
[ ]
1 00
120
. (Note
and x 1 =
1 00
80
that the components of each vector are the numbers of Brand A and Brand B users,
Let,s call the matrix P and label the vectors Xo =
in that order, after the number of months indicated by the subscript.) Thus, we have
x1 = Px 0 .
Extending the notation, let xk be the vector whose components record the distri­
bution of toothpaste users after k months. To determine the number of users of each
brand after 2 months have elapsed, we simply apply the same reasoning, starting with
x 1 instead of x 0 . We obtain
0.70 0.20 1 00
90
X2 = Px1 =
0.30 0.80 1 00
1 10
][ ] [ ]
[
from which we see that there are now 90 Brand A users and 1 1 0 Brand B users.
The vectors xk in Example 3.64 are called the state vectors of the Markov chain,
and the matrix P is called its transition matrix. We have just seen that a Markov chain
satisfies the relation
xk + 1 = Pxk for k = 0, 1 , 2, . . .
From this result it follows that we can compute an arbitrary state vector iteratively
once we know x 0 and P. In other words, a Markov chain is completely determined by
its transition probabilities and its initial state.
R e m arks
Suppose, in Example 3.64, we wanted to keep track of not the actual numbers
of toothpaste users but, rather, the relative numbers using each brand. We could con­
vert the data into percentages or fractions by dividing by 200, the total number of
users. Thus, we would start with
0.60
Xo = �s�o� =
0.40
2 00
to reflect the fact that, initially, the Brand A-Brand B split is 60%-40%. Check by
0.50
, which can then be taken as x1 (in agreement
direct calculation that PXo =
0.50
with the 50-50 split we computed above) . Vectors such as these, with nonnegative
components that add up to 1 , are called probability vectors.
•
Observe how the transition probabilities are arranged within the transition
matrix P. We can think of the columns as being labeled with the present states and the
rows as being labeled with the next states:
•
[ ]
Next
[] [ ]
[
Present
B
A
A 0.70 0.20
B 0.30 0.80
]
Chapter 3 Matrices
232
The
fromword
the Greekmeaning
adjectiis "capabl
vderie vede of
aiming"
(or guessi
ng). nItghasthatcome
togoverned
be applied
to
anythi
is
by
the
laws
of
probability
in the sense about
that probabi
lelitiyhood
makesof
predictions
the
l
i
k
things
happening.
Inprocesses"
probabiliform
ty
theory,
"stochastic
a generalization of Markov chains.
stochastic
stokhastikos,
Note also that the columns of P are probability vectors; any square matrix with this
property is called a stochastic matrix.
We can realize the deterministic nature of Markov chains in another way. Note
that we can write
and, in general,
xk = P k x0 for k =
0, 1, 2, ...
This leads us to examine the powers of a transition matrix. In Example
have
[ 0.0.3700 0.0.8200 ] [ 0.0.3700 0.0.8200 ] [ 0.0.4555 0.0.3700 ]
1.
14.)
0.45.
3.20
22 2
1
0.
7
(
0
.
3
)
0.
2
1)
,
1
0.5. 3(0.8) 0.24).
0.
4
exactly
2 0.45
p2 =
7
�
Y
� B 0.2 1 '
A 0.49
A
A
Y
� B 0.24*
A 0.06
B
figure 3 . 2 0
3.64,
we
=
What are we to make of the entries of this matrix? The first thing to observe is that P 2
is another stochastic matrix, since its columns sum to (You are asked to prove this
in Exercise
Could it be that P 2 is also a transition matrix of some kind? Consider
one of its entries-say, (P 2 ) 2 1 =
The tree diagram in Figure
clarifies where
this entry came from.
There are four possible state changes that can occur over months, and these
correspond to the four branches (or paths) of length in the tree. Someone who
initially is using Brand A can end up using Brand B months later in two different
ways (marked * in the figure) : The person can continue to use A after month and
then switch to B (with probability
=
or the person can switch to B after
month and then stay with B (with probability
=
The sum of these
probabilities gives an overall probability of
Observe that these calculations are
what we do when we compute (P 2 ) n.
It follows that (P 2 ) n =
represents the probability of moving from state
(Brand A) to state (Brand B) in two transitions. (Note that the order of the sub­
scripts is the reverse of what you might have guessed.) The argument can be general­
ized to show that
1
k
(P ) ij is the probability of moving from state j to state i in k transitions.
3.64,
In Example
what will happen to the distribution of toothpaste users in the
long run? Let's work with probability vectors as state vectors. Continuing our calcula­
tions (rounding to three decimal places), we find
x0 -
[ 0.0.3700 0.0.8200 ] [ 0.0.5500 ] [0.0.4555 ],
[ 0.0.4600 ] [ 0.0.5500 ]
[ 0.0.3700 0.0.8200 ] [ 0.0.4555 ] [ 0.0.457525 ] [ 0.0.458812 ] [0.0.459406 ] '
[ 0.0.459703 ] [ 0.0.450298 ] [ 0.0.459901 ] [ 0.0.460000 ] [ 0.0.460000 ]
'
x1 -
'
x2 - Px 1 -
X3 = Px2 =
X6 =
=
' X?
=
' Xg
=
=
'
x4 =
' x5 =
=
' X io
' X9
=
Section 3. 7 Applications
233
40%
60%
[ 0.0.3700 0.0.8200 ] [ 0.0.46] [0.0.46 ]
and so on. It appears that the state vectors approach (or converge to ) the vector
[ 0.0.46 ]
,
implying that eventually
of the toothpaste users in the survey will be using
Brand A and
will be using Brand B. Indeed, it is easy to check that, once this
distribution is reached, it will never change. We simply compute
=
4,
A state vector x with the property that Px = x is called a steady state vector. In
Chapter we will prove that every Markov chain has a unique steady state vector. For
now, let's accept this as a fact and see how we can find such a vector without doing any
iterations at all.
We begin by rewriting the matrix equation Px = x as Px = Ix, which can in turn be
rewritten as (I - P)x = 0. Now this is just a homogeneous system of linear equations
with coefficient matrix I - P, so the augmented matrix is [I - P I OJ . In Example
we have
[I
3.64,
- I [ 1 --0.0.3070 1 --0.0.2080 I 0 J - [ - 0.0.3300 - 0.0.2200 l 0o ]
O
p OJ =
which reduces to
So, if our steady state vector is x =
solution is
[ :: ]
, then x2 is a free variable and the parametric
1 X1 Xz t t t
x2 t 3 0. 6 x 1 2 0. 4 ,
[ 0.0.46]actual
x1 x2 200,
If we require x to be a probability vector, then we must have
Therefore,
= =5=
=t + =�
=
+
and
=5=
so x =
iterative calculations above. (If we require x to contain the
in this example we must have
Exa m p l e 3 . 6 5
+
, in agreement with our
from which it follows that x =
=
[ 12080 ] .)
3.2 1.
distribution, then
A psychologist places a rat in a cage with three compartments, as shown in Figure
The rat has been trained to select a door at random whenever a bell is rung and to
move through it into the next compartment.
2
1,
(a) If the rat is initially in compartment what is the probability that it will be in
compartment after the bell has rung twice? three times?
(b) In the long run, what proportion of its time will the rat spend in each compartment?
Solution
Let P = [p;j J be the transition matrix for this Markov chain. Then
P21 = P31 = 2I , P12 = Pn = 3I , P32 = p23 = 32 , and P11 = P22 = p 33 = O
234
Chapter 3 Matrices
Figure 3 . 2 1
�
(Why? Remember that pij is the probability of moving from j to i.) Therefore,
and the initial state vector is
(a) After one ring of the bell, we have
Continuing (rounding to three decimal places), we find
[
x2 = Px1 =
[ l m11 [!]
x3 = Px2 =
[ l ! ] [!] [ [
and
l
3
0
2
3
I
3
0
2
3
�l
18
l
=
=
0333 0.333
0.333
0 222
0.389
0.389
l
Therefore, after two rings, the probability that the rat is in compartment 2 is !
0.333, and after three rings, the probability that the rat is in compartment 2 is
fs 0.389. [Note that these questions could also be answered by computing ( P 2 )n
and ( P 3 ) 2 1 .]
=
=
Section 3. 7 Applications
-
235
(b) This question is asking for the steady state vector x as a probability vector. As we
saw above, x must be in the null space of I P, so we proceed to solve the system
[
J
x, x,
-i 0
_l O
Hence, if x �
[ :: l x,
x1 x2 x3
then
1 0
� l idm and
ability vector, we need 1 =
+
+
-----*
� l t,
1 0
0
0 0
-� O
-1 0
0 0
J
� l. Since x mu•t be ' prnb·
= � t. Thus, t = i and
which tells us that, in the long run, the rat spends � of its time in compartment 1 and
i of its time in each of the other two compartments.
linear Economic Models
We now revisit the economic models that we first encountered in Section 2.4 and
recast these models in terms of matrices. Example 2.33 illustrated the Leontief closed
model. The system of equations we needed to solve was
] [xx21 ]
In matrix form, this is the equation Ex = x, where
E=
[ � �: � � !
1 /2
1 /3
1 /2
1 /4 and x =
1 /4
X3
The matrix E is called an exchange matrix and the vector x is called a price vector.
In general, if E [ e;j ] , then e;j represents the fraction (or percentage) of industry j's
output that is consumed by industry i and X; is the price charged by industry i for its
output.
In a closed economy, the sum of each column of E is 1 . Since the entries of E are
also nonnegative, E is a stochastic matrix and the problem of finding a solution to the
equation
=
Ex = x
(1)
is precisely the same as the problem of finding the steady state vector of a Markov
chain! Thus, to find a price vector x that satisfies Ex = x, we solve the equivalent
homogeneous equation ( I E ) x = 0. There will always be infinitely many solu­
tions; we seek a solution where the prices are all nonnegative and at least one price
is positive.
-
236
Chapter 3 Matrices
2.34,
xXz=1 = 0.0.42xx1 0.0.25Xx2 O0..2lX3x3 1010
1
2
x3= O. lx1 0.3x2 0.3x3 30
x = Cx
(2)
- C) x =
0.
1
x
10
0.
2
0.
5
1
= [ 0.0.41 0.0.23 0.0.23 ] = [ X3Xz ] = [ 3100 ]
C
C= x
X;
d;
xx
The Leontief open model is more interesting. In Example
system
+
+
+
+
+
+
+
+
+
+d
or ( I
we needed to solve the
In matrix form, we have
where
, X
c
d
, d
The matrix is called the consumption matrix, x is the production vector, and d is
the demand vector. In general, if
[c;j ] ,
[x;J , and d [d;J , then c;j represents
the dollar value of industry i's output that is needed to produce one dollar's worth of
industryj's output, is the dollar value (price) of industry i's output, and is the dol­
lar value of the external demand for industry i 's output. Once again, we are interested
in finding a production vector with nonnegative entries such that at least one entry
is positive. We call such a vector a feasible solution.
=
Exa m p l e 3 . 6 6
=
Determine whether there is a solution to the Leontief open model determined by the
following consumption matrices:
(a)
C = [ 1/1/24 1/1/33 ]
Solulion
(a) We have
I-
so the equation ( I
-C
C= [ � �� 2/1/23 ]
C = [� 1 J [ 1/1/24 1/1/33 ] [ - 31/2/4 -1/2/33 ]
- C)x 3/4 1/3 X1 d1 ]
[ - 1/2 - 2/3 ] [x2 ] = [dz
(b)
O
_
= d becomes
3.7.
[XX21 ] = [ - 31/2/4 - 2/1/33 ] - 1 [dd21 ] - [ 3/22 9/l 4 ] [dd:]
d 1 , d2 ,
x1 x2•
- C) - 1
In practice, we would row reduce the corresponding augmented matrix to determine
a solution. However, in this case, it is instructive to notice that the coefficient matrix
I
is invertible and then to apply Theorem
We compute
Since
and all entries of ( I
are nonnegative, so are
find a feasible solution for any nonzero demand vector.
and
Thus, we can
Section 3.7 Applications
(b) In this case,
I- C=
so that
[
1 /2
- 1 /2
- 1 /2
2/3
]
(I - c) - l =
and
x = (I - C) - 1d =
[ -- 46 -- 66 ]d
231
[ -- 46 -- 66 ]
Since all entries of (I - C) - 1 are negative, this will not produce a feasible solution for
any nonzero demand vector d.
4
Motivated by Example 3.66, we have the following definition. (For two m X n
matrices A = [a ij ] and B [b;j ] , we will write A 2 B if a iJ 2 b;Jor all i and j. Similarly,
we may define A > B, A :::::: B, and so on. A matrix A is called nonnegative if A 2 0
and positive if A > 0.)
=
Definition
A consumption matrix C is called productive if I - C is invertible
and (I - c) - l 2 0.
We now give three results that give criteria for a consumption matrix to be
productive.
Theorem 3 . 3 4
Let C be a consumption matrix. Then C is productive if and only if there exists a
production vector x 2 0 such that x > Cx.
Proof
Assume that C is productive. Then I - C is invertible and (I - C) - 1
2 0. Let
Then x = (I - c) - 1j 2 0 and (I - C)x = j > 0. Thus, x - Cx > 0 or, equiva­
lently, x > Cx.
Conversely, assume that there exists a vector x 2 0 such that x > Cx. Since
C 2 0 and C * 0, we have x > 0 by Exercise 35. Furthermore, there must exist a
real number ,\ with 0 < ,\ < 1 such that Cx < ,\x. But then
C 2 x = C( Cx) :::::: C(,\x) = ,\( Cx) < ,\ (,\x) = ,\ 2 x
�
By induction, it can be shown that 0 :::::: C "x < ,\ " x for all n 2 0. (Write out the de­
tails of this induction proof.) Since 0 < ,\ < 1, ,\ " approaches 0 as n gets large. There­
fore, as n ---+ oo , ,\ " x ---+ 0 and hence C"x ---+ 0. Since x > 0, we must have C " ---+ 0 as
n ---+ oo .
Now consider the matrix equation
(I - C) ( I + C + C2 +
·
·
·
+ C" - 1) = I - C "
238
Chapter 3 Matrices
As n � oo , e n � 0 , s o we have
(I - C) (I + C + C2 + . . . ) = I - 0 = I
Therefore, I - C is invertible, with its inverse given by the infinite matrix series
I + C + C2 + . . . . Since all the terms in this series are nonnegative, we also have
(I - c) - 1 = r + c + c2 +
. . .
:::: o
Hence, C is productive.
R e m a rks
The infinite series I + C + C2 + . . . is the matrix analogue of the geomet­
ric series 1 + x + x2 + . . . . You may be familiar with the fact that, for l x l < 1 ,
1 + x + x2 + . . . = 1 / ( 1 - x) .
•
Since the vector Cx represents the amounts consumed by each industry, the in­
equality x > Cx means that there is some level of production for which each industry
is producing more than it consumes.
•
For an alternative approach to the first part of the proof of Theorem 3.34, see
Exercise 42 in Section 4.6.
•
corollarv 3 . 3 5
TheLatin
wordword comes from
therefers
which
to
a
garland
given
as
a
re­tle
ward.
Thus,
a
corollary
is
a
l
i
t
extra reward that follows from a
theorem.
corollary
corollarium,
Corollarv 3 . 3 6
Let C be a consumption matrix. If the sum of each row of C is less than 1, then
C is productive.
Proof
If
then Cx is a vector consisting of the row sums of C. If each row sum of C is less than
1 , then the condition x > Cx is satisfied. Hence, C is productive.
Let C be a consumption matrix. If the sum of each column of C is less than 1 , then
C is productive.
Proof
If each column sum of C is less than 1, then each row sum of cT is less than 1 .
Hence, CT is productive, by Corollary 3.35. Therefore, by Theorems 3.9(d) and 3.4,
It follows that (I - c) - l 2: 0 too and, thus, c is productive.
You are asked to give alternative proofs of Corollaries 3.35 and 3.36 in Exercise 52 of
Section 7.2.
It follows from the definition of a consumption matrix that the sum of column
j is the total dollar value of all the inputs needed to produce one dollar's worth of
industry j 's output-that is, industry j 's income exceeds its expenditures. We say that
such an industry is profitable. Corollary 3.36 can therefore be rephrased to state that
a consumption matrix is productive if all industries are profitable.
Section 3. 7 Applications
P.Matrices
H. Leslie,in Certain
"On thePopulation
Use of
Mathematics;'
(1945), pp. 183-212.
Biometrika 33
Exa m p l e 3 . 6 1
Population G rowlh
239
1945.
One of the most popular models of population growth is a matrix-based model, first
introduced by P. H. Leslie in
The Leslie model describes the growth of the fe­
male portion of a population, which is assumed to have a maximum lifespan. The
females are divided into age classes, all of which span an equal number of years. Using
data about the average birthrates and survival probabilities of each class, the model is
then able to determine the growth of the population over time.
1
3 (0-1
( 1 -2
(2-3
A certain species of German beetle, the Vollmar-Wasserman beetle (or VW beetle,
for short), lives for at most years. We divide the female VW beetles into three age
classes of year each: youths
year), juveniles
years), and adults
years).
The youths do not lay eggs; each juvenile produces an average of four female beetles;
and each adult produces an average of three females.
The survival rate for youths is
(that is, the probability of a youth's surviving to
become a juvenile is
and the survival rate for juveniles is
Suppose we begin
with a population of
female VW beetles: youths, juveniles, and adults.
Predict the beetle population for each of the next years.
Solution
that year:
0.1005), 50% 40 40 25%. 20
5
1
40 4 20 3 220
40 0.5 20
40 0.25 10
4
[ � s 00.25
2
;n,
[
[ :�
1
0, 1 , 2,
0, 1 , 2,
110
220
3
4
[ � s 00.25 00 ] [ 2010 ] [ 1105 ]
4s 0 03 110110 45555 l
[ � 0.25 0 ][ 5 ] [ 27.5
After year, the number of youths will be the number produced during
+
x
=
The number of juveniles will simply be the number of youths that have survived:
x
=
Likewise, the number of adults will be the number of juveniles that have survived:
x
=
We can combine these into a single matrix equation
x
° ' Lx o
�
x,, whm x,
�
the initilli populotion di'tcibution vcrtor nnd x,
�
-
is the distribution after year. We see that the structure of the equation is exactly the
same as for Markov chains: xk + i = Lxk for k =
. . . (although the interpretation
is quite different) . It follows that we can iteratively compute successive population
distribution vectors. (It also follows that xk = L k x0 for k =
. . . , as for Markov
chains, but we will not use this fact here.)
We compute
x, � L x,
x, � Lx, �
240
Chapter 3 Matrices
40 3 l [ 455 l [ 227.302.55 l
0.4 25 �3 302.��.55 13.951.752
00.25 00 ][ 227.13.755 l [ 151.56.828 l
Ther
e
for
e
,
t
h
e
model
pr
e
di
c
t
s
t
h
at
af
t
e
r
year
s
t
h
er
e
wi
l
be
appr
o
xi
m
at
e
l
y
5
You exampl
could are,g28ue
e VWrobeetundedles,to151thjeuveninearleesst, iandnteger57 adulat eachts. (Note:
t951adulhat young
wets aftsehroulfestmedpalhave3-whi
s
t
e
p-for
ch would have affected the subsequent iterations. We elected
not to do this, since the calculations are only approxim ations anyway and it is much
easier to use a calculator or if you do not round as you go.)
Theatiomatn wirixth ninageExampl
es 3.of6equal
7 is caldurledataion, wil be anInn genern matal, irfixwewihaveth thae
popul
cl
a
s
e
following structure: b b b b b
, 2 3
n-1 n
s, 0 0
0
0
00 S02 S03 00 00
0 0 0 Sn - 1 0
Here, b1by, beach2, fearmale teheinbirtclahs parameters
(b; the average numbers of females pro­
duced
i ) and s1 , s2 , are the survival probabilities (s; t h e
probabiWhatlityartehweat atofemalmakee inofclouras cali sucrulviavtesionsin?toOverclasali , th1)e beet. le population appears
t225o befrionmcreyearasing,1 talotyearhough2. tFihgerure ear3.e 2s2omeshowsfluctthueatchange
ions, suichn tasheapopul
decreaastieonfroinmeach250 tofo
the three age clas es and clearly shows the growth, with fluctuations.
=
CAS
L
Leslie matrix.
L
X
L=
•
•
=
•
•
•
=
•
+
4000
3000
·�
:;
§' 2000
i::
0
Cl.,
1 000
I
Adults
o l..:::::+:::::
::
:i:::
:: ___,_�::;..._.
:::
"""'*���=::::::i:
: ::::
::: ::::=:::+:+
_.
2
4
8
6
10
0
Figure 3 . 2 2
Time (in years)
Section 3. 7 Applications
241
0.9
0.8
0.7
-�
c
0
:;
0..
0
0..
4-<
0
...
c
0)
(.)
....
0)
0...
0.6
0.5
0.4
0.3
0.2
I
Juveniles
I
Adults
0. 1
Time (in years)
5
10
Figure 3 . 2 3
20
15
Ifcl, iansst, eaaddiffoferplentotpatingtetrhneemergespopul
attihoin,s, wewe needplot ttohecompute tpopul
acttiioonn iofn
eachthe popul
.
To
do
h
e
fr
a
as in each year;s. Forthexampl
at is, wee, afneedter 1toyeardiv,idwee eachhavedistribu­
tion vectoartbyionthine eachsum ageof itcls component
-2501-x1 250 [ 2202010 ] [ 0.0.0.800488 ]
sgetts ofayoutgraphhsl,ik8%e tihsejuoneveniinleFis, gandure43%.23,is
adulwhiwhiccthhs.stIehflowswes usplcltohetaratthl88%yisthtyatpeofthtofheeprdatpopuloaporovertaitoiontinmofconse,thweeipopul
each cleasis is approaching
a steady state. turns out that the steady0.st7a2te vectoraitniotnhiisnexampl
[ 0.0.0244 ]
Thatadultsis. ,(iInn totheherlongworrudn,s, 7th2%e popul
of theatpopul
aditiostnriwibutl ebed among
youths,t2h4e%thjruevenie agelescl, aands es4i%n
i
o
n
i
s
the ratio 18: 6 : 1 .) We wil see how to determine this ratio exactly in Chapter 4.
Therrelatieonsarheipmanys amongsituata ifionnsiteinsetwhiofcobjh itectiss.imForporexampl
tant toe,beweablmiegthto model
thdese icnrtiebre­
wi
s
h
t
o
varies,iocommuni
us types cofatineton wlinorksksconnect
(roads iconnect
iintegs,toetwnsc.) ,orairrleilnateiornsoutheipssconnect
igrngoupscit­
n
g
s
a
t
e
l
among
or individuals (friendship relationships in a society, predator-prey relationships in
actual
=
It
Graphs and Digraphs
relative
l
__
=
242
Chapter 3 Matrices
B
A
an ecosinygstseum,chdominetwnoranceks andrelarteiloanstiohnsiphsipins,aandsporittt,uetrncs.).outGraphsthat matareriidceales arlyesuaiustedefulto
model
tool in their stconsudy.ists of a finite set of points (called vertices) and a finite set of
connectif theys twaroe (tnhote endpoi
necessnaritslofy dianstedge.
inct) verFiguretices3.. 2We4 shsayowsthanat
tedges,
wexampl
o verteacheiofcestofharewhiesameadcjhacentgraph
drawn
i
n
t
w
o
di
ff
e
r
e
nt
ways.
The
gr
a
phs
ar
e
t
h
e
"
s
a
me"
i
n
t
h
e
s
e
ns
e
t
h
at
al
l
we
care
about
ar
e
t
h
e
adj
a
cency
r
e
l
a
t
i
o
ns
h
i
p
s
t
h
at
i
d
ent
i
f
y
D
c
edges.We can record the essential information about a graph in a matrix and use mattrhixe
questioensrs canabouthandlthe egrthaph.e calThiculsaitsioparns tveryiculaquirlycuskley­.
fulalgebrif thaetogrhelaphsp usareanslawrgere,cersintceaincomput
i s the
A B c D n n matrix If[or i s a grdefiaphnedwitbyh n vertices, then its
{
1
i
f
t
h
er
e
i
s
an
edge
bet
w
een
ver
t
i
c
es
i and j
Two
representations
of
the
same
a
graph
otherwise
The termcomes from the Latin
is theverb Figure 3. 2 5 shows a graph and its associated adj acency matrix.
plural)
which means "to turn:' In
A graph
G
A(G)]
Definition
X
Figure 3 . 2 4
A
..
'1
=
adjacency matrix
0
vertex (vertices
theetry),context
of graphs
(and geom­
a
vertex
is
a
corner-a
poidif­nt
where
an
edge
"turns"
i
n
to
a
ferent edge.
vertere,
Vl
V2
A=
V4
V3
graph with adjacency matrix
Figure 3 . 2 5
A
A
1
1
[ tl
0
0 0
sleysmmettherericismataObser
rloopix. atv(Weverhy?thtate)x Noti.thIenicsadjeomealascency
ositthuatatimatoansdir,aiaxgonalgrofaphaentmaygrryapha;;haveofis mornecesise tzerhsaanroilyoneun­a
of vertmaticesr.ixInsosutchhatcasaijeequal
s, it mays themakenumbersensofe edges
to modibetfyweenthe
defiveredgetnicibetestioinwandeenof jth.ae paiadjracency
ofofedgesa paththisatthale lnumber
ows us tofo tedgesravel
frit ocontmWeoneaidefinvers, nandteexa weto anotwilinhrereafegrcontr atophiantuouspato behlwiya. Thetshequence
as a is a 5-pForath.exampl
et,hiatn tthhee
Not
i
c
e
grfirastphofofthFiesgeuriseclosed
3.25, (it beginiss aand3-pendsath, andat thedges
e sametwiverce; taepatx); hsutchhatadoespathnotis calinclleuddea
The
s
e
cond
us
e
s
t
h
e
edge
bet
w
een
and
the same edge more than once is called a path.
Remark
�
A
path
length
k
v 1 v3 v2 v 1
circuit.
v1
k-path.
v4 v 1 v2 v2 v 1 v3
simple
v2
Section 3. 7 Applications
243
x to gievofe ustheinadjforamcencyationmataboutrix
tinheFipatWegurhcanes of3.2usevar5: itohuse power
lengthssofinathgreagrph'aph.s adjConsacencyidermattherisquar
represtehntat? Look at the (2, 3) entry. From the definition of
matWhatrixdomulthteipentlicatrieiosn,ofwe know
The onlys way thtihsatexprmakees iupon tcanhe sruemsulits inonzer
n a nonzero. Buto numberis inonzer
s if at loeasiftandoneonlof tyhief
product
welbot3 (vhliaasveranandteedgex k) .betIarnweournonzer
eenexamplando, whie, thcThushismeans
,happensthertheatforkwitlherbeeais2-andanpatedgeforkh betbetweenw2,eensvero ticandes 2 andas
2
whithe crhemaitel sniusngthentat rtiheers ofe are tcorrwo 2-ectplatyhgis vbete 2-wpeenathvers inticthese 2grandaph.3.) (TheCheckargtument
o see thweat
haveas Exerjusctisgie v72.en can be generalized to yield the following result, whose proof we leave
If is thofe adjk-pacency
number
aths betmatweenrix verof atigrcesaphand then the entry of is equal to the
A2
a2k
a 2 ka k 3
a k3
a 2 k a k3
vk
v3 .
=1
v2
vk
=
(A 2 ) 23 = a 2 , a l3 + a 22 a 23 + a 23 a 33 + a 24a43
= l·l + l·l + l·O + O·O
=
�
A2
A
Exa m p l e 3 . 6 8
i
G,
Ak
(i, j)
j.
How manyWe3-pneedaths arthee ther2)e betentwryeenof andwhichinis Fithgeurdote 3.pr25?oduct of row of and
column 2 of The calculation gives
3 2
6
so there are six 3-paths between vertices and 2, which can be easily checked.
beposmodel
verticesForareexampl
orderede,
bydirectsIonmeemany
td edgesypeapplofmiicgrathteliaotbensionusthteathdatcantoimrepres
eesnta one-diedrectbywiayaongraph,
onroutetshthieneedges.
graphmodel
that model
s
aecostransystpeorm.taAtiographn netwiworthkdiorrectpredated edgesor-priesycalrelleadtioanships in aFiagraph
i
n
g
an
examplAn eeas. y modification to the definition of adjacency matricgesureall3.ows26 sushowsto usane
them with digraphs.
Solution
A.
v1
A3,
(1,
(A 3 ) 1 2
=
v2
1
·1 + ·1 + 1·1 + 0·0
=
1
digraph.
A digraph
Figure 3 . 2 6
A2
Chapter 3 Matrices
244
i s the
matrix If[or is a didefigraphnedwibyth vertices, then its
otif thhererweiissean edge from vertex to vertex j
Thus, the adjacency matrix for the digraph in Figure is
Definition
n
Xn
A
G
A(G)]
adjacency matrix
n
i
3.26
Not(Whensurwoulprisindglity,be?th)e Youadjascency
mathaverixnoofdiaffidiculgrtayphseeiisngnotthatsymmetnowriccontin agenerins thale.
h
oul
d
sist thate algilvedges
pattnumber
hion offlotwhs iofisnidthea.e samek-dipratecthsiobetn. (wSeenee Exervertciicsese , wherThee wenextinexampl
es an applalongica­a
Firovund-e tennirobisnpltoayerurnsament(Djokoviin whic, Federch eacher, Nadal
, plRoddiays everck, yandothSafier pln)ayercompetonce.e iThen a
pl
a
yer
ditexgrjameans
ph in Fithgurat eplayersudefmmareateizdespltahyere rej.su(lAts.diAgdiraphrectiendwhiedgechfrthomereveris texactex tloy verone­
direThectedadjedgeacencybetwmateen reverix fory paither diofgverraphticiesn Fiis gcalurleed a is
Ak
directed
Exa m p l e 3 . 6 9
3.27
D
F
tournament
Figure 3 . 2 1
A
i
i
tournament.)
3.27
s
R
72.)
N
0
0
A= 1
0
0
1 0
0
0 0 1 0
0 0 0 1
0 1 0 0
wher
e thiecalorlyd.erThusof th, eFedervertiecrescor(arndesphence
toherorwowsandandcolcoluumnmns offor exampl
is deteer.mined
alphabetSuppos
onds
t
ayers, basofewid nons fortheeachresulpltsaofyerth. eiObsr matervcehesthat. Onethe
waynumberto dooftwiehiwesnmis wieachgshth tplbeoaryertaonkcounthadtheifistvhjueespltnumber
ng roew;
equivalently, the vector containing all tthheersouwmsuofmstheisentgivreniesbyin tthhee prcororductespondiwher
2
2,
A)
Aj,
j=
1
1
1
1
1
Section 3. 7 Applications
245
In our case, we have 0 1 0
3
01 00 01 1 01 23
00 00 01 00 01 11
which produces the followinFigrrsat:nkinDjg:okovic, Federer (tie)
Second:
Nadal
Thi
r
d
:
Roddi
c
k,
Safi
n
(
t
i
e
)
e thdefe pleatayered sFederwho etire, dheindesthiesrrvaesnkifinrsgtequalplace.lyRoddi
strong?ckDjwoulokovid usc mie tghhte same
arguettyhpeat
sofinarceArgheument
the tiee hewitbeath SafiNadal
n. However
,defeat
Safin ecould d arotghueertsh; atfurhetherhasmtorweo,
"ihendimiregctht" notvictteoorbrtiheeatsakbecaus
,
who
Roddick has only indirect victory (over Safin, who then
defeatSineced Nadal
)
.
grcororup,esptondsheinnotatogrioaonup2-ofpofatinhtdiieinrsettcthherewidie nmaygsrasph,eemsnotsobemorwea canple ausyerusefulewhot.hMore defesquareoverateedof, alanthl teihnadjediotraehcency
ctervis icntma­othrye
tofrixth. Toe matcomput
rix e bothwhiwincshandare giinvdienrebyct wins for each player, we need the row sums
00 01 01 1 01 00 21 1 21
001 000 00 001 00 001 001 00 01 020
0 01 22 22 23 11 87
01 001 011 02 201 11 23
Thus
Unfor, tweunatwoulely,dthriasnkapprtheoachplayeris nots as guarfolloaws:nteDjedotkovio brecak, Federall tieers,.Nadal, Safin, Roddick.
Aj
=
two
one
A + A2,
(A + A 2 )j
+
=
6
I
Exercises 3 . 1
M a rkov C h a i n s
[ 0.0.55 0.0.73 ]
[ 0.0.55 ]
In Exercises 1 -4, let P
be the transition mabe
trix for a Markov chain with two states. Let x0
the initial state vector for the population.
=
=
1.
2.
3.
4.
Computpreoporandtion of the state 1 population wil be in
What
state 2prafoteporr twtioonstofeps?the state 2 population wil be in
What
sFitantde 2thafetsetreadytwossttaetps?e vector.
x1
x2 •
Chapter 3 Matrices
246
P [! ! I]
: � [ :m
Comput
e
and
What
ation wil be in
sWhattate 1prprafooteporporr twttiiooonnstofofeps?tthhee ssttaattee 21 popul
sFitantde t3hafetsetreadytwossttaetps?e vector. population wil be in
Supposeaccor
that tdhiengweatto ahMarer inkaovparchaiticnul. aSpecir regifiocnally,
behaves
sauwetpposdaye thisat0.t6h62e prifotbabiodaylitiys wetthatandtomor0.2r50owifwitoldaybe
idays driys. 0.The750prifotbabiodaylitiys drthaty andtomor0.3r38owifwitoldaybe aisdrwety .
studyR.ofGabrrainfaliell
iand[nThiTels exerAviNeumann,
vcioverse is abas27-eMardyearonkovperian Chaiactod.ualnSeeModel
Rainfall Occurrence at Tel Aviv;' ( 1 962),for Daily
pp. 90-95.
]
Wrchaiitne. down the transition matrix for this Markov
If Mondaydayiswialdrbey day,wet?what is the probability that
Wednes
In thdre lyodaysng rube?n, what wil the distribution of wet
and
ated onetthheatheithgehtprs oofbabichillidtireesn
rtDathelatataiavhaveteatlopartbeenheientr paraccumul
ewilnthaves. Suppos
sabihorlittiechis tlhdataraemedi0.6, 0.u2m-, andheiag0.thta2l,par,rmediesepntectuwim-ivlelhhaveyei; tghhtea,prtoraol b­,
medi
shavepectuaivm-telaylh;, eiandmedightt,huorem-prshhooreibabigthtchil,iortlidessarhtheorat0.t 1chia, s0.hld7or,artandpare 0.0.e2nt,20., wire­4,l
andWr0.4,itreesdownpectivtehley.transition matrix for this Markov
chain.is the probability that a short person wil
What
havef 20%a tofal thgreacurndchirentldpopul
? ation is tal , 50% is of
Imedi
is shaortiotns, what? wil the
diWhatstriubpropor
utmiheiongbethtio,innandofthrt30%eheegener
popul
a
t
i
o
n
wi
l
be
t
a
l
,
of
medium height, and short in the long run?
In Emdm 5-8, let
�
11.
be the tmn;;t;on ma-
i,;xfor n Ma,kov cha;n wUh th " al" Let x,, �
H.
be
6.
7.
8.
9.
x1
J.
x2 .
''A
(a)
(b)
K.
Quarterly Journal of
the Royal Meteorological Society, 88
''A
Models in Archaeology
the initial state vector for the population.
5.
Asoutstuhdywesoft pifrofmwn1940(pine)tonut1947cropshypotinhtesheizAmer
ican
ed
t
h
at
nutD. prThomas
oduction, folComput
lowed a eMarr Sikmovulachaitionn.Model
[See of
etaternBass;' iinn ShosD. L.hCloneanarke,Subsed., istence and Settlement
PatGr(London:
.] tThehendattheaprsuoggesbabitleidtieths atthat
itfhonee folyearlowiMet'ns gcryearhouen,p was's cr1972)good,
wercrope was0.08,fa0.ir0, 7,thanden th0.eo8prp5,woul
robabiespdectlitbeiievsgood,
eltyh;atiftonefaheirfol,yearorlopoorw-'s
0.itnh1gen1,yeartandhe'spr0.cro8obabi0,p woulrelsiptiecteds betihvatelgood,yt;hief onefolfailro,yearwiornpoor'gs yearcropwer'swascreo0.ppoor09, ,
woul
respWrectd beiivteegood,
lydown. faithre, ortrapoornsitiowern mate 0.ri1x1,fo0.r0t5,hisandMar0.k8ov4,
chaiIf thne .pifion nut crop was good in 1940, find the
probabi
crop in the years 1941
tIhnrtohughe lolngi1945.tiesruofn,awhatgoodpropor
tion of the crops wil
be
good,
fai
r
,
and
poor?
Robot
e mazey
schoos
hownesiwhihaven Ficghbeenurwaye 3.pr2to8ogo.andgrammed
at eachtojutrnctaversionertahndoml
(c)
12.
(a)
(b)
( c)
10.
Figure 3 . 2 8
(b)
(b)
( c)
(d)
Cons
he trasnsthitiisosnitmatuatiroixn.for the Markov
chaiSupposntrthueatctwetmodel
t(iAosn.uFimendththatsetiasttrettaadywikesthseacht1a5terobotdirosbottrisbatuttheacheiosnameofjurnc-amount
obots.
of
t
i
m
e
t
o
t
r
a
vel
bet
w
een
t
w
o
adj
a
cent
j
u
nct
i
o
ns.
)
e a roiwvevectmatorrixconsPisisatisntogchasentitrieclymatof rlis.x Prif ove
tandLethatjonladenotnonnegat
y if j P = j .
(a)
(a)
13.
Section 3. 7 Applications
02
0.
1
0.
4
04
]
Show
t
h
at
t
h
e
product
of
t
w
o
2
2
s
t
o
chas
t
i
c
[
0.
2
[
0
35
5
0.0.41 55 0.0.3505 �0.6350 : 0.0 3 0.0.42 0.0.25 0.0.31
Prmatmatoverriiccteseshatiisstalhalsesoproaaosducttsotochaschasoftittcwicmatomatrirxi.x.stochastic
0.5 0 0.2 0.2
IPf-a1 2is als2osatoschastochastictmatic matrixrPixi. s invertible, prove that
C
x
1/21/2 1/41/2 ] [ 3l ]
[
P
0.31 0.0.42 ] , [ 21 ]
[
0.
In Exerecdisnumber
e if Monda
y iunts a dril ya wetday,day?what is the
5 0.0.42 010.2 ] '2 ]
expect
of
days
[
[
�
ItinonsExeruntciislea10,shorwhatt perissothnehasexpecta tael ddesnumber
of
gener
a
­
0
0.
5
4
c
endant
?
IisntExerhe expectcise 11,ed inumber
fthe pifiofonyearnutscruntopiilsafagoodir onecryearop occur, whats? C [ 010 0.0.42 010.2 ] [ 3.u5 ]
ItinonsExer, whatcise i12,s thsetaexpect
rting fredomnumber
each ofofthmoves
e otheruntjunc­il a Let A be0.3an 0.2 mat0.3 rix, A 2 2.0 Suppose that
robot reaches junction 4?
Ax<
x
fo
r
s
o
me
x
i
n
x
2 Prove that x
and ixtiandes:
vectLet A,IofrAs in2C, and2ProvebeandtheCfol2lomatwin2rgicinesequal
ACIf A 2 and2 x 2 x thentAxhen
[ 11/22 3/41/4 ]
[ 1/21/3 2/31 2 ]
[ 0.0.46 0.0.47]
[ 0.0. 1 0.0.46]
A
popul
a
t
i
o
n
wi
t
h
t
w
o
age
cl
a
s
e
s
has
a
Les
l
i
e
mat
r
i
x
2
[ 1/31/3 3/20
1/2
1
�] . If the initial population vector is
[
�
0.
6
[
0
0
1
3
L
]
�
1/2
1/3
[ 030.3 0.0 5 0.023 ] 0500.1/225 00.0.37002/30350.l 25 ] x0 [ 1 � ], compute x1, x2, and x3.
0.4 0.5 0.5 [ 0.25 0 0.40 A population with three age clas es has a Leslie matrix
L � H 0.05 1 � l If the m;t;ru populahon
00.2300 0.0. 1105 0100.45 vector is x0 [ : . compute x,. x,, ond x,.
[ 0.0.25 0.0.63 ]
[ 0. 1 5 0.30 0.50 ]
]
14. (a)
X
(b)
nXn
In Exercises 31 -34, a consumption matrix and a demand
vector d are given. In each case, find a feasible production
vector that satisfies Equation (2).
Suppose we want to know the average (or expected) number
of steps it will take to go from state i to state j in a Markov
chain. It can be shown that the following computation
answers this question: Delete the jth row and the jth column
of the transition matrix to get a new matrix Q. (Keep
the rows and columns of Q labeled as they were in P.) The
expected number of steps from state i to state j is given by
the sum of the entries in the column of (I Q) - 1 labeled i.
15.
9,
17.
18.
30.
29.
X
(c)
16.
CAS
31. c =
,d =
32. c =
d=
33. c =
,d =
34.
,d =
=
nXn
35.
36.
Linear E c o n o m i c M o d e l s
In Exercises 1 9-26, determine which of the matrices are
exchange matrices. For those that are exchange matrices,
find a nonnegative price vector that satisfies Equation (1).
19.
21.
23.
I
20.
22.
I
9
B,
D
!R n .
ll�r,
B 0
D 0,
BD 0.
>B
0, * 0,
(a)
(b)
P o p u l at i o n G r owth
37.
24.
26.
38.
1
·
In Exercises 27-30, determine whether the given consumption matrix is productive.
27.
28.
0.
0.
=
> 0.
y
nXn
=
25.
241
> Bx.
Chapter 3 Matrices
248
39.
A popul1 ation1 with3three age clas es has a Leslie matrix
[ 00. 00.5 00 l . If the initial population vector is
[ 100100 ] , compute and
100
A population with four age clas es has a Leslie matrix
-_[ �.5 0.0� 7 0.0 3 00 . Ifthe initial population
� � �i
voctodno rn l rnmpute Md
nasspuecirvievsalwiprthobabitwolageity ofcla80%s es frofom1 yearclas's dur1 toa­
tAclioacerns has2.taiEmpi
eachtwo posfemalsibleegiLesrivcesalliebievimatrtdhencertiocesfivshareowsfee maltheats ,peronyearaver.aThusge, ,
[ 0.0 8 05] and [ 40. 8 01 ]
Steacharticasng ewi. th [ � � l compute , in
of eachdoageyour
clgrForaasphseachoversucasggestime,etpl?(aost itnheFirgeurlateiv3.e2s3)ize. What
Suppose the2Leslie matrix for the VW beetle is
[ �.! L � l sta,ting with acb;tmy Xo, detec­
miSupposne theetbehavi
o
r
of
t
h
i
s
popul
a
t
i
o
n.
he Leslie matrix for the VW beetle is
[ 0� �0.5 20� l Investigate the effect of varying
tWoodl
he suravndivalcarpriobbabiou arelityfoundof thpre young
beet
l
e
s
.
imaricianly inorn ththeweswestt.ern
prTheoviaverncesageoflCanada
and
t
h
e
Amer
ThegivenbiirnthTablandeifs3.esu4rpv,anwhiivalofcrhaatfseehsmalowsforeeachtihsataboutcarageibbr1ou4ayearcketcowssar. doe
not give birth at all during their first 2 years and give
L
=
7
x0 =
40.
x1, x2 ,
biyearrths.tTheo aboutmoronetalitcaly rfatpere foryearyoungduricalngvtesheiisr vermidydlhiegh.
x3 .
L
x,. x,.
�
41.
LI
L2
=
(a)
x0
x,
=
x1,
=
.
•
.
x1 0
(b)
42.
L
·
43.
L=
cAs 44.
AA
·
s
=
Ta b l e 3 . 4
Age
(years)
Birth
Rate
Survival
Rate
2-44-60-2 0.0.1.408 0.0.0.93
6-88-10 1.1.88 0.0.99
10-12
12-14 0.1.66 0.0.60
JTablaspTheere 3.Nat5number
i. UsonalingParsaofCAS,kwoodl
in Alprbeadierndcttacartihnei1990bcarouibrareoupore popul
shteownd inatiinon
ctyoutheconclpopulude?atio(nWforhatthe
yearforassu1992mpts 2010iandonsanddoes1994.2020.thThenisWhatmodelprodojemake,
and how could it
be improved?)
7
Ta b l e 3 . 5
Age
(years)
0-22-4
4-66-8
8-10
10-12
12-14
Woodland Caribou
Population in Jasper
National Park. 1 9 9 0
102
85
120
Number
Source: World Wildlife Fund Canada
Section 3. 7 Applications
Graphs and Digraphs
In Exercises 45-48, determine the adjacency matrix of the
given graph.
45
VI
In Exercises 53-56, determine the adjacency matrix of the
given digraph.
V2
.--------.
V4
46.
249
V4
V3
V1
V2
54.
V3
v,
V2
V4
47.
55.
V4
V5
56. V J
V2
V4
V3
In Exercises 49-52, draw a graph that has the given adjacency matrix.
49.
51.
001 001
[ 0� 00 01 �]1 0
01 00 00 01 11
10 0 00 00
50.
52.
0
[ 0; 0 00 ll1 1
00 00 00 1 1
11 00 00
In Exercises 57-60, draw a digraph that has the given adjacency matrix.
57.
10 00
�[ 01 0 ; ]
58.
10 00
0
0
[: 0 1 �:
Chapter 3 Matrices
250
0 1 0 1
0 0 1 0
0 0 0 1
0 1 0 0
0 1 0
0
1
59. 0
1
0
60.
sadjweracency
the folmatlowirinxg quesfor tthioisns.digraph and use it to an­
A
0 1 0 0 1
0 0 0
0
0 0
0 1 0 0
1 0 0 0
Rodent
In Exercises 61-68, use powers of adjacency matrices to
determine the number ofpaths of the specified length
between the given vertices.
61.
50,
2, v 1
v2
62.
2, v 1
v2
63.
50,
3, v 1
v3
64.
52,
4, v2
v2
65.
57,
2, v 1 v3
66.
57,
3, v4 v 1
67.
60,
3, v4 v 1
68.
60,
4, v 1 v4
69. A
(a)
i A
ExerExercciissee 52, lleengtngthh andand
ExerExercciissee lleengtngthh andand
Fish
Bird
ExerExercciissee lleengtngthh ttoo
WhiHowcdoesh speciesshhasow tthhies?most direct sources of food?
ExerExercciissee lleengtngthh ttoo
Whimostchotshperecispeseciiseas?diHowrect sdoesource ofshfoodow thforis? the
Let Ifberowtheofadjaicency
mat
r
i
x
of
a
gr
a
ph
G.
s all zeros, what does this imply
Iinf direatects sbandourcebeatof food.s weHowsaycanthatwehasuse astoande­
about
G?
IfaboutcoluG?mn of is all zeros, what does this imply
tseormiurcesne?whiWhichchspspeciecieseshashasththe emosmost tindidirreectctandfood
i
n
di
r
e
ct
food
s
o
ur
c
es
combi
n
ed?
Let Ifberowtheofadjacency
mat
r
i
x
of
a
di
g
r
a
ph
D.
Suppos
e
t
h
at
pol
l
u
t
a
nt
s
ki
l
t
h
e
pl
a
nt
s
i
n
t
h
i
s
food
i
s
al
l
zer
o
s
,
what
does
t
h
i
s
i
m
pl
y
web,
and
we
want
t
o
det
e
r
m
i
n
e
t
h
e
eff
e
ct
t
h
i
s
aboutcoluD?mn ofA2 is all zeros, what does this imply change
wi
l
have
on
t
h
e
ecos
y
s
t
e
m.
Cons
t
r
u
ct
a
Ifabout
new
adj
a
cency
mat
r
i
x
fr
o
m
by
del
e
t
i
n
g
t
h
e
D?
r
o
w
and
col
u
mn
cor
r
e
s
p
ondi
n
g
t
o
pl
a
nt
s
.
Repeat
a tourmatnamentrices,wiratnkh stixhe
parthetmoss (a)t tando (c)leandast affdetecteremdibyne twhihe change.
ch species are
plplFiaagyeryeruress,firsttoibys thdeteUsdierignmrgaiphnadjinofgacency
What wimatl rtihxecallong-culatetiromnseffwielctsofhowthethpolis?lution be?
usas iinngExampl
the noteion of combinedwiwinsnonls andy andindithreenctbywins, FiveWhat
peopl
e
ar
e
al
l
connect
e
d
by
em
ai
l
.
Whenever
tithalemonghearbyse-amjuaiiclyinpigeitcetoofsogosmeonesip, heelsore isnhtehe
pasgroneoupseofs accor
d
i
n
g
t
o
Tabl
e
netDrawworthke" diandgrafiphndtihtats adjmodelacencys thmatis "grosixsip
(b)
70.
71.
Figure 3 . 3 0
(a)
A
(b)
(c)
a
A
c,
a
j A
A
(a)
i A2
(h)
j
3.29
P1
c
A
(d)
A*
A
P6.
(e)
3.69.
73.
3.6.
(a)
A.
Ta b l e 3 . 6
amfoodtowebinidin ­
aFicatsgmeursaltehl atecoshasiyssatedim.asgrAaaphdisourrreectprceeedofsedgeentfood.infrg oCons
truct the
Figure 3 . 2 9
72.
3.30
a
BerAnnCartla
Dana
Ehaz
Sender
p6
b
a
b
CarCarllaa,, Ehaz
Dana
Ehaz
Ann,
Car
l
a
Bert
Recipients
Chapter Review
evergosDefisyinponeegeta ons frhioasms torhAnnehertimtoleisbotitt. t(ahTkeshusCara,lpersainandoneonEhaz.sttoep,e-)mIfail
steps wiWhatl it tmatakerix
forcalBercteverulhearsatyioonenarreeluvealmor,se tsothhowhearis? many
the rumor?
mat
r
i
x
i
n
Exer
c
i
s
e
49
forcalIf Annceverulathearyioonen rseaelvealrsue mortsothheari,show? themanyrumor?stepsWhatwil matit tarkeix TheThe adjadjaacency
cency
mat
r
i
x
i
n
Exer
c
i
s
e
52
The
adj
a
cency
mat
r
i
x
i
n
Exer
c
i
s
e
51
Idinggeneral
,
i
f
i
s
t
h
e
adj
a
cency
mat
r
i
x
of
a
ed to 00 00 00 11
ver[Thetreaxgosph,byshowipanetpatcanwhor(wekofinstoetlmehiisf verexerlengttecxhis)e?iiss connect
r
e
mi
n
i
s
c
ent
ofplatyheandnotfiiolmn ofby"tshixatdegrname)ees, ofwhisecparh sautggesion"ts(fothundat anyin the 0 0 01 01 01 01
0
0
0
1
1
twhoswo people lengte arh eisconnect
e
d
by
a
pat
h
of
acquai
n
t
a
nces
at mose frivt ol6.oTheuslygameassert"sStihxatDegrall acteesoofrs are 1 Prove0that1 a 0graph0 is bipartite if and only if its
Keviconnectn Bacon"
mor
e
d
t
o
t
h
e
act
o
r
Kevi
n
Bacon
i
n
s
u
ch
a
wa
y
.
]
vercantibecesparcantitbeionedlabelased so that its adjacency matrix
Let Bybeintducthe adjion,acencyprovematthatrixforofalalgraph1, the
of n-paths
[�--H�-]
betentHowwryeendoof tverheitssitcequal
esatementandto tandhe number
tIf ios tbehemodiadjacency
fied ifmatisriaxdiofgraproof
adiph?graphin partwhat(a) havedoes grUsaiphng hasthe rnoesuciltrcinuiparts oft (odda), prleongtvehth. at a bipartite
the entry of represent if ?
(b)
A graph is called bipartite if its vertices can be subdi­
vided into two sets U and V such that every edge has one
endpoint in U and the other endpoint in V For example,
the graph in Exercise 48 is bipartite with U = { v 1 , v2 , v3 }
and V = {v4 , v5 }. In Exercises 76-79, determine whether a
graph with the given adjacency matrix is bipartite.
step
(c)
(d)
76.
77.
78.
A
i
j
79.
80. ( a)
74.
251
(a)
G.
A
An
(b)
75. A
n 2::
i
(i, j)
(b)
G
(i, j )
A=
j.
AA T
i *j
G,
Chapter Review
basBasiiss,Theor198em, 202
colcoluumnmn matspacerixof(vaector), 138
matrixi,tion195oflinear
compos
coortbasransdisi,nfoatrm208e vectationsor,wit219h respect to a
didiamgonalensiomatn, r203ix, 139
elementary matrix, 170
Kev Defi nitions and concepts
Fundament
al172,Theor206em of Invertible matmatrriixx,addi138tion, 140
Mat
r
i
c
es
,
iindverentisteyofmata srqixuar, e139
matmatrriixx mulfactotirpizliatcatioin,on, 180141
matnegatriixvpowers
, r149ix, 140
intmatverranssreifoxof,rma163atlinioearn, 221
e
of
a
mat
lliinnearear dependence/
combinationiofndependence
matrices, 154 nulnuloutlleirtsyppraceofoductaofmata, matrix147,rix204, 197
linofearfactmattrorarinsiczesatfo,iromn,157atio181n, 213 perparmulmtituttiiopanedltiicoatnmatiomatn)r,irciesx145,, (bl187o1ck48
LU
Chapter 3 Matrices
252
sttarndaransfodrmatmatrioixn,of a216linear
ssyummet
bspace,ric mat192rix, 151
tzerranso matposeriofx, a mat141 rix, 151
MarFork eachanyofmattherifolx lowibotngh statementand s truearore defifalsne:ed. uctIf posofselibelment
e, exprareys mattherimatces.rix [41 ! ] as a prodIf andthBenareBmatrices such that AB and If is a square matrix such that show that
If A, B,B,andthen are invertible matrices such that
Thetary matinverrisxe. of an elementary matrix is an elemen­ Find an faotmization of [: - l :J
Find bases for the row space, column space, and null
Theelementtransarpyosmate ofrixan. elementary matrix is an
Theelementproductary matof trwixo. elementary matrices is an sp�<ofA [� =� � � n
aEverIfsubsiyspplanaceaneofin matis aritxw, o-thendimthense nulionall spsaceubsofpaceis haveSupposthee smatamericroesw spandace?BarWhye roorwwhyequinotvale?ntDo. Do andthey
BIfhaveis antheinsverametibcolle umatmnrisxp, ace?explaWhyin whyor whyandnot? must
ofThe transformation T: defined by
nulmatl srpixace.? ExplIs tahinis. true if is a nonin­
T(If xT:) - xis aislianlearineartratnsrafonsrfomratmiaton.ion, then verhavetibtlheessame
q
uar
e
whose berowsinveraddtibuple.to the zero
talhlerxeinistahe4 domai5 matn ofrix such that T(x) Ax for vectIf oisr,aexplsquarainewhymatrixcannot
Let
be
an
mat
r
i
x
wi
t
h
l
i
n
ear
l
y
i
n
dependent
col
u
mns
.
Expl
a
i
n
why
mus
t
be
an
i
n
ver
t
i
b
l
e
]
-�
�].
[� � [�
mat
r
i
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.
Mus
t
al
s
o
be
i
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ver
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i
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?
Expl
a
i
n
.
Find a linear transformation T: such that
A(B2BBT) - 1 A(B2TBB2) - 1 B -1B
�
]
].
]
J
and
r
r
�
[
[
[
[
_�
�
The outer product expansion of
Fi
n
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e
s
t
a
ndar
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t
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l
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ear
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ns
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2
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If is a matrix such that [ - 3 /2 4 ], find rT:otation of 45°thataboutcorrtehspeondsorigintofola count
ebyrcloackwiprojseec­
l
o
wed
ion onteoththate lTine is a linear transformation
If [ � : = : ] ond X is a matdx such thot tSuppos
andT2 (vs)uppos(ewtherhatevTis2 a vectT orT)su. PrchotvehatthTat(vv) and T(butv)
-[ � - �], findX.
are linearly independent.
3 -2
pr158,oper159,ties of167matrix algebra, 154,
rank ofTheora matem,rix, 205204
Rank
reprproeductsentast, ions146-148
of matrix
rrooww matspacerixof(vaectmatorr)i,x, 138195
ssspccalalanaarrofmulmata setrtiipxofl,e ofmat139a rmaticesr,ix, 156140
square matrix, 139
Review Questions
1.
(a)
(b) A
A * 0,
(c)
XA =
A,
AA T
= 0.
X
A TA
=0
X = A - 1 B.
(d)
10.
A=
11. A
(I - A) - 1 = I + A + A 2 •
A 3 = 0,
12.
(e)
LU
13.
(f)
�
mXn
(g) A
(h)
(i)
(j)
A
ll�r .
14.
IR 3
IR 3 .
=
IR 4 ---+ IR 5
X
A
9. A �
AX =
A
A
A
16. A
=
17.
In Exercises 2-7, let A =
and B =
Compute the indicated matrices, ifpossible.
4. TA
3.
2.
5.
6.
7.
AA T
A-1 =
A
15. A
IR 2 ---+ IR 2
T.
8. A
A�
mXn
A
AA T
A
A TA
IR 2 ---+ IR 2
18.
A.
19.
AT
IR 2 ---+ IR 2
y = - 2x.
: !R n ---+ !R n
20.
=O
=
0
*0
E i genva l ues and
E i genvectors
latent,secular,
characteristic,proper,
eiroot,
gen number
value,
Almost every combination of
the adjectives
and
with the nouns
and
has been used in
the literature for what we call a
proper value.
4.0
Introduction : A ovnamical svstem o n Graphs
Weestinsgawreisnultths.eBotlasthchapt
eovr tchaihat intesrandatintghmate Lesrilxiemulmodeltipliofcatpopul
ion ofatteionnprgrooduces
intebri­t
Mar
k
wt
h
exhi
steadystsatndatessuichn cerbehavitain osri.tuFiatrsiot ns.we wiOnel loofokthate anotgoalhserofittehraistivchapte proceser iss,toro help you
under
-Paul R. Halmos
t
h
at
us
e
s
mat
r
i
c
es
.
(
I
n
t
h
e
pr
o
bl
e
ms
t
h
at
fol
l
o
w,
you
wi
l
fi
n
d
i
t
hel
p
ful
t
o
us
e
a
CAS
or
a
cal
c
ul
a
t
o
r
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mat
r
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capabi
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i
t
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o
faci
l
i
t
a
t
e
t
h
e
comput
a
t
i
o
ns.
)
(2nd
edition)
Our
exampl
e
i
n
vol
v
es
gr
a
phs
(
s
ee
Sect
i
o
n
A
i
s
any
gr
a
ph
Van Nostrand, 1958, p. 102 which every vertex is adj acent to every other vertex. If a complete graph has vertiin­
ces, it is denotePid cbyk anyKw vectFor exampl
e
,
Fi
g
ur
e
s
h
ows
a
r
e
pr
e
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of
K4
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4
or x ison IRthatwithisnonnegat
ivethentriandes andso on.labelComput
the veretitchese
ofadjK4acencywith matthe rcomponent
s
ofx,
l
a
bel
e
d
wi
iAx.x A TrofyK4thandis forrelsaebelvertahlevectvertoircsesxofandtheexplgraphainwi, inthtethrme cors ofrethspeondigraph,ng
component
s
of
how the new laNowbels can be dettheerprmoinceseds frinomProthbleeolmd laThatbels. is, for a given choice of x,
ces asges.descSirnibceedcomponent
above and tshofentapplhe vecty Aoagairs thnems(andelvagaies win, landgetagaiquinte)
untrlaerlagibele,l awetpathewitverelrntiemer
hem byatdiiovnidriensguleachts in tvecthe vector byor its
after each
iteration. Thus, if a tcomput
CAS
dynami­
Finite Dimensional Vector Spaces
cal system,
3.7).
complete graph
n
4. 1
Problem 1
v1
x1,
1.
iterate
Problem 2
scale
VJ
V2
------
we wil replace it by
largest component
.!. [ � ] [ � ]
4 1
Figure 4 . 1
K4
1
·5
0.25
0.25
253
Chapter 4 Eigenvalues and Eigenvectors
254
thisisforprocestshenguarantandees thatUsteheatlalerasgest ttecomponent
ofandeachtwo-vectdeciormwiall-pnowlace
beNotaccure Dotahcy.atthWhat
n
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o
ns
appearshouls todbehavehappeninotincedg? that, in each case, the labeling vector is
You
grappr(waiphsthooutachiwistnchgaltiahnicerg)s s.ttWhat
eaadyin vectsitsatotehrevect(aresltoaetradyioandnsshtapplaiptebetlaybeltwheene!)adj. Labelthaecencynewthelmatverabeltrisicxandes ofonetthheemorolcompl
ed onestimet?ee
Make aifconjwe elactbelure aboutwithththeegenersteadyal casstatee vectWhator andis tapplhe sytetadyhe adjstaatelcency
abel? What
happens
matrix Thewithout scaling? is shown in Figure Repeat the process in
ProblWeemswil tnowhroughexplowiretthhtehpris ogrcesaph.s with some other clas es of graphs to see if they
behave
me way.e, Theis the graphisshthowne grainphFiwigurthe 4.3ver. tices arranged in a cyclic
fashion.thFore saexampl
Repeat
oblemsthe tgener
hroughal caswie.th cycles for various
values of Repeat
and makePrthoeblaprconjeomcesesctwiofutrhPre about
val
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of
What
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?
A
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ar
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such thhaveat eververy vertictesex, tihnen
the graph is denotRepeated bythe prForocesexampl
e,blemsis thtehrgroughaph in wiFigthurcompl
e ete bipartite
s
of
Pr
o
graphsBy the endforofvarthioisuschaptvalueesr, youof wiWhatl be ihappens
?
have made in this Introduction. n a position to explain the observations you
1.
K4 ,
K3
Ks .
Problem 3
A
Kw
Problem 4
Kn
A
Figure 4 . 2
1
3
Figure 4 . 3
Problem 1
cycle Cn
Cs
Problem 6
odd
n
1
n
6
3
even
Cn
n.
74-78
complete bipartite graph
37
Kn , n -
Problem 8
K3 , 3
Kn , n
Figure 4 . 4
4.2.
Petersen graph
Problem 5
n.
U
U
1
n
U
4.4.
3
I n t ro d u ct i o n to E i g e n v a l u e s a n d E i g envectors
eady stofatepopulvectoartioi nn tgrheowtconth. eForxt ofa
tInMarwoChapt
applkov chaiiecratinons:wewithMarencount
ktransovitiechaiornedmatntsheandrinotx tihoaensLestofeadylaiestmodel
or atihadon vecttheoprr opersatitsyfythinatg
for, whera Lese lireematpresreintx edathsteeadysteadystatsetavectte groorwtswastahteravectapopul
te. For example, we saw that
]
]
]
and
[�5
[
[
[
The
German
adj
e
ctive
means
.
"own'' orand"characteristicareof'charac­
I
n
t
h
i
s
chapt
e
r
,
we
i
n
ves
t
i
g
at
e
t
h
i
s
phenomenon
mor
e
gener
a
l
y
.
That
i
s
,
for
a
s
q
uar
e
ma­
teristic
of
a
matrix
in
the
sense
that
t
r
i
x
we
as
k
whet
h
er
t
h
er
e
exi
s
t
nonzer
o
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s
s
u
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at
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i
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t
a
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c
al
a
r
mul
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i
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they
contai
n
i
m
portant
informa­
of
Thi
s
i
s
t
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e
and
i
t
i
s
one
of
t
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e
mos
t
cent
r
a
l
pr
o
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e
ms
i
n
l
i
n
ear
tion
aboutThetheletter,.\
nature(lambda),
of the the algebra. has applications throughout mathematics and in many other fields as well.
matrix.
Greek equiis used
valentforofeigenvalues
the English
letter
Let
be
an
s
c
al
a
r
i
s
cal
l
e
d
an
of
i
f
mat
r
i
x
.
A
because
at
one
time
they
were
also
t
h
er
e
i
s
a
nonzer
o
vect
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s
u
ch
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at
Such
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vect
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xi
s
cal
l
e
d
an
knownis aspronounced "EYE-gun:'
The prefix
of corresponding to
3,
Px = x;
Lx = rx
Eigen­
x
x
4
0
0.25
eigenvectors
A,
x.
L,
It
D e fi n it i o n
latent values.
eigen
r
0.4
0.7 0.2 0.4
=
0.6
0 3 0.8 0.6
eigen
values
P,
L,
tor
A
x
eigenvalue problem,
A
x
nXn
A.
A x = Ax.
A
eigenvalue A
eigenvec­
Section 4.1 Introduction to Eigenvalues and Eigenvectors
Exa m p l e 4 . 1
255
Show that x [ � ] is an eigenvector of [ � � ] and find the corresponding
eigenvalue.
We compute
[ � � ] [ � ] [:] [ � ]
from which it follows that xis an eigenvector of corresponding to the eigenvalue
A=
=
Solution
= 4x
=4
Ax =
A
Exa m p l e 4 . 2
4.
4
genvalue.ue of [ � ] and determine all eigenvectors corresShowponditnhgatto tihs ians eigeienval
Weequimusvaletntshtowo ththeatequattherieonis a vectosroxwesuchneedthatto comput
x Sx.e Butthe nulthisl
equat
i
o
n
i
s
space of the matrix We find that
[ I 23 ] [ J [ ]
Since ethme ofcolIunmnsvertibofletMathis matricesriximarpleiecls etarhatly iltisnnulearllyspdependent
, theo.Fundament
al
Theor
ace
i
s
nonzer
Thus
,
x
x has ainnontg therinulviall sspoace:lution, so is an eigenvalue of We find its eigenvectors by
comput
[ 1 ] [I t 1 J
Thus, if x [:J is an eigenvector corresponding to the eigenvalue it satisfies
t or t so these eigenvectors are of the form
5
A=
nonzero
(A - SI)x = 0,
Solution
A - 51.
A
_
SJ =
S
1
4
_
4
5 O
0 5
=
5
A =
-4 2
4 -2
A =
A.
-4 2 0
- 0
[A - SI I O J =
�
4 -2 0
0
0 0
=
x 1 - x2 = 0,
5,
x 1 = x2 ,
That is, they are the nonzero multiples of [ t ] (or, equivalently, the nonzero multiples
of [ :Ji
.+
Thest thseetsetofofall eigenvectvectorsocorrs inretshpeondinulnl gsptaceo anofeigenval,\Iu. eIt folofloanws that tmathis setrix of
ieisgjuenvect
ors, together with the zero vector in the null space of
1
,\
nonzero
!R n , is
A-
nXn
A - AI.
A
256
Chapter 4 Eigenvalues and Eigenvectors
Let
be
an
mat
r
i
x
and
l
e
t
be
an
ei
g
enval
u
e
of
The
colcallleedcttihone of all eigenvectof oandrs coris rdenot
espondied nbyg to together with the zero vector, is
Therefore, in Example { t[ � ] } .
D e fi n it i o n
A
nxn
4.2, E5
Exa m p l e 4 . 3
A,
=
eiShowgenspthace.at is an eigenvalue of
produces As in Example we compute the null space of Row reduction
[ - � � - !] [ � � - � i
ee thatotrhsecornulrelssppondiace ofng to this ieis gnonzer
oe. satHence,isfy x1 is an ei-g2x3envalue
oforfroxm1 andwhicthheweeig2x3senvect
enval
u
. It follows that
A
=
6
A
=
4.2,
Solulion
A
-
6I
A
=
-
2
2
-4
A
A,
=
-
-
-----+
0
0
-
6I.
0
6I
6
x2 +
+ x2
=
0
,
Inion we can gisavyse athgeomet
ric ionrtserpreandtationarofe tparhe anotl eli.oThusn of ,an ieis ganenvecteigenvect
or. Theor
equat
at
t
h
e
vect
ofif if andis paronlalyeilfto trwher
ansfoerms is itnhteomata parrixaltrealnsvectforomrat[oior,nequicorvrealspentondily, nifgandto only
Find the eigenvectors and eigenvalues of [ � O ] geometrically.
Example We rTheecognionlzyevectthators thisatthemapsmatparrix aofl elatoretfhleemselctionves arein tvecthe x-orasxipars a(sleeel
to the y-axis (i.e., multiples of [ �] ), w�ich are reversed (eigenvalue and vectors
paral el to the x-axis (i.e., multiples of [ ] ), which are sent to themselves (eigenvalue
cor(seereFispgondiure n.g ei. gAccor
enspacesdinglarye, and are the eigenvalues of and the
span([ � ] ) and span([ � ] )
IR 2 ,
Ax Ax
Ax
=
A
TA (x)
Exa m p l e 4 . 4
A.
EA-
A
eigenspace
A
A
x,
TA
x
A
3.56 ) .
x
A] .
A
Solulion
x
=
-1
F
F
- 1 ),
1)
0
4 5)
A
E_ 1
=
=
-1
A
=
E1
1
A,
=
Section 4.1 Introduction to Eigenvalues and Eigenvectors
251
y
3
-3
The eigenvectors of a reflection
Figure 4 . 5
y
4
3
2
3
4
Figure 4 . 6
The discussion
is basedPictur­
on the
article
"Eigenpictures:
ingSteventhe Schonefeld
EigenvectorinProblem" by
(1996),
pp. 316-319.
The College
Mathematics Journal 26
of eigoenvect
oirfsandgeometonlyriifcalx landy is Axto drareawalxigandnedAxin ahead-straigthto­
tlianile..AnotThenIn Fihgxerurwiwayel4.be6to, xianthseiinangkenvect
r
of
A
ei
g
envect
o
r
of
A
but
i
s
not
.
eigenvalorsuegeometthenricsoalliys,anywe need
non­
zeronloyIfconsmulxistiiandperleeitofghenvect
x.e effSo,ectoiroffofweAAonwantcorretsopvectsearondiocrnhsg. forFitogurteihgeeenvect
4.7(a) shows what happens when
we transform unit vectors with the matrix A [ � �] of Example 4.[\�ny,]isplay
the results head-to-tail, as in Figure 4.6. We can see that the vector x 1 is an
eigenvector, but we also notice that there appears to be an eigenvector in the second
quadrant. Indeed, this is the case, and it turns out to be the vector [ � j �] .
y
A,
unit
=
=
-
/ v'2
258
Chapter 4 Eigenvalues and Eigenvectors
y
y
(b)
(a)
Figure 4 . 1
1
(b), weorsseate what
happens
when
we
us
e
t
h
e
mat
r
i
x
A
ITherne arFiegurnoeei4.g7envect
[
1
al
l
!
to firincdineitegrenvect
oiorns onceofthem-but
we have tonehe corrquesestpioondin renmaig eingsenval
­
uesdo we,Weandfinowrwest fihavenknowd thaehowgeomet
p
r
e
t
a
t
:
How
an eigenvalue ofA ieif gandenvalonluyesifofthaeginulvenl smatpaceriofx?AThe- keyisisnontthe robsiviaelr.vation that is
Recall from Section that the determinant of a matrix A [ ; �] is
tFurhe texprhermesorioe,nthdete Fundament
A -al TheorandeAmiofls innververtitbibleleifMatandriconles yguarif detantAeesisthnonzer
o
.
atonla ma­y if
titrsixdethaserma nontinantriivsizeral nulo. Putl sptaceing itfhandese factonlys tiofgetit ishernoni, wenverseettibhlate-hence,
i
f
and
(fosr fact charmatactriecresizesat
leiegasenvalt) iusesan, andeigenval
u
e
of
A
i
f
and
onl
y
i
f
det(
A
0.
Thi
moment, though,weletwi's sleesohowon generto usaeliiztewiitttho squarematmatrirciceses. ofarbitrary size. For the
Find all of the eigenvalues and corresponding eigenvectors of the matrix A
[ � �] from Example 4. 1 .
tion det(A The- precedi0. nSignrceemarks show that we must find all solutions of the equa­
det (A det [ 1 l ]
-1
wetionneedare easto isloylvfound
e the quadr
to be atic equat
4 andion These are th0.erTheeforesotlhuetieionsgenvalto thuiess equa­ofA.
=
A
AI
3.3
=
2 X 2
=
be,
ad
A
-
2X2
AI ) =
2X2
Exa m p l e 4 . 5
A
Solulion
AI )
- AI)
=
=
3 -A
A
=
=
3-A
(3 - A ) ( 3 - A )
A 2 - 6A + 8 =
A 2.
=
=
A 2 - 6A + 8
Section 4.1 Introduction to Eigenvalues and Eigenvectors
259
To find the eigenvectors corresponding to the eigenvalue A = 4, we compute the
null space of A - 4I. We find
l
[-1 1 J [1 1 J
[:J
{ [:: ] } { [ � ] }
[A - 4I oJ =
from which it follows that x =
l
1 O
-+
-1 0
0
-1 O
0 0
is an eigenvector corresponding to A = 4 if and
only if x 1 - x2 = 0 or x 1 = x2 • Hence, the eigenspace E4 =
span
([ � ] )
.
Similarly, for A = 2, we have
so y =
[A - 2I I 0 l =
[;J
= x2
[ � � I �J [� � I �J
{ [ �2 ] } { [ � ] } ([ � ] )
=
---+
is an eigenvector corresponding to A = 2 if and only if y 1 + y2 = 0 or
y 1 = -Yi- Thus , the eigenspace E2 =
= y2
= span
.
Figure 4.8 shows graphically how the eigenvectors of A are transformed when
multiplied by A: an eigenvector x in the eigenspace E4 is transformed into 4x, and an
eigenvector y in the eigenspace E2 is transformed into 2y. As Figure 4.7(a) shows, the
eigenvectors of A are the only vectors in IR 2 that are transformed into scalar multiples
of themselves when multiplied by A.
y
Ay = 2y
-4
-3
-2
2
-1
-1
-2
-3
-4
How transforms eigenvectors
Figure 4 . 8
A
3
4
Chapter 4 Eigenvalues and Eigenvectors
260
Remark
You will recall that a polynomial equation with real coefficients (such as
the quadratic equation in Example 4.5) need not have real roots; it may have complex
roots. (See Appendix C.) It is also possible to compute eigenvalues and eigenvectors
when the entries of a matrix come from "ll_P ' where p is prime. Thus, it is important to
specify the setting we intend to work in before we set out to compute the eigenvalues
of a matrix. However, unless otherwise specified, the eigenvalues of a matrix whose
entries are real numbers will be assumed to be real as well.
Exa m p l e 4 . 6
Interpret the matrix in Example 4.5 as a matrix over "11_ 3 and find its eigenvalues in
that field.
A2 = -2 = 1, A2 - A=6A 1 = A= -1 = 2A2 2 =
The solution proceeds exactly as above, except we work modulo 3. Hence,
the quadratic equation
+ 8 0 becomes +
0. This equation is the
same as
giving
and
as the eigenvalues in Z 3 • (Check
that the same answer would be obtained by first reducing A modulo 3 to obtain
S o lulion
[� �]
Exa m p l e 4 . 1
and then working with this matrix.)
= [� -�]
= - AI) = [ -1A --1A ] = ,\2 1
A= A=
Find the eigenvalues of A
S o lulion
(a) over IR and (b) over the complex numbers C.
We must solve the equation
O
det (A
det
+
(a) Over IR, there are no solutions, so A has no real eigenvalues.
(b) Over C, the solutions are
i and
- i. (See Appendix C.)
In the next section, we will extend the notion of determinant from
2 2
X
to
n X n matrices, which in turn will allow us to find the eigenvalues of arbitrary square
..
I
matrices. (In fact, this isn't quite true-but we will at least be able to find a polynomial
equation that the eigenvalues of a given matrix must satisfy.)
Exercises 4 . 1
In Exercises 1 -6, show that v is an eigenvector ofA and find
the corresponding eigenvalue.
[� �l = [ � ]
= [� �l v = [ -�]
= [ - ! �l = [ _� ]
1. A =
2. A
3. A
v
v
4. A
= [! = �l = [�]
= [ 1 1 -; ] .F [ -: ]
-1
-2
= [ : 2 1 ] = [ 11 ]
v
0
5. A
6. A
0
0
,v
•
Section 4.1 Introduction to Eigenvalues and Eigenvectors
A
In Exercises 7- 12, show that is an eigenvalue of A and
find one eigenvector corresponding to this eigenvalue.
7. A =
8. A =
9. A =
IO. A =
11. A =
12. A �
[� 2] A
[� 2] ,
[ _� 4] A
[! -2]
0 : J A -I
H 0 -I ] �
[: 2 � , A = 2
, =3
-1
3
IR2
In Exercises 1 9-22, the unit vectors x in and their
images Ax under the action of a 2 X 2 matrix A are drawn
head-to-tail, as in Figure 4.7. Estimate the eigenvectors and
eigenvalues ofA from each ''eigenpicture."
19.
y
,\ = - 1
,
5
-7 ,
=1
A = -6
1
In Exercises 1 3 - 1 8, find the eigenvalues and eigenvectors of
A geometrically.
13. A =
14. A =
15. A =
[ -� �]
[� �]
[� �]
[il� �d]
[� �]
[� -�]
(reflection in the y-axis)
20.
(reflection in the line y = x)
(projection onto the x-axis)
(projection onto the line through the
2
origin ith irection vector
16. A =
17. A =
[i])
and a factor of 3 vertically)
18. A =
2
(stretching by a factor of horizontally
(counterclockwise rotation of
about the origin)
90°
261
y
262
Chapter 4 Eigenvalues and Eigenvectors
21.
y
29. A =
[� �]
30. A =
[1
0
1 + i
1
i
]
In Exercises 3 1 -34, find all of the eigenvalues of the ma­
trix A over the indicated Zp .
31. A =
33. A =
[ � �]
[ ! �]
over Z 3
32. A =
over Zs
34. A =
[�
[: �]
over Zs
35. (a) Show that the eigenvalues of the 2 X 2 matrix
A=
-2
22.
[ : �]
are the solutions of the quadratic equation
A 2 - tr(A)A + det A = 0, where tr (A) is the trace
of A. (See page 1 62.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
y
A = H a + d ± V(a
-
d) 2 + 4bc )
( c) Show that the trace and determinant of the matrix A
in part (a) are given by
tr(A) = A 1 + A 2 and det A = A 1 A 2
35.
In Exercises 23-26, use the method of Example 4.5 to find
all of the eigenvalues of the matrix A. Give bases for each of
the corresponding eigenspaces. Illustrate the eigenspaces and
the effect of multiplying eigenvectors by A as in Figure 4. 8.
23. A =
[� - � ]
[� �]
24. A =
[! �]
[ � �]
where A 1 and A 2 are the eigenvalues of A.
36. Consider again the matrix A in Exercise Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.
37. Show that the eigenvalues of the upper triangular
matrix
A=
[ � �]
are A = a and A = d, and find the corresponding
eigenspaces.
25. A =
26. A =
,E:S738. Let a and b be real numbers. Find the eigenvalues and
,E:S7
corresponding eigenspaces of
In Exercises 27-30, find all of the eigenvalues of the matrix
A over the complex numbers C. Give bases for each of the
_
corresponding eigenspaces.
27. A =
[ � �]
_
28. A =
[ -3]
2
l
O
over the complex numbers.
11 f
Section 4.2 Determinants
263
Determ i n a nts
Historically, determinants preceded matrices-a curious fact in light of the way linear
algebra is taught today, with matrices before determinants. Nevertheless, determi­
nants arose independently of matrices in the solution of many practical problems,
and the theory of determinants was well developed almost two centuries before
matrices were deemed worthy of study in and of themselves. A snapshot of the his­
tory of determinants is presented at the end of this section.
Recall that the determinant of the 2 X 2 matrix A =
[ aa2l 1l
det A = a l l a 22 - a 1 2 a 2 1
We first encountered this expression when we determined ways to compute the
inverse of a matrix. In particular, we found that
[ aa2111 aa2212 ]
I
The determinant of a matrix A is sometimes also denoted by I A , so for the 2 X 2
matrix A =
Warning
we may also write
. I aa ll1 aa 12 I
2 22
tation. It 1s. easy to mistake
·
[ aa2ll1 aa2212 ]
This notation for the determinant is reminiscent of absolute value no-
. for determmant,
.
, the notat10n
for
,
the notation for the matrix itself. Do not confuse these. Fortunately, it will usually
be clear from the context which is intended.
11
We define the determinant of a X matrix A = [a] to be
det A = l a l = a
I I
(Note that we really have to be careful with notation here: a does not denote the
absolute value of a in this case.) How then should we define the determinant of a
3 X 3 matrix? If you ask your CAS for the inverse of
A=
[
the answer will be equivalent to
A-1 =
[� � ;]
g h
l
� Jg - di ai - cg cd - af
ei - fh
ch - bi
bf - ce
l
dh - eg bg - ah ae - bd
where Li = aei - afh - bdi + bfg + cdh - ceg. Observe that
Li = aei - afh - bdi + bfg + cdh - ceg
= a ( ei - fh) - b (di - Jg) + c(dh - eg)
=a
I � �I - b I � �I c I � �I
+
264
Chapter 4 Eigenvalues and Eigenvectors
and that each of the entries in the matrix portion of A - i appears to b e the determi­
nant of a 2 X 2 submatrix of A. In fact, this is true, and it is the basis of the definition
of the determinant of a 3 X 3 matrix. The definition is recursive in the sense that the
determinant of a 3 X 3 matrix is defined in terms of determinants of 2 X 2 matrices.
Defi n ition
[
a ll
Let A = a 21
a 31
]
a 13
a 23 . Then the determinant of A is the scalar
a 33
(1)
Notice that each of the 2 X 2 determinants is obtained by deleting the row and col­
umn of A that contain the entry the determinant is being multiplied by. For example,
the first summand is a 11 multiplied by the determinant of the submatrix obtained by
deleting row 1 and column 1 . Notice also that the plus and minus signs alternate in
Equation ( 1 ) . If we denote by A;j the submatrix of a matrix A obtained by deleting row
i and column j, then we may abbreviate Equation ( 1 ) as
det A = a ll det A ll - a 12 det A 12 + a 13 det A 13
3
2: ( - l ) 1 +ja 1j det A 1j
j= l
For any square matrix A, det A;j is called the (i, j)-minor of A.
Exa m p l e 4 . 8
Compute the determinant of
Solution
We compute
det A = 5
1 � �I
-
- ( - 3)
I� �I I� �I
+2
_
= 5 ( 0 - ( - 2 )) + 3 ( 3 - 4 ) + 2 ( - 1 - 0 )
= 5 (2 ) + 3 ( - 1 ) + 2 ( - 1 ) = 5
:+
With a little practice, you should find that you can easily work out 2 X 2 determinants
in yom head. Wciting out the secood line in the above solution is then unnecess
Another method for calculating the determinant of a 3 X 3 matrix is analogous
to the method for calculating the determinant of a 2 X 2 matrix. Copy the first two
columns of A to the right of the matrix and take the products of the elements on the six
Section 4.2 Determinants
265
diagonals shown below. Attach plus signs to the products from the downward-sloping
diagonals and attach minus signs to the products from the upward-sloping diagonals.
(2)
+
This method gives
In Exercise 19, you are asked to check that this result agrees with that from Equa­
tion ( 1) for a 3 X 3 determinant.
Exa m p l e 4 . 9
Calculate the determinant of the matrix in Example 4.8 using the method shown in (2).
Solution
products:
We adjoin to A its first two columns and compute the six indicated
0
- 10
-9
-2
Adding the three products at the bottom and subtracting the three products at the
top gives
det A
=
0 + ( - 1 2) + ( - 2) - 0 - ( - 1 0) - ( - 9)
=
5
as before.
Warning
We are about to define determinants for arbitrary square matrices.
However, there is no analogue of the method in Example 4.9 for larger matrices. It is
valid only for 3 X 3 matrices.
Delerminanls of n x n Malrices
The definition of the determinant of a 3 X 3 matrix extends naturally to arbitrary
square matrices.
D e fi D i l i O D
Let A [a;) be an n x n matrix, where n 2: 2. Then the determinant of A is the scalar
=
n
_L ( - l ) 1 +ia 11 det A 11
j� I
(3)
266
Chapter 4 Eigenvalues and Eigenvectors
It is convenient to combine a minor with its plus o r minus sign. To this end, we define
the (i, j)-cofactor ofA to be
With this notation, definition (3) becomes
<let A
2: a 1j C ij
j= l
n
=
(4)
Exercise 20 asks you to check that this definition correctly gives the formula for the
determinant of a 2 X 2 matrix when n 2.
Definition ( 4) is often referred to as cofactor expansion along the first row. It is
an amazing fact that we get exactly the same result by expanding along any row (or
even any column) ! We summarize this fact as a theorem but defer the proof until the
end of this section (since it is somewhat lengthy and would interrupt our discussion
if we were to present it here).
=
Theorem 4 . 1
The Laplace Expansion Theorem
The determinant of an n X n matrix A
=
[a;) , where n 2: 2, can be computed as
2: a ;j Cij
j= l
n
(5)
(which is the cofactor expansion along the ith row) and also as
2: a ij cij
i= l
n
=
(6)
(the cofactor expansion along the jth column).
A
Since Cij = ( - l ) i +j <let A;j , each cofactor is plus or minus the corresponding minor,
with the correct sign given by the term ( - 1 y +j . quick way to determine whether
the sign is + or - is to remember that the signs form a "checkerboard" pattern:
+
+
+
+
+
+
+
+
Section 4.2 Determinants
Exa m p l e 4 . 1 0
261
Compute the determinant of the matrix
by (a) cofactor expansion along the third row and (b) cofactor expansion along the
second column.
Solution
(a) We compute
= 2 1 -� � 1 - (- 1) 1 � � 1 + 3 1 � -� 1
2 ( - 6) + + 3 ( 3 )
=
=5
8
PierrebornSimonin Normandy,
Laplace (17France,
49-1827) (b) In this case, we have
was
and
was expected
tos mathematical
become a
clergyman
unti
l
hi
talents
at school.
Hecontributions
madeweremanynoticed
important
= - ( - 3) 1 � � 1 + 0 1 � � 1 - ( - 1 ) 1 � � I
to
calculus,
probabi
He was
3( - 1) + 0 + 8
=
anBonaparte
examilitny,eratandofthetheastronomy.
young
Napoleon
=
5
RoyalNapoleon
Artillery was
Corps
and
later,
when
inof thepower,Interior
servedandbrithen
efly asChancellor
Minister
Laplace
wasEmpire
granted Notice that in part (b) of Example 4. 1 0 we needed to do fewer calculations than
theinof the1806
titlSenate.
e and
of Count
of
the
in part (a) because we were expanding along a column that contained a zero entry­
receivedinthe1817.title of namely,
a 22 ; therefore, we did not need to compute C22 • It follows that the Laplace
Marquis de Laplace
Expansion Theorem is most useful when the matrix contains a row or column with
lots of zeros, since, by choosing to expand along that row or column, we minimize the
number of cofactors we need to compute.
Exa m p l e 4 . 1 1
Compute the determinant of
3
+
First, notice that column has only one nonzero entry; we should there­
fore expand along this column. Next, note that the / - pattern assigns a minus sign
Solution
268
Chapter 4 Eigenvalues and Eigenvectors
to the entry a 2 3 = 2 . Thus, we have
det A = a 13 C 13 + a 23 C 23 + a 33 C 33 + a 43 C43
= O(C 13 ) + 2C 23 + O(C 33 ) + O(C43 )
2 -3 1
-2 1 - 1 3
-2
0
We now continue by expanding along the third row of the determinant above
(the third column would also be a good choice) to get
( I � � I - I � � I)
det A = - 2 - 2 =
- 2 ( - 2 ( - 8) 5)
- 2 ( 1 1 ) = - 22
-
(Note that the + I pattern for the 3 X 3 minor is not that of the original matrix but
that of a 3 X 3 matrix in general.)
-
4
The Laplace expansion is particularly useful when the matrix is (upper or lower)
triangular.
Exa m p l e 4 . 1 2
Compute the determinant of
2
0
A= 0
0
0
Solution
0
-3
3 2 5
0 1 6
0 0 5
0 0 0
4
7
0
2
-1
We expand along the first column to get
3
0
det A = 2
0
0
2
1
0
0
7
5
0
6
5
2
0 -1
(We have omitted all cofactors corresponding to zero entries.) Now we expand along
the first column again:
0
1 6
2
det A = 2 · 3 0 5
0 0 -1
Continuing to expand along the first column, we complete the calculation:
det A = 2 · 3 · 1
I� 1
2
= 2 . 3 . 1 . ( 5 ( - 1 ) - 2 . 0 ) = 2 . 3 . 1 . 5 . ( - 1 ) = - 30
-1
4
Section 4.2 Determinants
269
Example 4. 1 2 should convince you that the determinant of a triangular matrix is the
product of its diagonal entries. You are asked to give a proof of this fact in Exercise 2 1 .
We record the result as a theorem.
Theorem 4 . 2
The determinant of a triangular matrix is the product of the entries on its main
diagonal. Specifically, if A = [a;) is an n X n triangular matrix, then
<let A = a 1 1 a 22 · • · a ""
Nole
In general (that is, unless the matrix is triangular or has some other special
form), computing a determinant by cofactor expansion is not efficient. For example,
the determinant of a 3 X 3 matrix has 6 = 3 ! summands, each requiring two multipli­
cations, and then five additions and subtractions are needed to finish off the calcula­
tions. For an n X n matrix, there will be n! summands, each with n - 1 multiplications,
and then n ! - 1 additions and subtractions. The total number of operations is thus
T(n) = (n - l ) n ! + n ! - 1 > n!
Even the fastest o f supercomputers cannot calculate the determinant o f a mod­
erately large matrix using cofactor expansion. To illustrate: Suppose we needed to
calculate a 50 X 50 determinant. (Matrices much larger than 50 X 50 are used to store
the data from digital images such as those transmitted over the Internet or taken by a
digital camera.) To calculate the determinant directly would require, in general, more
6
than 50! operations, and 50! 3 X 1 0 4 . If we had a computer that could perform
1
2
a trillion ( 1 0 ) operations per second, it would take approximately 3 X 1 0 52 sec­
onds, or almost 1 0 4 5 years, to finish the calculations. To put this in perspective, con­
sider that astronomers estimate the age of the universe to be at least 10 billion ( 1 0 1 0 )
years. Thus, on even a very fast supercomputer, calculating a 50 X 50 determinant by
cofactor expansion would take more than 1 0 30 times the age of the universe!
Fortunately, there are better methods-and we now turn to developing more
computationally effective means of finding determinants. First, we need to look at
some of the properties of determinants.
=
Properties of Determinants
The most efficient way to compute determinants is to use row reduction. However,
not every elementary row operation leaves the determinant of a matrix unchanged.
The next theorem summarizes the main properties you need to understand in order
to use row reduction effectively.
Theorem 4 . 3
Let A = [a ij ] be a square matrix.
a. If A has a zero row (column), then <let A = 0.
b. IfB is obtained by interchanging two rows (columns) of A, then <let B = - <let A.
c. If A has two identical rows (columns), then <let A = 0.
d. If B is obtained by multiplying a row (column) of A by k, then <let B = k <let A.
e. If A, B, and C are identical except that the ith row (column) of C is the sum of
the ith rows (columns) of A and B, then <let C = <let A + <let B.
f. If B is obtained by adding a multiple of one row (column) of A to another row
(column), then <let B = <let A.
210
Chapter 4 Eigenvalues and Eigenvectors
Proof We will prove (b) as Lemma 4. 14 at the end of this section. The proofs of
properties (a) and (f) are left as exercises. We will prove the remaining properties in
terms of rows; the corresponding proofs for columns are analogous.
(c) If A has two identical rows, swap them to obtain the matrix B. Clearly, B = A, so
<let B = det A. On the other hand, by (b), det B = - det A. Therefore, det A = - det A,
so det A = 0.
( d) Suppose row i ofA is multiplied by k to produce B; that is, b ij = ka;j for j = 1 , . . . , n.
Since the cofactors Cij of the elements in the ith rows of A and B are identical (why?),
expanding along the ith row of B gives
n
n
n
det B = � b ;j Cij = � ka ij cij = k � a ij cij = k det A
j= l
j= l
j= l
(e) As in (d), the cofactors Cij of the elements in the ith rows of A, B, and C are
identical. Moreover, c ij = a;j + b;j for j = 1 , . . . , n. We expand along the ith row of C
to obtain
n
n
n
n
det C = � c ;j Cij = � ( au + b ;)Cu = � a ;j CiJ + � bu Cu = det A + det B
j= l
j= l
j= l
j= l
Notice that properties (b), ( d), and ( f) are related to elementary row operations.
Since the echelon form of a square matrix is necessarily upper triangular, we can
combine these properties with Theorem 2 to calculate determinants efficiently. (See
Exploration: Counting Operations in Chapter 2, which shows that row reduction of
an n X n matrix uses on the order of n 3 operations, far fewer than the n ! needed for
cofactor expansion.) The next examples illustrate the computation of determinants
using row reduction.
Exa m p l e 4 . 1 3
[
20 35 - 31 ]
A
Compute <let A if
(a)
=
(b) A �
Solution
-4
-6
2
[ i -� j il
(a) Using property (f) and then property (a), we have
2
3
5
<let A = 0
-4 -6
- 1 R, + 2R1 2 3
3
0 5
2
0 0
-1
3 =O
0
Section 4.2 Determinants
211
(b) We reduce A to echelon form as follows (there are other possible ways to do this):
1
2 -4 5
0
3
0 -3 6
2 - 4 5 Ri/ 3
2
0 - 3 6 R ,<->R, 0
0
= -3
2
4
5 7
4
2
4
5 7
5 -1 -3 1
5 -1
- 1 -3
1
1
2
2
0 -1
0 -1
R3 - 2R 1
1tt
4
R4 - SR 1
R
R
2 -4
5
2 -9
0
Q -1
= -3
= - ( - 3)
0
4
4
0
7
7
3
3
2 -9
0 -1
5
2 -4
0
0
3
<let A =
2
5
-1
-4
5
-3
2
5
7
1
1
0 -1
2
Q -1
2 -9
=3
0 1 5 - 33
0
0
0
0 - 13
= 3 . 1 . ( - 1 ) . 1 5 . ( - 1 3 ) = 585
R, + 4R,
R4 + 2 R2
Remark By Theorem 4.3, we can also use elementary column operations in the
process of computing determinants, and we can "mix and match'' elementary row and
column operations. For example, in Example 4. l 3(a), we could have started by adding
column 3 to column 1 to create a leading 1 in the upper left-hand corner. In fact, the
method we used was faster, but in other examples column operations may speed up
the calculations. Keep this in mind when you work determinants by hand.
oe1erminan1s of Eleme n1arv Malrices
Recall from Section 3.3 that an elementary matrix results from performing an ele­
mentary row operation on an identity matrix. Setting A = In in Theorem 4.3 yields
the following theorem.
Theorem 4 . 4
Let E be an n X n elementary matrix.
a. If E results from interchanging two rows of In , then <let E = - 1 .
b. If E results from multiplying one row of In by k, then <let E = k.
c. If E results from adding a multiple of one row of In to another row, then
det E = 1 .
TheGreek
word verb is derivedwhifromch
themeans
"toisgrasp:
'pIner mathematics,
athatlemma
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"hel
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we
"grasp
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andmoreuse
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lemma
lambanein,
Since <let In = 1 , applying (b), (d), and (f) of Theorem 4.3 immediately gives
(a), (b), and (c), respectively, of Theorem 4.4.
Proof
Next, recall that multiplying a matrix B by an elementary matrix on the left per­
forms the corresponding elementary row operation on B. We can therefore rephrase
(b), (d), and (f) of Theorem 4.3 succinctly as the following lemma, the proof of which
is straightforward and is left as Exercise 43.
212
Chapter 4 Eigenvalues and Eigenvectors
Lem m a 4 . 5
Let B b e an n X n matrix and let E b e an n X n elementary matrix. Then
det ( EB )
=
( det E ) ( det B )
We can use Lemma 4.5 to prove the main theorem of this section: a characteriza­
tion of invertibility in terms of determinants.
Theorem 4 . 6
A
square matrix A is invertible if and only if det A * 0.
Let A be an n X n matrix and let R be the reduced row echelon form of A.
We will show first that det A * 0 if and only if det R * 0. Let E 1 , E2 ,
, Er be the
elementary matrices corresponding to the elementary row operations that reduce
A to R. Then
Proof
•
•
•
Taking determinants of both sides and repeatedly applying Lemma 4.5, we obtain
( det E) · · · ( det E 2 ) ( det E , ) ( det A )
det R
By Theorem 4.4, the determinants of all the elementary matrices are nonzero. We
conclude that det A * 0 if and only if det R * 0.
Now suppose that A is invertible. Then, by the Fundamental Theorem oflnvertible
Matrices, R In , so det R 1 * 0. Hence, det A * 0 also. Conversely, if det A * 0,
then det R * 0, so R cannot contain a zero row, by Theorem 4.3 (a). It follows that
R must be In (why?), so A is invertible, by the Fundamental Theorem again.
=
...--
=
=
Determinants and Matrix Operations
....,...
Theorem 4 . 1
Let's now try to determine what relationship, if any, exists between determinants and
some of the basic matrix operations. Specifically, we would like to find formulas for
det (kA), det(A + B), det(AB), det(A - i ), and det(A T ) in terms of det A and det B .
Theorem 4.3(d) does not say that det(kA) = k det A. The correct relationship
between scalar multiplication and determinants is given by the following theorem.
If A is an n x n matrix, then
det ( kA )
....,...
=
kn det A
You are asked to give a proof of this theorem in Exercise 44 .
Unfortunately, there is no simple formula for det (A + B), and in general,
det (A + B) * det A + det B. (Find two 2 X 2 matrices that verify this.) It therefore
comes as a pleasant surprise to find out that determinants are quite compatible with
matrix multiplication. Indeed, we have the following nice formula due to Cauchy.
Section 4.2 Determinants
wasth. bornbriliniaParis
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to the newuntiking.l the university dropped its requirement that faculty swear an oath of
213
Augustin Louis Cauchy ( 1 789- 1 857)
A
X
Theorem 4 . 8
If A and B are n X n matrices, then
det (AB ) = ( det A ) ( det B )
We consider two cases: A invertible and A not invertible.
If A is invertible, then, by the Fundamental Theorem of lnvertible Matrices, it can
be written as a product of elementary matrices-say,
Proof
A = E , E2 • · · Ek
Then AB = E 1 E2
· · ·
EkB, so k applications of Lemma 4.5 give
Continuing to apply Lemma 4.5, we obtain
det (AB ) = det ( E 1 E 2 • • • E k ) det B = ( det A ) ( det B )
On the other hand, if A is not invertible, then neither is AB, by Exercise 47
in Section 3.3. Thus, by Theorem 4.6, det A = 0 and det(AB) = 0. Consequently,
det (AB) = (det A)(det B), since both sides are zero.
Exa m p l e 4 . 1 4
Applying Theorem 4.8 to A =
[ 22 �]
[ � � ],
[ 1216 � ]
12
and B =
AB =
and that det A = 4, det B = 3, and det(AB) =
(Check these assertions! )
we find that
= 4 · 3 = (det A) (det B), as claimed.
4
The next theorem gives a nice relationship between the determinant of an invertible
matrix and the determinant of its inverse.
214
Chapter 4 Eigenvalues and Eigenvectors
Theorem 4 . 9
If A is invertible, then
1
det (A - 1 ) = -det A
Since A is invertible, AA - I = I, so <let (AA - I ) = <let I = 1 . Hence,
(<let A)(det A - 1 ) = 1 , by Theorem 4.8, and since <let A * 0 (why?), dividing by
<let A yields the result.
Proof
.......
Exa m p l e 4 . 1 5
Verify Theorem 4.9 fo r the matrix A of Example 4.14.
Solulion
We compute
so
det A - 1 =
Remark
(�) (�) - ( - �) ( - �) = % - t = � = �
de A
The beauty of Theorem 4.9 is that sometimes we do not need to know
what the inverse of a matrix is, but only that it exists, or to know what its determinant
is. For the matrix A in the last two examples, once we know that <let A = 4 * 0, we
immediately can deduce that A is invertible and that <let A - l = � without actually
computing A - l .
We now relate the determinant of a matrix A to that of its transpose A T_ Since the
rows of A T are just the columns of A, evaluating <let A T by expanding along the first
row is identical to evaluating <let A by expanding along its first column, which the
Laplace Expansion Theorem allows us to do. Thus, we have the following result.
Theorem 4 . 1 0
For any square matrix A,
det A = det AT
Gabriel
Cramer (1704-1752)
was
athatSwissbearsmathematician.
The
rule
in 1750, inhishisname
treatisewas published
Ascasesearlyof theas 1730,
however,
special
formula
were
known
other
mathematicians,
including
theto(1698-1746),
Scotsman
Colin
Maclaurin
perhaps the greatest
ofwerethetheBritish
mathematicians
"successors of Newton:who'
Introduction
to the Analysis ofAlgebraic Curves.
Cra m e r 's Rule and lhe Adioinl
In this section, we derive two useful formulas relating determinants to the solution
of linear systems and the inverse of a matrix. The first of these, Cramer's Rule, gives
a formula for describing the solution of certain systems of n linear equations in n
variables entirely in terms of determinants. While this result is of little practical use
beyond 2 X 2 systems, it is of great theoretical importance.
We will need some new notation for this result and its proof. For an n X n ma­
trix A and a vector b in !R n , let A;(b) denote the matrix obtained by replacing the ith
column of A by b. That is,
Column i
-1.-
A ; ( b ) = [ a 1 • · · h · · · an l
Section 4.2 Determinants
Theorem 1 . 1 1
215
Cramer's Rule
Let A be an invertible n X n matrix and let b be a vector in !R n . Then the unique
solution x of the system Ax = b is given by
X·l =
Proof
e2 ,
.
•
•
det (A; ( b ))
for i = 1 , . . . , n
det A
The columns of the identity matrix I = In are the standard unit vectors e 1 ,
, en - If Ax = b, then
AI; ( x) = A [ e 1
x · · · en ] = [Ae 1
= [ a 1 · · · b · · · an ] = A; ( b )
•
•
•
Ax · · · Ae n ]
Therefore, by Theorem 4.8,
( detA ) ( det I; ( x)) = det (AI; ( x)) = det (A ; ( h ))
Now
1
0
0
1
X1
Xz
0
0
0
0
det I; ( x) = 0
0
X;
0
0 = X;
0
0
0
0
Xn - 1
Xn
1
0
0
as can be seen by expanding along the ith row. Thus, (<let A) x; = det(A; (h)), and the
result follows by dividing by <let A (which is nonzero, since A is invertible) .
Exa m p l e 4 . 1 6
Use Cramer's Rule t o solve the system
X 1 + 2X 2 = 2
- x 1 + 4x 2 = 1
Solution
det A
We compute
= 1-� !I =
6, det (A 1 ( b ))
= I� !I =
= 1-� �I
6, and det (A 2 ( b ))
=3
By Cramer's Rule,
det (A 2 ( b ))
det (A 1 ( b )) 6
X 1 = ---- = - = 1 and x2 = ---­
det A
6
det A
3
6
2
216
Chapter 4 Eigenvalues and Eigenvectors
As noted previously, Cramer's Rule is computationally inefficient for
all but small systems of linear equations because it involves the calculation of many
determinants. The effort expended to compute just one of these determinants, using
even the most efficient method, would be better spent using Gaussian elimination to
solve the system directly.
Remark
The final result of this section is a formula for the inverse of a matrix in terms of
determinants. This formula was hinted at by the formula for the inverse of a 3 X 3
matrix, which was given without proof at the beginning of this section. Thus, we have
come full circle.
Let's discover the formula for ourselves. If A is an invertible n X n matrix, its
inverse is the (unique) matrix X that satisfies the equation AX = I. Solving for X one
column at a time, let "i be the jth column of X. That is,
X·1 =
Therefore, Axj = ej , and by Cramer's Rule,
det (A/e) )
det A
However,
i th column
a 11 a i 2
a z, a 22
<let (A; ( e) ) =
-J,
0
0
ai , a1 2
a n , an 2
a 1n
a2n
aJn
0
= ( - l )j + i det A = C
JI
JI
a nn
which is the (j, i ) -cofactor of A.
It follows that xij = ( 1 /det A)Cji • so A - i = X = ( 1 / <let A) [Cji ] = ( 1 /det A) [Cij f.
In words, the inverse of A is the transpose of the matrix of cofactors of A, divided by
the determinant of A.
The matrix
is called the adjoint (or adjugate) of A and is denoted by adj A. The result we have
just proved can be stated as follows.
Section 4.2 Determinants
Theorem 4 . 1 2
Let A be an invertible n X n matrix. Then
A-1 =
Exa m p l e 4 . 1 1
211
1 adj A
--
det A
Use the adjoint method to compute the inverse of
A=
Solution
C1 1 =
C2 1 =
C31 =
We compute det A =
+ I � 4 1 -18
- I� 1 1
+ I � -14 1
3
-3
2 and the nine cofactors
- I� 41
+I� 11
- I � -14 1
+ I� �I 4
I � � I -1
+I� �I
[ -1 : �� � J [ �4� -1- � �� 1
1
1 [ -18l � �� 1 [ - � - : 1
-3
-
-
[ �1 � -�i
-3
=
C1 2 =
=3
C22 =
= 10
C3 2 =
-3
-
-3
= 10
C1 3 =
= -2
C2 3 =
= -6
C33 =
=
=
-
= -2
The adjoint is the transpose of the matrix of cofactors-namely,
r
adj A =
Then
A-1 =
det A
10
-6 -2
adj A = - 2
=
-
3
-2
-1 - 2
-2
=
-t
-2
t
which is the same answer we obtained (with less work) in Example 3.30.
1
Proof of lhe Laplace Expansion Theorem
Unfortunately, there is no short, easy proof of the Laplace Expansion Theorem. The
proof we give has the merit of being relatively straightforward. We break it down into
several steps, the first of which is to prove that cofactor expansion along the first row
of a matrix is the same as cofactor expansion along the first column.
Le m m a 4 . 1 3
Let A be an n X n matrix. Then
218
Chapter 4 Eigenvalues and Eigenvectors
We prove this lemma by induction on n . For n = 1, the result is trivial. Now
assume that the result is true for (n - 1) X ( n - 1) matrices; this is our induction
hypothesis. Note that, by the definition of cofactor (or minor), all of the terms con­
taining au are accounted for by the summand au Cu. We can therefore ignore terms
containing au.
The ith summand on the right-hand side of Equation (7) is a; 1 C; 1 = a; 1 ( - 1 ) i + l
det A; 1 . Now we expand det A; 1 along the first row:
Proof
a 13
a 11
a ln
a ; - 1, 2 a ; - 1,3
a ; +1, 2 a ; +1,3
a ; - 1,J
a ; +1,J
a ; - 1,n
a ; +J,n
a n3
a n)
a n,n
a12
an2
1
The jth term in this expansion of det A; 1 is a 11( - 1) +J - l det A 1 ;, 11, where the nota­
tion A kl, rs denotes the submatrix of A obtained by deleting rows k and l and columns
r and s. Combining these, we see that the term containing a; 1 a 11 on the right-hand
side of Equation ( 7) is
a ; 1 ( - l ) ; +1 a 1/ - l ) 1 +J - I det A 1 u1 = ( - l ) i +J+ 1 a ; 1 a 11 det A 1 ;,11
What is the term containing a i l a 11 on the left-hand side of Equation (7)? The
factor a 11 occurs in the jth summand, a 11 C 1i = a 1/ - 1) 1 +J det A 11 . By the induction
hypothesis, we can expand det A 1i along its first column:
a z1
a 31
a z, J - 1 a z, J+ 1
a 3, j - I a 3, j+ I
a2n
a 3n
a;1
a ;,j - 1 a ;, J+ l
a ;n
an !
a n, j - 1 a n, )+ !
a nn
The ith term in this expansion of det A 11 is a i l ( - l ) (i - l ) + l det A 1 ;, 11 , so the term con­
taining a; 1 a 1i on the left-hand side of Equation (7) is
i
a 1J ( - 1 ) 1 +Ja ; 1 ( - 1 ) ( ; - i ) + i det A 1 ;, 11 - ( - l) +J+ i a il a 1J det A 1 ;, 11
which establishes that the left- and right-hand sides of Equation (7) are equivalent.
Next, we prove property (b) of Theorem 4.3.
Le m m a 4 . 1 4
Let A be an n X n matrix and let B be obtained by interchanging any two rows
(columns) of A. Then
det B = - det A
Section 4.2 Determinants
219
Proof
Once again, the proof is by induction on n. The result can be easily checked
when n = 2, so assume that it is true for ( n - 1 ) X (n - 1 ) matrices. We will prove
that the result is true for n X n matrices. First, we prove that it holds when two adja­
cent rows of A are interchanged-say, rows r and r + 1 .
By Lemma 4. 13, we can evaluate <let B by cofactor expansion along its first col­
umn. The ith term in this expansion is ( - 1 ) 1 + i b; 1 <let B; 1 . If i * r and i * r + 1, then
b; 1 = a; 1 and B; 1 is an (n - 1) X (n - 1) submatrix that is identical to A; 1 except that
two adjacent rows have been interchanged.
al l
a12
aln
a; 1
a;2
a; n
a r + l, l a r + l, 2
a rl
a r2
a r + l ,n
a rn
a n2
a nn
an l
Thus, by the induction hypothesis, <let B; 1 = - <let A; 1 if i * r and i * r + 1 .
If i = r, then b; 1 = a r + l , l and B i l = A r + l , l ·
al l
Row
a12
aln
i -+ a r + l, l a r + l, 2
a rl
a r2
a r + l ,n
a rn
an ]
an2
a nn
Therefore, the rth summand in <let B is
( - l ) r + I b r l <let B r ! = ( - l ) r + I a r + I,I <let A r + I, I = - ( - l ) ( r + I) + I a r + I,I <let A r + I, I
Similarly, if i = r + 1 , then b; 1 = a r 1 , B; 1 = A r 1 , and the (r + l )st summand in <let B is
( - l ) ( r + I) + I b r I,I det B r I,I = ( - l ) 'a r l det A r l = - ( - l ) r + l a r l det A r l
+
+
In other words, the rth and (r + l)st terms in the first column cofactor expansion of
<let B are the negatives of the ( r + 1 )st and rth terms, respectively, in the first column
cofactor expansion of <let A.
Substituting all of these results into <let B and using Lemma 4. 1 3 again, we obtain
n
� ( - 1 ) i + l b; 1 <let B; 1
<let B = ,L,.;
i= l
n
� ( - l ) i + l b; 1 <let B; 1 + ( - l ) r+ l b rl <let B r1 + ( - l ) ( r + J ) + l b r + l, l <let B r + l,l
i= l
i ;" r, r + l
n
� ( - l ) i + 1 a; 1 ( - det A; 1 ) - ( - l ) ( r + l) + l a r + l , l det A r + l, l - ( - l ) r + l a r1 det A r1
i= l
i t= r,r + l
n
- ,L,.;
� ( - 1 ) i + l a; 1 <let A; 1
i= l
- det A
280
Chapter 4 Eigenvalues and Eigenvectors
This proves the result for n X n matrices if adjacent rows are interchanged. To
see that it holds for arbitrary row interchanges, we need only note that, for example,
rows r and s, where r < s, can be swapped by performing 2( s - r) - 1 interchanges of
adjacent rows (see Exercise 67) . Since the number of interchanges is odd and each one
changes the sign of the determinant, the net effect is a change of sign, as desired.
The proof for column interchanges is analogous, except that we expand along
row 1 instead of along column 1 .
We can now prove the Laplace Expansion Theorem.
Proof of Theorem 4 . 1
Let B be the matrix obtained by moving row i of A to the top,
using i - 1 interchanges of adjacent rows. By Lemma 4. 14, <let B = ( - 1 ) i - l det A. But
b 1j = a;j and B 1j = A;j for j = 1 , . . . , n.
a;1
a in
a ;j
a ll
a 1j
a ln
Thus,
<let B = a ; - 1,1
a; + 1,1
a ; - 1,j
a; + 1,j
a i 1, n
ai+ l, n
a nl
a nj
a nn
-
det A = ( - l ) i - 1 det B = ( - l ) i - 1 2: ( - 1 ) 1 +j b 1j det B 1j
j� I
n
n
n
= ( - 1 ) ; - i 2: ( - 1 ) 1 +ja ij <let A ij = 2: ( - l ) i +ja ij <let A ij
j� I
j� I
which gives the formula for cofactor expansion along row i.
The proof for column expansion is similar, invoking Lemma 4. 1 3 so that we can
use column expansion instead of row expansion (see Exercise 68).
A Brief Hislorv of oe1erm ina n1s
ATakakazu
self-taughtSekichiKowa
ld prodi(1642-1708)
gy,
was
descended
fromIn aaddition
family of
samurai
warriors.
discovering
determinants,
hetoequations,
wrote about
diophantine
magic
squares,
and
Bernoulli
numbers
(before
Bernoulli) andin calculus.
quite likely made
discoveries
As noted at the beginning of this section, the history of determinants predates that of
matrices. Indeed, determinants were first introduced, independently, by Seki in 1 683
and Leibniz in 1 693. In 1 748, determinants appeared in Maclaurin's Treatise on A lge ­
bra, which included a treatment of Cramer's Rule up to the 4 X 4 case. In 1 7 50, Cramer
himself proved the general case of his rule, applying it to curve fitting, and in 1 772,
Laplace gave a proof of his expansion theorem.
The term determinant was not coined until 1 80 1 , when it was used by Gauss.
Cauchy made the first use of determinants in the modern sense in 1 8 1 2 . Cauchy, in
fact, was responsible for developing much of the early theory of determinants, in­
cluding several important results that we have mentioned: the product rule for de­
terminants, the characteristic polynomial, and the notion of a diagonalizable matrix.
Determinants did not become widely known until 1 84 1 , when Jacobi popularized
them, albeit in the context of functions of several variables, such as are encountered in
a multivariable calculus course. (These types of determinants were called "Jacobians"
by Sylvester around 1 850, a term that is still used today.)
Section 4.2 Determinants
Gottfried phiWilhelm
vonandLeibniz
(1646-1716)
wasprobabl
bornyinbestLeipzig
andforstudied
law, (with
theology,
l
osophy,
mathematics.
He
is
known
developing
Newton,
independentl
ybranches
) the mainof imathematics
deas of differenti
alsand
integralve.calculus.
However,the notion
his
contributions
to
other
are
al
o
impressi
He
developed
ofothers
a determinant,
knew
versions
ofandCramer'
s Rulefoundation
and the Laplace
Expansion
Theoremworkbeforehe
were
given
credi
t
for
them,
laid
the
for
matrix
theory
through
didHe believed
on quadratic
forms.
Leibnizofalsogoodwasnotation
the firstand,to develop
the binary
systemnotation
of arithmetic.
i
n
the
importance
along
with
the
familiar
fora
deri
v
ati
v
es
and
integrals,
i
n
troduced
a
form
of
subscri
p
t
notation
for
the
coefficients
of
linear system that is essential y the notation we use today.
281
By the late 1 9th century, the theory of determinants had developed to the stage
that entire books were devoted to it, including Dodgson's An Elementary Treatise on
Determinants in 1 867 and Thomas Muir's monumental five-volume work, which
appeared in the early 20th century. While their history is fascinating, today deter­
minants are of theoretical more than practical interest. Cramer's Rule is a hopelessly
inefficient method for solving a system of linear equations, and numerical methods
have replaced any use of determinants in the computation of eigenvalues. Determi­
nants are used, however, to give students an initial understanding of the characteristic
polynomial (as in Sections 4. 1 and 4.3).
..
I
Exercises 4 . 2
•
Compute the determinants in Exercises 1 -6 using cofactor
expansion along the first row and along the first column.
1 0 3
1. 5
0
2
1
3. - 1
0
-1
0
2 3
5. 2 3 1
2
3
0 1
2. 2 3
-1 3
0
-1
-1
-2
0
1 1 0
4. 1 0
0
2 3
6. 4 5 6
7 8 9
8. 2
3
1
0
-2
-1
1
2
0
tan e
- sin e
cos e
12. b c d
0 e 0
-1 0 3
5 2 6
0 0
4 2 1
0 0 0 a
0 0 b c
15.
0 d e f
g h
cos e sin e
10.
0 cos e
0 sin e
0 a 0
a b 0
11. 0 a b
a 0 b
13.
Compute the determinants in Exercises 7-15 using cofactor
expansion along any row or column that seems convenient.
5 2 2
7. - 1
2
3 0 0
1 3
-4
9. 2 - 2 4
-1 0
j
2
1
14.
0
2
0 3
0 2
-1
0
-1
2
4
-3
Chapter 4 Eigenvalues and Eigenvectors
282
In Exercises 1 6- 1 8, compute the indicated 3 X 3 determi­
nants using the method of Example 4. 9.
16. The determinant in Exercise 6
17. The determinant in Exercise 8
18. The determinant in Exercise 1 1
19. Verify that the method indicated in (2) agrees with
Equation ( 1 ) for a 3 X 3 determinant.
20. Verify that definition ( 4) agrees with the definition of a
2 X 2 determinant when n = 2.
2 1 . Prove Theorem 4.2. [Hint: proof by induction would
be appropriate here.]
A
In Exercises 22-25, evaluate the given determinant using
elementary row and/or column operations and Theorem 4.3
to reduce the matrix to row echelon form.
22. The determinant in Exercise 1
23. The determinant in Exercise 9
24. The determinant in Exercise 1 3
25. Th e determinant i n Exercise 14
1
-2
2
1 0
3
27. 0 - 2 5
0 4
0
0 1
0
28. 0
5 2
3 -1 4
29.
2 3
30. 0 4 1
6 4
4 1
31. -2 0
5 4
1
0
32.
0
0
0
-3
33.
0
0
0 0 0
0 1 0
1 0 0
0 0 1
1 0 1 0
0
0 1
34.
1 1 0 0
1
0 0
2a 2b 2c
2a b/ 3 - c
36. 2 d e/ 3 -f
e f
g h
2g h/ 3 - i
a-c b c
d e f
37. a b c
38. d - f e f
g- i h
g h
2c b a
39. 2f e d
2i h g
a + 2g b + 2h c + 2i
40. 3d + 2g 3e + 2h 3f + 2i
h
g
35. d
41. Prove Theorem 4.3(a).
43. Prove Lemma 4.5.
In Exercises 26-34, use properties of determinants to
evaluate the given determinant by inspection. Explain
your reasoning.
1 1
26. 3 0
2 2
Find the determinants in Exercises 35-40, assuming that
a b c
d e f =4
g h
2
-1
3
-3
5
-4
-2
2
3
-2
1
2 0 0
0 0 0
0 0 4
0
0
42. Prove Theorem 4.3(f) .
44. Prove Theorem 4.7.
In Exercises 45 and 46, use Theorem 4. 6 to find all values of
k for which A is invertible.
-k
k+1
45. A =
-8 k 1
k
46. A �
[1 � :
[ � �]
2
k
In Exercises 47-52, assume that A and B are n X n matrices
with det A = 3 and det B = - 2. Find the indicated
determinants.
47. det (AB )
50. det ( 2A )
48. det (A 2 )
5 1 . det ( 3B r )
49. det ( B- 1 A )
52. det (AA r)
In Exercises 53-56, A and B are n X n matrices.
53. Prove that det (AB) = det (BA).
54. If B is invertible, prove that det (B - 1AB) = det (A) .
5 5 . I f A i s idempotent (that is, A 2 = A), find all possible
A
values of det (A) .
56. square matrix A i s called nilpotent i f A m = 0 for
some m > 1. (The word nilp otent comes from the
Latin nil, meaning "nothing;' and potere, meaning
"to have power:' nilpotent matrix is thus one that
A
Section 4.2 Determinants
becomes "nothing" -that is, the zero matrix-when
raised to some power.) Find all possible values of
det (A) if A is nilpotent.
In Exercises 57-60, use Cramer's Rule to solve the given
linear system.
58. 2x - y = 5
57. x + y = 1
x + 3y = - 1
x-y=2
59. 2x + y + 3z = 1
60. x + y - z = 1
y+ z=1
x+y+z=2
=3
z=1
x-y
In Exercises 61-64, use Theorem 4. 12 to compute the in­
verse of the coefficient matrix for the given exercise.
62. Exercise 58
61. Exercise 57
63. Exercise 59
64. Exercise 60
65. If A is an invertible n X n matrix, show that adj A is
where P and S are square matrices. Such a matrix is
said to be in block (upper) triangularform. Prove that
det A = ( det P) ( det S)
[Hint: Try a proof by induction on the number of rows
of P.]
70. (a) Give an example to show that if A can be
partitioned as
A =
(b) Assume that A is partitioned as in part (a) and
that P is invertible. Let
B=
[-�-f-�-]
Writi n g Project
[��-�� -d-?-]
Compute det (BA) using Exercise 69 and use the
result to show that
det A = det P det ( S - RP - 1 Q )
[The matrix S - RP - 1 Q is called the Schur com­
1
( adj A ) - 1 = --A = adj (A - 1 )
det A
66. If A is an n X n matrix, prove that
det ( adj A ) = ( det A ) n - I
A =
[�- f{j
where P, Q, R, and S are all square, then it is not
necessarily true that
det A = ( det P) ( det S) - ( det Q ) ( det R )
also invertible and that
67. Verify that if r < s, then rows r and s of a matrix can
be interchanged by performing 2 (s - r) - 1 inter­
changes of adjacent rows.
68. Prove that the Laplace Expansion Theorem holds for
column expansion along the jth column.
69. Let A be a square matrix that can be partitioned as
283
plement of P in A, after Issai Schur ( 1 875- 1 94 1 ),
who was born in Belarus but spent most of his
life in Germany. He is known mainly for his fun­
damental work on the representation theory of
groups, but he also worked in number theory,
analysis, and other areas.]
(c) Assume that A is partitioned as in part (a), that
P is invertible, and that PR = RP. Prove that
detA = det ( PS - RQ )
Which Came First: The Matrix or the Determinant?
The way in which matrices and determinants are taught today-matrices before
determinants-bears little resemblance to the way these topics developed histori­
cally. There is a brief history of determinants at the end of Section 4.2.
Write a report on the history of matrices and determinants. How did the nota­
tions used for each evolve over time? Who were some of the key mathematicians
involved and what were their contributions?
1. Florian Cajori, A History of Mathematical Notations (New York: Dover, 1 993).
2. Howard Eves, An Introduction to the History of Mathematics (Sixth Edition)
(Philadelphia: Saunders College Publishing, 1 990).
3. Victor J. Katz, A History of Mathematics: An Introduction (Third Edition)
(Reading, MA: Addison Wesley Longman, 2008).
4. Eberhard Knobloch, Determinants, in Ivor Grattan-Guinness, ed., Compan­
ion Encyclopedia of the History and Philosophy of the Mathematical Sciences
(London: Routledge, 2013).
Vi g n ette
L e wis C arro ll's C o n d e n s at i o n M e th o d
In 1 866, Charles Dodgson-better known by his pseudonym Lewis Carroll­
published his only mathematical research paper. In it, he described a "new and brief
method" for computing determinants, which he called "condensation:' Although not
well known today and rendered obsolete by numerical methods for evaluating determi­
nants, the condensation method is very useful for hand calculation. When calculators or
computer algebra systems are not available, many students find condensation to be their
method of choice. It requires only the ability to compute 2 X 2 determinants.
We require the following terminology.
If A is an n x n matrix with n 2: 3, the interior of A, denoted
int(A), is the (n - 2 ) X (n - 2 ) matrix obtained by deleting the first row, last
row, first column, and last column of A.
D e fi n it i o n
is muchLewisbetterCarroll,
knownunderby his
pen
name,
which he wroteand
He aland
so wrote
several
mathematics
books
collections
oflogic puzzles.
Charles Lutwidge Dodgson ( 1 8321 898)
in Wonderland
Looking Glass.
Alice's Adventures
Through the
This
visgCarrol
nette ils'sbased
on theonarticle
"Lewi
Condensati
atiEveng Determi
nants"
byMethod
Adriaforn RiEvalce anduNovember
Torrence
in
2006,
12-15. For ofurther
detaiseels of
thepp.Davicondensati
n
method,
d M. Bressoud,
We will illustrate the condensation method for the 5 X 5 matrix
0
2
3 -1 2
2
3 1 -4
A =
1
2 -1
2 1
3
1 - 1 2 -2
-4
2
0
Begin by setting A 0 equal to the 6 X 6 matrix all of whose entries are 1 . Then, we set
A 1 = A. It is useful to imagine A 0 as the base of a pyramid with A 1 centered on top of
A 0 . We are going to add successively smaller and smaller layers to the pyramid until
we reach a 1 X 1 matrix at the top-this will contain <let A. (Figure 4.9)
Math Horizons,
Proofs and
Confirmations: The Story of the
Alternating Sign Matrix Conjecture,
MAA
UniversiSpectrum
ty Press, Series
1999).(Cambridge
284
Ao
Figure 4 . 9
alNextl , we s"ucondens
bmatricees" Aof1 iAnto a matrix A; whose entries are the determinants of
2X2
1:
4X4
3
-� 1 1 -� � I I � �I
1
1
2
I �
3
-�1
�
�
1
1
1
1
1
1
2
2
�
I
�
�
A; = 2 - - 2 1 1
I 3 - 11 1 1 - 11 - 21 1 1 - 1 2 1 1 2 - � I
1
1 -! � 1 1 1 -� 1 1 -� � I I� -� 1
2
[ -j
11
7
-1
1
-7
1
5
-1
-
:
-4
6
1
of A;Aby2 =theA;corr. esponding entry of int(A0) to get matrix
ANow2. WeSiwenceredipeatA0viidstehaleachelprso, cedur
tenthisrymeans
dividing each entry ofA� bye,thconse cortrruectspinondig A�ngfrentomrtyhofe int(Asubmatand sroicon.es ofWeA2obtandaitnh: en
1 - � �l I � - ; 1 1 - ; - : 1
]
A� = 1 : _; 1 1 _; � I I � ! I
,
1
2X2
1
1),
1
-
A'
4
1
1
11
11
60/3
36/2
- 4/ - 1
�= ��� l = [ �� �� -- 2927 ] ,
26/2
12
4
- 27
- 29
26
13
1 = [ - 42 - 94 ] ,
350
- 96
30 1 8 1 8 - 29
4
13 1
12 4
3 1 20
30 1 8
20
18
- 27
- 29
]=[
]
[
A � = [ 1 : -�� I ] =
A5 = =
be checked
by hotodherwimetl prhoodsduce, <leat A = matIrnixgener
al, aforinianng <let A. matrix A,
tAshecancondens
a
t
i
o
n
met
A
cont
n
y, thedmetthenhodbe brtryeaksing down
idfethbye inzerteoriotor ofconsanytruofctthA;e +A; sHowever
contain,s cara zerefulo,
susineceClofweeelarelwoulment
t
o
di
v
i
that we can proaceed.ry row and column operations can be used to eliminate the zeros so
- 42/7
- 96/ ( - 1 )
6
[ 8604/ 1 8 ]
- 94/ 1
350/5
-6
96
- 94
70
[ 8604 ] ,
[ 478 ]
478.
1X1
nXn
1.
285
Exp loration
G e o m etric App l i c at i o n s of D eterminants
This exploration will reveal some of the amazing applications o f determinants to
geometry. In particular, we will see that determinants are closely related to area and
volume formulas and can be used to produce the equations of lines, planes, and
certain other curves. Most of these ideas arose when the theory of determinants was
being developed as a subject in its own right.
The Cross Product
[ :: ]
Recall from Exploration: The Cross Product in Chapter 1 that the cross product
of u �
[ ::J
ond v �
;,
thrndor n X v defined by
U
XV
=
[
U 2 V 3 - U 3 V2
U 3 V 1 - U 1 V3
U 1 V2 - U 2 V 1
]
If we write this cross product as ( u 2 v3 - u 3 v2 ) e1 - ( u1v3 - u 3 v1 ) e2 + ( u1v2 - u 2 v1 ) e3 ,
where e1, e2 , and e3 are the standard basis vectors, then we see that the form of this
formula is
if we expand along the first column. (This is not a proper determinant, of course,
since e1, e2 , and e 3 are vectors, not scalars; however, it gives a useful way of remem­
bering the somewhat awkward cross product formula. It also lets us use properties of
determinants to verify some of the properties of the cross product.)
Now let's revisit some of the exercises from Chapter 1 .
286
1. Use the determinant version of the cross product to compute u X v.
[} HJ [ l m
[
[
i
l
n =:
lm
[:l [ :: l [ :l
[ U3UU21
( a) u �
�
(b) u �
u�
v�
( d) u �
(o)
2. If u �
and w �
�
U •
-
�
�
how that
(V X W) = det
3. Use properties of determinants (and Problem 2 above, if necessary) to prove
the given property of the cross product.
( a ) v X u = - ( u X v)
(b) u x 0 = 0
( c) u X u = 0
( d ) u X kv = k ( u X v)
( e ) u X ( v + w) = u X v + u X w ( f ) u · ( u X v) = 0 and v · ( u X v) = 0
( g ) u · ( v X w) = ( u X v) · w ( the triple scalar product identity)
Area a n d Vol u m e
We can now give a geometric interpretation of the determinants of 2 X 2 and 3 X 3
matrices. Recall that if u and v are vectors in IFR 3 , then the area A of the parallelogram
determined by these vectors is given by A = II u X v 11 . (See Exploration: The Cross
Product in Chapter 1 . )
4. Let u =
y
[ �:]
[ ::] .
Show that the area A o f the parallelogram
determined by u and v is given by
[H;nt w,;,, u and v "'
Fioure 4 . 1 0
and v =
[ �:] [�:Ji
and
5 . Derive the area formula in Problem 4 geometrically, using Figure 4. 1 0 as a
guide. [Hint: Subtract areas from the large rectangle until the parallelogram remains.]
Where does the absolute value sign come from in this case?
281
h{
vxw
v
Figure 4 . 1 1
6. Find the area of the parallelogram determined by u and v.
( a) u =
Figure 4 . 1 2
[ � ], v [ - � ]
=
(b) u =
[!l [�]
v=
Generalizing from Problems 4-6, consider a parallelepiped, a three-dimensional
solid resembling a "slanted" brick, whose six faces are all parallelograms with oppo­
site faces parallel and congruent (Figure 4. 1 1 ). Its volume is given by the area of its
base times its height.
7. Prove that the volume V of the parallelepiped determined by u, v, and w is
given by the absolute value of the determinant of the 3 X 3 matrix [ u v w] with u,
v, and w as its columns. [Hint: From Figure 4. 1 1 you can see that the height h can be
expressed as h = ll u ll cos e, where e is the angle between u and v X w. Use this fact to
show that V = l u · (v X w) I and apply the result of Froblem 2.]
8. Show that the volume V of the tetrahedron determined by u, v, and w
(Figure 4. 12) is given by
V = i l u · (v X w) I
[Hint: From geometry, we know that the volume of such a solid is V = t (area of the
base) (height).]
Now let's view these geometric interpretations from a transformational point of
view. Let A be a 2 X 2 matrix and let P be the parallelogram determined by the vectors
u and v. We will consider the effect of the matrix transformation TA on the area of P.
Let TA ( P) denote the parallelogram determined by TA (u) = Au and TA ( v) = Av.
9. Prove that the area of TA ( P ) is given by I det Al (area of P ).
10. Let A be a 3 X 3 matrix and let P be the parallelepiped determined by the
vectors u, v, and w. Let TA ( P ) denote the parallelepiped determined by TA (u) = Au,
TA (v) = Av, and TA (w) = Aw. Prove that the volume of TA (P) is given by l det A I
(volume of P ).
The preceding problems illustrate that the determinant of a matrix captures what
the corresponding matrix transformation does to the area or volume of figures upon
which the transformation acts. (Although we have considered only certain types of fig­
ures, the result is perfectly general and can be made rigorous. We will not do so here.)
288
lines and Planes
Suppose we are given two distinct points (x1, y 1) and (x2 , y2 ) in the plane. There is a
unique line passing through these points, and its equation is of the form
ax + by + c = 0
Since the two given points are on this line, their coordinates satisfy this equation.
Thus,
ax1 + by 1 + c = 0
ax2 + by2 + c = 0
The three equations together can be viewed as a system of linear equations in the vari­
ables a, b, and c. Since there is a nontrivial solution (i.e., the line exists), the coefficient
matrix
cannot be invertible, by the Fundamental Theorem of Invertible Matrices. Conse­
quently, its determinant must be zero, by Theorem 4.6. Expanding this determinant
gives the equation of the line.
The equation of the line through the points (x1, y 1) and (x2 , y2 ) is given by
1 1 . Use the method described above to find the equation of the line through the
given points.
( a ) ( 2, 3 ) and ( - 1 , 0 )
( b ) ( 1 , 2 ) and ( 4, 3 )
12. Prove that the three points (xi, Y i) , ( x2 , y2 ), and (x3 , y3 ) are collinear (lie on
the same line) if and only if
X i Yi 1
Xz Y2 1 = 0
X3 Y3
1 3 . Show that the equation of the plane through the three noncollinear points
(x1, Y 1 , z 1 ), (x2 , y2 , z2 ), and (x3 , y3 , z3 ) is given by
x
Xi
Xz
X3
y z
Y1 Z i
=O
Y2 Zz
Y3 Z3
What happens if the three points are collinear? [Hint: Explain what happens when
row reduction is used to evaluate the determinant.]
289
14. Prove that the four points (x1, y 1 , z 1), (x2 , y2 , z2 ), (x3 , y3 , z3 ), and ( x4 , y4 , z4 ) are
coplanar (lie in the same plane) if and only if
X1
Xz
X3
X4
Y1
Y2
Y3
Y4
Z1
Zz
Z3
Z4
=O
Curve Filling
y
�+--+--+---+---. x
-2
2
4
6
Figure 4 . 1 3
When data arising from experimentation take the form of points (x, y) that can be
plotted in the plane, it is often of interest to find a relationship between the variables x
and y. Ideally, we would like to find a function whose graph passes through all of the
points. Sometimes all we want is an approximation (see Section 7. 3 ), but exact results
are also possible in certain situations.
1 5 . From Figure 4. 1 3 it appears as though we may be able to find a parabola
passing through the points A( - 1 , 10), B(O, 5), and C(3, 2). The equation of such a
parabola is of the form y = a + bx + cx 2 . By substituting the given points into this
equation, set up a system of three linear equations in the variables a, b, and c. Without
solving the system, use Theorem 4.6 to argue that it must have a unique solution. Then
solve the system to find the equation of the parabola in Figure 4. 13.
1 6. Use the method of Problem 1 5 to find the polynomials of degree at most 2
that pass through the following sets of points.
(a) A ( l , - 1 ), B ( 2 , 4), C(3, 3)
l
(b) A ( - 1 , - 3 ), B ( , - 1 ), C(3, 1 )
1 7. Generalizing from Problems 1 5 and 1 6, suppose a 1 , a 2 , and a 3 are distinct
real numbers. For any real numbers b 1 , b 2 , and b 3 , we want to show that there is a
unique quadratic with equation of the form y = a + bx + cx 2 passing through the
points (a1, b 1) , (a 2 , b 2 ), and (a 3 , b 3 ). Do this by demonstrating that the coefficient
matrix of the associated linear system has the determinant
a 1 ai
a 2 a� = ( a 2 - a 1 ) ( a 3 - a 1 ) ( a 3 - a z )
a 3 a�
which is necessarily nonzero. (Why?)
1 8. Let a 1 , a 2 , a 3 , and a 4 be distinct real numbers. Show that
a1
az
a3
a4
ai
a�
a�
a4z
ai
a 23
= (a 2 - a 1 ) ( a 3 - a 1 ) ( a4 - a 1 ) ( a 3 - a 2 ) ( a4 - a 2 ) ( a4 - a 3 ) * 0
a�
a 43
For any real numbers b 1 , b 2 , b 3 , and b4 , use this result to prove that there is a unique
cubic with equation y = a + bx + cx 2 + dx 3 passing through the four points (a1 , b 1) ,
(a 2 , b2 ), (a 3 , b 3 ) , and (a4 , b4 ) . (Do not actually solve for a, b, c, and d. )
290
19. Let a1, a 2 ,
•
•
•
, a n be n real numbers. Prove that
a 1 ai
a 1 a�
a 3 a�
a n1 - 1
a zn - 1
a 3n - 1
l
II (aj - a ;)
5' i < j -:5 n
a n a zn
a nn - 1
where TI1 :s i <J:s n (a1 - a;) means the product of all terms of the form (a1 - a;), where
i < j and both i and j are between 1 and n. [ The determinant of a matrix of this form
(or its transpose) is called a Vandermonde determinant, named after the French
mathematician A. T. Vandermonde ( 1 735- 1 796).]
Deduce that for any n points in the plane whose x-coordinates are all distinct,
there is a unique polynomial of degree n - 1 whose graph passes through the given
points.
291
292
Chapter 4 Eigenvalues and Eigenvectors
'
Eigenvalues a n d E i g envectors o t n x n M atrices
Now that we have defined the determinant o f an n X n matrix, we can continue our
discussion of eigenvalues and eigenvectors in a general context. Recall from Section 4. 1
that A is an eigenvalue of A if and only if A - AI is noninvertible. By Theorem 4.6, this
is true if and only if det(A - AI) = 0. To summarize:
The eigenvalues of a square matrix A are precisely the solutions A of the equation
det (A - AI) = 0
When we expand det(A - AI ), we get a polynomial in A, called the characteristic
polynomial of A. The equation det(A - AI ) = 0 is called the characteristic equation
of A. For example, if A =
det (A - AI) =
[ : �],
its characteristic polynomial is
l a -e A d -b i\ = (a - i\)(d - A) - be = A2 - (a + d)i\ + (ad - be)
I
If A is n X n, its characteristic polynomial will be of degree n. According to the Fun­
damental Theorem of Algebra (see Appendix D), a polynomial of degree n with real
or complex coefficients has at most n distinct roots. Applying this fact to the charac­
teristic polynomial, we see that an n X n matrix with real or complex entries has at
most n distinct eigenvalues.
Let's summarize the procedure we will follow (for now) to find the eigenvalues
and eigenvectors ( eigenspaces) of a matrix.
Let A be an n X n matrix.
1 . Compute the characteristic polynomial det(A
-
U ) of A.
2. Find the eigenvalues ofA by solving the characteristic equation det (A - AI ) = 0
for i\.
3. For each eigenvalue A, find the null space of the matrix A - i\I. This is
the eigenspace EA , the nonzero vectors of which are the eigenvectors of A
corresponding to A.
4. Find a basis for each eigenspace.
Exa m p l e 4 . 1 8
Find the eigenvalues and the corresponding eigenspaces of
Section 4.3 Eigenvalues and Eigenvectors of
Solution
nX n
Matrices
293
We follow the procedure outlined previously. The characteristic polynomial is
det (A - AI) =
0
1
0 - ;\
- ;\
1
2 - 5 4 - ;\
1
0
1
= - ;\
- 5 4 - ;\
2 4 - ;\
2
= - ;\(;\ - 4;\ + 5 ) - ( - 2 )
= - +4 2
+2
-I A
I 1
I
A3 A - SA
To find the eigenvalues, we need to solve the characteristic equation det(A - AI) = 0
for A. The characteristic polynomial factors as (A - 1 ) 2 (;\ - 2). (The Factor
- the characteristic equation is
Theorem is helpful here; see Appendix D.) Thus,
- (;\ - 1 ) 2 (;\ - 2) = 0, which clearly has solutions ;\ = 1 and ;\ = 2. Since ;\ = 1 is
a multiple root and A = 2 is a simple root, let us label them ;\1 = ;\ 2 = 1 and A3 = 2.
To find the eigenvectors corresponding to ;\1 = ;\ = 1 , we find the null space of
A - II =
Row reduction produces
� J n !]
[
]
n � : �]
n
[A - I I O J =
9--!>-
2
1
-1
-1
-5 4
-5
1 0
-1 1
0
-----+
-5 3
-1
1 -1
0
0
(We knew ;n odvan<e thot we mu" get ot lmt one mo rnw. Why?) Thu', x �
=
[:: l'
in the eigenspace E1 if and only if x1 - x3 = 0 and x2 - x3 = 0. Setting the free variable
x3 = t, we see that x1 = t and x2 t, from which it follows that
-2
[A - 2I I O J =
So x �
[:: l
A3 = 2, we find the null space of A - 21
[ � � � ----+ [ � =i �
i � i
To find the eigenvectors corresponding to
by row reduction:
1
-2
-5 2 0
0 0
0 0
; , ;n the e;gmpoce E, ;f ond only ;f x, � )x, ond x, � !x,. Sethng the
free variable x3 = t, we have
294
Chapter 4 Eigenvalues and Eigenvectors
where we have cleared denominators in the basis by multiplying through by the least
common denominator 4. (Why is this permissible?)
4
Remark
Notice that in Example 4. 1 8, A is a 3 X 3 matrix but has only two distinct
eigenvalues. However, if we count multiplicities, A has exactly three eigenvalues (A = 1
twice and A = 2 once). This is what the Fundamental Theorem of Algebra guarantees.
Let us define the algebraic multiplicity of an eigenvalue to be its multiplicity as a root
of the characteristic equation. Thus, A = 1 has algebraic multiplicity 2 and A = 2 has
algebraic multiplicity 1 .
Next notice that each eigenspace has a basis consisting of just one vector. In other
words, dim E 1 = dim E2 = 1 . Let us define the geometric multiplicity of an eigenvalue
A to be dim E;"' the dimension of its corresponding eigenspace. As you will see in
Section 4.4, a comparison of these two notions of multiplicity is important.
Exa m p l e 4 . 1 9
Find the eigenvalues and the corresponding eigenspaces of
A=
Solulion
[ -� � -�]
1 0
-1
The characteristic equation is
-1 - A
0 = det (A - AI) =
3
1
1
1
-1 - A
-3 = -A
1
-1 - A
0
-A
0
Hence, the eigenvalues are A 1 = A 2 = 0 and A 3 = -2. Thus, the eigenvalue 0 has alge­
braic multiplicity 2 and the eigenvalue - 2 has algebraic multiplicity 1 .
For A 1 = A 2 = 0 , we compute
[A - OJ I O J
[A I OJ
0
0
0
&om which it follow' th"t '"' eigenwdm x �
-� �i [� � J
[ :: l
-1 0
�
in E0 "ti'fi" x, � x,. Thecefoce,
both x2 and x3 are free. Setting x2 = s and x3 = t, we have
For A 3 = - 2,
[A - ( - 2)I I O J
[A + 2I I O J
0 0
-1 O
0 0
0 0
Section 4.3 Eigenvalues and Eigenvectors of
nX n
Matrices
295
so x3 = t is free and x 1 = -x3 = - t and x2 = 3x3 = 3t. Consequently,
It follows that ,\ 1 = ,\ 2 = 0 has geometric multiplicity 2 and ,\ 3 = - 2 has geometric
multiplicity 1 . (Note that the algebraic multiplicity equals the geometric multiplicity
for each eigenvalue.)
4
In some situations, the eigenvalues of a matrix are very easy to find. If A is a
triangular matrix, then so is A - AI, and Theorem 4.2 says that det (A - AI ) is just
the product of the diagonal entries. This implies that the characteristic equation of
a triangular matrix is
(a ll - .\ )(a 22 - A) · · · (a nn - A) = 0
,
from which it follows immediately that the eigenvalues are A 1 = a 11 , A 2 = a 22 ,
A n = a n,,. We summarize this result as a theorem and illustrate it with an example.
•
Theorem 4 . 1 5
The eigenvalues of a triangular matrix are the entries on its main diagonal.
Exa m p l e 4 . 2 0
The eigenvalues of
.
•
are ,\ 1 = 2, ,\ 2 = 1, ,\ 3 = 3, and ,\4 = - 2, by Theorem 4. 15. Indeed, the characteristic
polynomial is just (2 - ,\)( 1 - ,\)(3 - ,\)( - 2 - A).
Note that diagonal matrices are a special case of Theorem 4.15. In fact, a diagonal
matrix is both upper and lower triangular.
Eigenvalues capture much important information about the behavior of a matrix.
Once we know the eigenvalues of a matrix, we can deduce a great many things without
doing any more work. The next theorem is one of the most important in this regard.
Theorem 4 . 1 6
A
square matrix A is invertible if and only if 0 is not an eigenvalue of A.
Let A be a square matrix. By Theorem 4.6, A is invertible if and only if
<let A * 0. But <let A * 0 is equivalent to <let (A - O J ) * 0, which says that 0 is not a
root of the characteristic equation of A (i.e., 0 is not an eigenvalue of A).
Proof
We can now extend the Fundamental Theorem of Invertible Matrices to include
results we have proved in this chapter.
296
Chapter 4 Eigenvalues and Eigenvectors
Theorem 4 . 1 1
The Fundamental Theorem of Invertible Matrices: Version 3
Let A be an n X n matrix. The following statements are equivalent:
a. A is invertible.
b. A x = b has a unique solution for every b in IJ�r.
c. A x = 0 has only the trivial solution.
d. The reduced row echelon form of A is In e. A is a product of elementary matrices.
f. rank(A) = n
g. nullity(A) = 0
h. The column vectors of A are linearly independent.
i. The column vectors of A span IJ�r.
j. The column vectors of A form a basis for IJ�r.
k. The row vectors of A are linearly independent.
1. The row vectors of A span !R n .
m. The row vectors of A form a basis for !R n .
n. det A * 0
o. 0 is not an eigenvalue of A.
The equivalence (a) � (n) is Theorem 4.6, and we just proved (a) � (o) in
Theorem 4. 16.
Proof
There are nice formulas for the eigenvalues of the powers and inverses of a matrix.
Theorem 4 . 1 8
Let A be a square matrix with eigenvalue A and corresponding eigenvector x.
a. For any positive integer n, A n is an eigenvalue of A n with corresponding
eigenvector x.
b. IfA is invertible, then 1 /A is an eigenvalue ofA - i with corresponding eigenvector x.
c. If A is invertible, then for any integer n, A n is an eigenvalue of A n with corre­
sponding eigenvector x.
Proof
We are given that A x = Ax.
(a) We proceed by induction on n. For n = 1 , the result is just what has been given.
Assume the result is true for n = k. That is, assume that, for some positive integer k,
A k x = A k x. We must now prove the result for n = k + 1 . But
A k + 1 x = A (A kx) = A(Akx)
by the induction hypothesis. Using property ( d) ofTheorem 3.3, we have
1
A(Akx) = A k (Ax) = Ak (Ax) = A k +I x
Thus, A k + l x = A k + x, as required. By induction, the result is true for all integers
n 2: 1 .
(b) You are asked to prove this property in Exercise 13.
(c) You are asked to prove this property in Exercise 14.
The next example shows one application of this theorem.
Section 4.3 Eigenvalues and Eigenvectors of
Exa m p l e 4 . 2 1
Compute
Solution
[o ] [ ]
[� � ]
2
1 10 5
.
1 1
Let A
[�]
=
eigenvalues of A are A 1
and v2
�
=
=
and x
=
[�l
nX n
Matrices
291
�
[_ ]
then what we want to find is A 1 0 x. T e
- 1 and A 2 = 2, with corresponding eigenvectors v 1
. That is,
=
1
(Check this.) Since {v1 , v2 } forms a basis for IR 2 (why?), we can write x as a linear com­
bination of v1 and v2 . Indeed, as is easily checked, x 3v1 + 2v2 .
Therefore, using Theorem 4.1 8(a), we have
10
A 1 0x A 10 ( 3v1 + 2v2 ) 3(A 10v 1 ) + 2 (A v2 )
3(A: 0 ) v1 + 2 ( A� 0 ) v2
1
3 + 211
205 1
3 ( - 1 ) 10
+ 2 ( 2 1 0)
1
2
-1
2
-3 + 2
4093
=
=
=
=
=
[ l]
[] [
] [ ]
=
=
This is certainly a lot easier than computing A 1 0 first; in fact, there are no matrix
multiplications at all!
When it can be used, the method of Example 4.2 1 is quite general. We summarize
it as the following theorem, which you are asked to prove in Exercise 42.
Theorem 4 . 1 9
Suppose the n X n matrix A has eigenvectors v1 , v2 ,
, vm with corresponding
eigenvalues A 1 , A 2 , . . . , A m . If x is a vector in !R n that can be expressed as a linear
combination of these eigenvectors-say,
•
.
.
then, for any integer k,
Warning
The catch here is the "if" in the second sentence. There is absolutely
no guarantee that such a linear combination is possible. The best possible situation
would be if there were a basis of !R n consisting of eigenvectors of A; we will explore
this possibility further in the next section. As a step in that direction, however, we
have the following theorem, which states that eigenvectors corresponding to distinct
eigenvalues are linearly independent.
Theorem 4 . 2 0
Let A be an n X n matrix and let A 1 , A 2 , . . . , A m be distinct eigenvalues of A with cor­
responding eigenvectors v1 , v2 , , vm . Then v1 , v2 , , vm are linearly independent.
.
.
•
.
.
•
Chapter 4 Eigenvalues and Eigenvectors
298
The proof is indirect. We will assume that v1 , v2 , , vm are linearly dependent
and show that this assumption leads to a contradiction.
If v1 , v2 , , vm are linearly dependent, then one of these vectors must be express­
ible as a linear combination of the previous ones. Let vk + 1 be the first of the vectors V;
that can be so expressed. In other words, v1 , v2 ,
, vk are linearly independent, but
there are scalars c 1 , c2 , , ck such that
(1)
Proof
•
•
.
.
.
•
•
•
.
.
.
•
Multiplying both sides of Equation ( 1 ) by A from the left and using the fact that Av; =
A;V; for each i, we have
A k+ i Vk+ i = Avk+ 1 = A(c 1v1 + c2v2 + · · · + ckvk )
= C 1AV1 + C2AV2 + . . . + ckAvk
= C 1 A 1 V1 + C2 A 2V2 + . . . + ck ,\ kvk
Now we multiply both sides of Equation ( 1 ) by A k + l to get
(2)
(3)
When we subtract Equation (3) from Equation (2), we obtain
0 = c 1 (,\ 1 - A k+1 )v1 + c 2 (A 2 - A k+ 1 )v2 + · · · + ck (,\ k - A k+1 )vk
The linear independence of v 1 , v2 ,
•
.
•
, vk implies that
c 1 (,\ 1 - A k+1 ) = c2 (A 2 - A k+1 ) = · · · = ck (,\ k - A k+1 ) = 0
Since the eigenvalues A; are all distinct, the terms in parentheses (A; - ,\ k + 1) ,
i = 1 , . . . , k, are all nonzero. Hence, c 1 = c2 = · · · = ck = 0. This implies that
vk+1 = c 1 v1 + c2v2 + · · · + ckvk = Ov1 + Ov2 + · · · + Ovk = 0
which is impossible, since the eigenvector Vk + 1 cannot be zero. Thus, we have a
contradiction, which means that our assumption that v1 , v2 , . . . , vm are linearly
dependent is false. It follows that v1 , v2 , . . . , vm must be linearly independent.
..
I
Exercises 4 . 3
In Exercises 1 - 12, compute (a) the characteristic poly nomial
of A, (b) the eigenvalues of A, (c) a basis for each eigenspace
ofA, and (d) the algebraic and geometric multiplicity of
each eigenvalue.
2
1 3
1. A
2. A =
-1 0
-2 6
=
3. A �
5. A �
[ ]
[� - � :J
[ - � - : :J
4. A �
6. A �
[ l]
[� : �]
[� -� :J
Section 4.3 Eigenvalues and Eigenvectors of
13. Prove Theorem 4. 1 8(b).
14. Prove Theorem 4. 1 8(c). [Hint: Combine the proofs of
parts (a) and (b) and see the fourth Remark following
Theorem 3.9 (page 1 69).]
[_�]
[�]
In Exercises 1 5 and 1 6, A is a 2 X 2 matrix with eigenvec­
tors v1 =
and v2 =
corresponding to eigenvalues
[ � ].
A 1 = t and A 2 = 2, respectively, and x =
15. Find A 1 0x.
16. Find Ak x. What happens as k becomes large (i.e., k ---+ oo)?
In Exercises 1 7 and 1 8, A is a 3 X 3 matrix with eigenvectors
v, �
[H [J
v, �
and v, �
[ :J
wm,pond;ng to
eigenvalues A 1 = -L A2 = t , and A 3 = 1 , respectively, and
Fm
17. Find A 2 0 x.
18. Find Ak x. What happens as k becomes large (i.e., k ---+ oo)?
19. (a) Show that, for any square matrix A, A T and A have
the same characteristic polynomial and hence the
same eigenvalues.
(b) Give an example of a 2 X 2 matrix A for which A T
and A have different eigenspaces.
20. Let A be a nilpotent matrix (that is, A m = 0 for some
m > 1 ) . Show that A = 0 is the only eigenvalue of A.
21. Let A be an idempotent matrix (that is, A 2 = A). Show that
A = O and A = 1 are the only possible eigenvalues ofA.
22. If v is an eigenvector of A with corresponding eigen value A and c is a scalar, show that v is an eigenvector
of A - cI with corresponding eigenvalue A - c.
23. ( a) Find the eigenvalues and eigenspaces of
A =
[� �]
(b) Using Theorem 4. 1 8 and Exercise 22, find the eigen­
values and eigenspaces of A - i , A - 2I, and A + 2I.
24. Let A and B be n X n matrices with eigenvalues A and
IL, respectively.
( a) Give an example to show that A + IL need not be
an eigenvalue of A + B.
(b) Give an example to show that AIL need not be an
eigenvalue of AB.
nX n
Matrices
299
(c) Suppose A and IL correspond to the same eigen­
vector x. Show that, in this case, A + IL is an eigen value of A + B and AIL is an eigenvalue of AB.
25. If A and B are two row equivalent matrices, do they
necessarily have the same eigenvalues? Either prove
that they do or give a counterexample.
Let p(x) be the poly nomial
The companion matrix ofp(x) is the n X n matrix
- an - I - an - 2
C(p) =
1
0
0
0
- a 1 - ao
0
0
0
0
0
0
0
0
(4)
26. Find the companion matrix ofp(x) = x 2 - 7x + 12 and
then find the characteristic polynomial of C( p).
27. Find the companion matrix of p(x) = x 3 + 3x 2 4x + 1 2 and then find the characteristic polynomial
of C(p).
28. ( a) Show that the companion matrix C (p) ofp (x) =
x 2 + ax + b has characteristic polynomial
A 2 + aA + b.
(b) Show that if A is an eigenvalue of the companion
matrix C(p) in part (a), then
of C ( p) corresponding to A.
[ �]
is an eigenvector
29. (a) Show that the companion matrix C(p) of p (x) =
x3 + ax 2 + bx + c has characteristic polynomial
- (A 3 + aA 2 + bA + c).
(b) Show that if A is an eigenvalue of the companion
[� l
2
matrix C(p) in part (a), then
of C ( p) corresponding to A.
is an eigenvector
1
30. Construct a nontriangular 2 X 2 matrix with eigen­
values 2 and 5. [Hint: Use Exercise 28.]
31. Construct a nontriangular 3 X 3 matrix with eigen­
values - 2, 1 , and 3. [Hint: Use Exercise 29.]
32. ( a) Use mathematical induction to prove that, for
n 2, theI companion matrix C(p) of p (x) = x n +
a n _ 1 x n - + · · · + a 1 x + a 0 has characteristic
polynomial ( - 1 ) np (A). [Hint: Expand by cofactors
along the last column. You may find it helpful to
introduce the polynomial q ( x) = (p ( x) - a 0 ) /x .]
2
Chapter 4 Eigenvalues and Eigenvectors
300
(b) Show that if A is an eigenvalue of the companion
matrix C(p) in Equation (4), then an eigenvector
corresponding to A is given by
An - I
,t n - 2
Ifp(x) = x n + a n _ 1 x n - ! +
+ a 1 x + a0 and A is a
square matrix, we can define a square matrix p(A) by
+ a 1A + a0I
p(A) = A n + a n _ 1A n - i +
An important theorem in advanced linear algebra says that
if cA (,\) is the characteristic poly nomial of the matrix A,
then cA (A) = 0 (in words, every matrix satisfies its charac­
teristic equation). This is the celebrated Cayley-Hamilton
Theorem, named after Arthur Cayley ( 1 82 1 - 1 895),
pictured below, and Sir William Rowan Hamilton (see
page 2). Cayley proved this theorem in 1 858. Hamilton
discovered it, independently, in his work on quaternions, a
generalization of the complex numbers.
·
·
·
·
·
·
and
A 3 = AA 2 = A( - aA - bl)
= - aA 2 - bA
= - a( - aA - bl) - bA
= (a 2 - b ) A + ab I
It is easy to see that by continuing in this fashion we can
express any positive power of A as a linear combination of I and A. From A 2 + aA + bl = 0, we also obtain
A(A + al ) = - bl, so
1
a
A - 1 = - -A - - I
b
b
provided b * 0.
35. For the matrix A in Exercise 33, use the Cayley­
Hamilton Theorem to compute A 2 , A 3 , and A 4 by
expressing each as a linear combination of I and A.
36. For the matrix A in Exercise 34, use the Cayley­
Hamilton Theorem to compute A 3 and A 4 by express­
ing each as a linear combination of I, A, and A 2 .
37. For the matrix A in Exercise 33, use the Cayley­
Hamilton Theorem to compute A - i and A - 2 by
expressing each as a linear combination of I and A.
38. For the matrix A in Exercise 34, use the Cayley­
Hamilton Theorem to compute A - i and A - 2 by
expressing each as a linear combination of I, A, and A 2 .
39. Show that if the square matrix A can be partitioned as
[-ij-i-;-]
A =
c
j
[ ]
1 -1
.
2
3
That is, find the characteristic polynomial cA ( ,\) of A
and show that cA(A) = 0.
34. Verify the Cayley-Hamilton Theorem for
33. Verify the Cayley-Hamilton Theorem for A =
A�
[� � :J
The Cayley-Hamilton Theorem can be used to calculate
powers and inverses of matrices. For example, ifA is a 2 X 2
matrix with characteristic poly nomial cA (,\) = ,\ 2 + a,\ + b,
then A 2 + aA + bl = 0, so
A 2 = - aA - bl
where P and S are square matrices, then the character­
istic polynomial of A is cA (,\) = cp(A) c5(,\). [Hint: Use
Exercise 69 in Section 4.2.]
40. Let A 1 , A 2 , . . . , A n be a complete set of eigenvalues (rep­
etitions included) of the n X n matrix A. Prove that
det (A ) = A 1 A 2 A n and
tr (A ) = A 1 + A 2 +
+ An
•
•
·
·
·
·
[Hint: The characteristic polynomial of A factors as
det (A - AI) = ( - l) n (,\ - ,\ 1 ) (,\ - A 2 ) (,\ - A n )
Find the constant term and the coefficient of A n - ! on
•
•
•
the left and right sides of this equation.]
41. Let A and B be n X n matrices. Prove that the sum of all
the eigenvalues of A + B is the sum of all the eigenval­
ues of A and B individually. Prove that the product of all
the eigenvalues of AB is the product of all the eigenval­
ues of A and B individually. (Compare this exercise with
Exercise 24.)
42. Prove Theorem 4. 19.
Section 4.4 Similarity and Diagonalization
Writi n g Proiect
301
The History of Eigenvalues
Like much oflinear algebra, the way the topic of eigenvalues is taught today does not
correspond to its historical development. Eigenvalues arose out of problems in sys­
tems of differential equations before the concept of a matrix was even formulated.
Write a report on the historical development of eigenvalues. Describe the types
of mathematical problems in which they originally arose. Who were some of the
key mathematicians involved with these problems? How did the terminology for
eigenvalues change over time?
1 . Thomas Hawkins, Cauchy and the Spectral Theory of Matrices, Historia Math­
ematica 2 ( 1 975), pp. 1 -29.
2. Victor J. Katz, A History of Mathematics: An Introduction (Third Edition) (Read­
ing, MA: Addison Wesley Longman, 2008) .
3. Morris Kline, Mathematical Thought from Ancient to Modern Times (Oxford:
Oxford University Press, 1 972) .
S i m i l a rilv and D i a u o n alizalion
As you saw in the last section, triangular and diagonal matrices are nice in the sense
that their eigenvalues are transparently displayed. It would be pleasant if we could
relate a given square matrix to a triangular or diagonal one in such a way that they
had exactly the same eigenvalues. Of course, we already know one procedure for con verting a square matrix into triangular form-namely, Gaussian elimination. Unfor­
tunately, this process does not preserve the eigenvalues of the matrix. In this section,
we consider a different sort of transformation of a matrix that does behave well with
respect to eigenvalues.
Similar Malrices
D e fi n it i o n Let A and B be n x n matrices. We say that A i s similar to B if
there is an invertible n X n matrix P such that P - 1AP = B. If A is similar to B,
we write A � B.
R e m a rks
If A � B, we can write, equivalently, that A = PBP - 1 or AP = PB.
•
Similarity is a relation on square matrices in the same sense that "less than or
equal to" is a relation on the integers. Note that there is a direction (or order) implicit
in the definition. Just as a :::; b does not necessarily imply b :::; a, we should not assume
that A � B implies B � A. (In fact, this is true, as we will prove in the next theorem,
but it does not follow immediately from the definition.)
•
The matrix P depends on A and B. It is not unique for a given pair of similar ma­
trices A and B. To see this, simply take A = B = I, in which case I � I, since p - i IP = I
for any invertible matrix P.
•
302
Chapter 4 Eigenvalues and Eigenvectors
Exa m p l e 4 . 2 2
Let A =
[ J [ J
[ � - � J [ � -l J [ l J [ 1 -l J [ O J
[ -l J
1
2
1
and B =
0 -1
-2
1
-
0
. Then A � B , since
-1
3
-1
-1
-
1
1
1
-2 - 1
1
. (Note that it is not necessary to compute P - 1 .
1
1
See the first Remark before Example 4.22. )
Thus, AP = PB with P =
Theorem 4 . 2 1
Let A, B, and C be n X n matrices.
4
a. A � A
b. If A � B, then B � A.
c. If A � B and B � C, then A � C.
(a) This property follows from the fact that r 1AI = A.
(b) If A � B, then P - 1AP = B for some invertible matrix P. As noted in the first
Remark on the previous page, this is equivalent to PBP - 1 = A. Setting Q = P - 1 , we
have Q - 1 BQ = (P - 1 ) - 1 BP - 1 = PBP - 1 = A. Therefore, by definition, B � A.
(c) You are asked to prove property (c) in Exercise 30.
Proof
Remark
Any relation satisfying the three properties of Theorem 4.2 1 is called
an equivalence relation. Equivalence relations arise frequently in mathematics, and
objects that are related via an equivalence relation usually share important properties.
We are about to see that this is true of similar matrices.
Theorem 4 . 2 2
Let A and B be n X n matrices with A � B. Then
a. det A = det B
b. A is invertible if and only if B is invertible.
c. A and B have the same rank.
d. A and B have the same characteristic polynomial.
e. A and B have the same eigenvalues.
f. A m � B m for all integers m 2::: 0.
g. If A is invertible, then A m � B m for all integers m .
Proof We prove (a) and (d) and leave the remaining properties as exercises. If A � B,
then P - 1AP = B for some invertible matrix P.
(a) Taking determinants of both sides, we have
det B = det ( P - 1AP) = ( det P - 1 ) ( det A ) ( det P)
1
= -- ( det A ) ( det P) = det A
det P
(d) The characteristic polynomial of B is
det ( B - AI) = det (p - i AP - AI)
= det ( P - 1 AP - AP- 1 IP)
( )
Section 4.4 Similarity and Diagonalization
303
det ( P - 1 AP - P - 1 ( AI) P)
det ( P - 1 (A - AI) P) = det (A - AI)
with the last step following as in (a) . Thus, det(B - i\I) = det(A - i\I); that is, the
characteristic polynomials of B and A are the same.
Remark
[ � � ] and B = [ � � ] both have de­
Two matrices may have properties (a) through ( e) (and more) in common
and yet still not be similar. For example, A =
terminant 1 and rank 2, are invertible, and have characteristic polynomial ( 1 - i\) 2
and eigenvalues i\ 1 = i\ 2 = 1 . But A is not similar to B, since P - 1 AP = P - 1 IP =
I * B for any invertible matrix P.
Theorem 4.22 is most useful in showing that two matrices are not similar, since
A and B cannot be similar if any of properties (a) through (e) fails.
Exa m p l e 4 . 2 3
[ � �] and B = [� � ] are not similar, since det A = - 3 but det B = 3.
1 3
1
1
] are not similar, since the characteristic polyno]
and B = [
(b) A = [
2 2
3 -1
(a) A =
mial of A is i\ 2 - 3i\ - 4 while that of B is i\ 2 - 4. (Check this.) Note that A and B do
have the same determinant and rank, however.
Diagonalization
The best possible situation is when a square matrix is similar to a diagonal matrix.
As you are about to see, whether a matrix is diagonalizable is closely related to the
eigenvalues and eigenvectors of the matrix.
An n x n matrix A is diagonalizable if there is a diagonal matrix
D such that A is similar to D -that is, if there is an invertible n X n matrix P such
that P - 1AP = D.
D e fi n it i o n
Exa m p l e 4 . 2 4
[l ]
[l ]
[
3
3
4 0
A=
is diagonalizable since, if P =
and D =
, then
0 -1
2 2
1 -2
1
P - AP = D, as can be easily checked. (Actually, it is faster to check the equivalent
statement AP = PD, since it does not require finding P - 1 .)
]
-+
Example 4.24 begs the question of where matrices P and D came from. Observe
that the diagonal entries 4 and - 1 of D are the eigenvalues of A, since they are the
roots of its characteristic polynomial, which we found in Example 4.23(b ). The origin
of matrix P is less obvious, but, as we are about to demonstrate, its entries are obtained
from the eigenvectors of A. Theorem 4.23 makes this connection precise.
304
Chapter 4 Eigenvalues and Eigenvectors
Theorem 4 . 2 3
Let A b e an n X n matrix. Then A is diagonalizable if and only if A has n linearly
independent eigenvectors.
More precisely, there exist an invertible matrix P and a diagonal matrix D such
that P - 1AP = D if and only if the columns of P are n linearly independent eigen­
vectors of A and the diagonal entries of D are the eigenvalues of A corresponding
to the eigenvectors in P in the same order.
Suppose first that A is similar to the diagonal matrix D via P - 1AP = D or,
equivalently, AP = PD. Let the columns of P be p1 , p2 ,
, P n and let the diagonal
entries of D be A 1 , A 2 , , A n . Then
A, 0
Az
A [ p 1 Pz
(1)
[ p l Pz
p. ]
Pn l
Proof
•
.
or
.
.
.
.
[ A p 1 A pz
.
.
A p n ] = [ A 1P1 A z pz
.
[
.
.
•
!
]J
0
A nPn ]
(2)
where the right-hand side is just the column-row representation of the product PD.
Equating columns, we have
A p 1 = A 1P1 , A p z = A z pz ,
·
·
·
,
A pn = A nPn
which proves that the column vectors of P are eigenvectors of A whose corresponding
eigenvalues are the diagonal entries of D in the same order. Since P is invertible, its col­
umns are linearly independent, by the Fundamental Theorem of lnvertible Matrices.
Conversely, if A has n linearly independent eigenvectors p1 , p2 ,
, P n with cor­
responding eigenvalues A 1 , A 2 , , A n , respectively, then
A pn = A nPn
A p 1 = A 1P1 , A p z = A z pz ,
This implies Equation ( 2) above, which is equivalent to Equation ( 1 ) . Consequently,
if we take P to be the n X n matrix with columns p1 , p2 , . . . , P n ' then Equation ( 1 )
becomes AP = PD. Since the columns of P are linearly independent, the Fundamental
Theorem of lnvertible Matrices implies that P is invertible, so P - 1AP = D ; that is, A
.
.
•
•
.
•
·
·
·
,
is diagonalizable.
Exa m p l e 4 . 2 5
If possible, find a matrix P that diagonalizes
A
� [� -� :J
We studied this matrix in Example 4. 1 8, where we discovered that it has
eigenvalues A 1 = A 2 = 1 and A 3 = 2. The eigenspaces have the following bases:
Solulion
Fod , � A, � 1 , E , has hasis
Fod, � 2, E, has basis
[n
[ :J
Section 4.4 Similarity and Diagonalization
305
Since all other eigenvectors are just multiples of one of these two basis vectors, there
cannot be three linearly independent eigenvectors. By Theorem 4.23, therefore, A is
not diagonalizable.
Exa m p l e 4 . 2 6
4
[� - � � =i l
If possible, find a matrix P that diagonalizes
A
Solution
This is the matrix of Example 4. 19. There, we found that the eigenvalues of
A are A 1 = A 2 = 0 and ,\ 3 = - 2, with the following bases for the eigenspaces:
Fod ,
Fod,
� �
�
A,
�m �m
[� - :J
0, E 0 h" b"i' p,
- 2, E_, h" b"i' P ;
ond p,
It is straightforward to check that these three vectors are linearly independent. Thus,
if we take
then P is invertible. Furthermore,
as can be easily checked. (If you are checking by hand, it is much easier to check the
equivalent equation AP = PD.)
R e m a rks
When there are enough eigenvectors, they can be placed into the columns of P
in any order. However, the eigenvalues will come up on the diagonal of D in the same
order as their corresponding eigenvectors in P. For example, if we had chosen
•
then we would have found
306
Chapter 4 Eigenvalues and Eigenvectors
In Example 4.26, you were asked to check that the eigenvectors p1, p 2 , and p 3
were linearly independent. Was it necessary to check this? We knew that { p1, p2 } was
linearly independent, since it was a basis for the eigenspace E0 We also knew that the
sets { p1, p 3 } and { p2 , p 3 } were linearly independent, by Theorem 4.20. But we could not
conclude from this information that { p 1 , p2 , p 3 } was linearly independent. The next
theorem, however, guarantees that linear independence is preserved when the bases of
different eigenspaces are combined.
•
•
Theorem 4 . 2 4
Let A be an n X n matrix and let A 1 , A 2 , , A k be distinct eigenvalues of A. If B; is
a basis for the eigenspace E;v then B B1 U B 2 U · · · UB k (i.e., the total collection
of basis vectors for all of the eigenspaces) is linearly independent.
•
•
•
=
Proof
Let Bi
=
{v;1 , v;2 ,
•
.
•
, v; n ) for i
=
1, . . . , k. We have to show that
is linearly independent. Suppose some nontrivial linear combination of these vectors
is the zero vector-say,
( C 1 1 V1 1 + . . ' + C 1 n,V1 n ) + ( C2 1 V2 1 + . ' . + C2 n2V2 n) + . ' . + ( ck! Vk l + . . . + Ckn,Vkn) 0
(3)
Denoting the sums in parentheses by x1 , x2 , . . . xk, we can write Equation (3) as
x 1 + x2 +
(4)
+ xk 0
=
·
�
·
·
=
Now each x, is in EA, (why?) and so either is an eigenvector corresponding to A; or
is 0. But, since the eigenvalues A; are distinct, if any of the factors X; is an eigenvector,
they are linearly independent, by Theorem 4.20. Yet Equation ( 4) is a linear depen­
dence relationship; this is a contradiction. We conclude that Equation (3) must be
trivial; that is, all of its coefficients are zero. Hence, B is linearly independent.
There is one case in which diagonalizability is automatic: an n X n matrix with
n distinct eigenvalues.
Theorem 4 . 2 5
If A is an n X n matrix with n distinct eigenvalues, then A is diagonalizable.
Let v1, v2 , . . . , vn be eigenvectors corresponding to the n distinct eigenvalues
of A. (Why could there not be more than n such eigenvectors?) By Theorem 4.20,
v1 , v2 , . . . , vn are linearly independent, so, by Theorem 4.23, A is diagonalizable.
Proof
�
Exa m p l e 4 . 2 1
The matrix
has eigenvalues A 1 2, A 2 5, and A 3 - 1 , by Theorem 4.15. Since these are three
distinct eigenvalues for a 3 X 3 matrix, A is diagonalizable, by Theorem 4.25. (If we actu­
1
ally require a matrix P such that p - AP is diagonal, we must still compute bases for the
eigenspaces, as in Example 4. 1 9 and Example 4.26 above.)
=
=
=
4
Section 4.4 Similarity and Diagonalization
301
The final theorem of this section is an important result that characterizes diagonaliz­
able matrices in terms of the two notions of multiplicity that were introduced following
Example 4. 1 8. It gives precise conditions under which an n X n matrix can be diago­
nalized, even when it has fewer than n eigenvalues, as in Example 4.26. We first prove a
lemma that holds whether or not a matrix is diagonalizable.
Le m m a 4 . 2 6
A
If is an n X n matrix, then the geometric multiplicity of each eigenvalue is less
than or equal to its algebraic multiplicity.
A
A
Q
Suppose 1 is an eigenvalue of with geometric multiplicity p; that is,
dim EA , = p. Specifically, let EA , have basis B 1 = {v1 , v2 , , vp }. Let be any invertible
n X n matrix having v1 , v2 , , vp as its first p columns - say,
Proof
Q ( Y1
.
•
=
or, as a partitioned matrix,
·
·
Yp
·
Let
C
U
Yp + l
Q [U VJ
=
where is p X n.
Since the columns of
also have
.
.
•
.
. . .
Yn
J
i
are eigenvectors corresponding to A 1 ,
AU A U.
= 1
CU= CV= DU= DV
�!.l:!!._l-�!.l:Y] LA[�_11-D�_f!U:_�-D�!.l:AVYJ [�-1�!'.L!DAV
��-�-]
Q- 1AQ [�lD ] A [ U :' V J [-DAU!DAV
1
(
)
(
)
det
Q
A
Q
A
P
V
A
AI
A
1
Q- 1AQ - AI) A,
Q- 1AQ,
A1
A
A ... , Ak.
A
A
from which we obtain
IP ,
=
=
0, and
0,
= In- p · Therefore,
=
=
By Exercise 69 in Section 4.2, it follows that
=
We
det ( DA
0
I)
(5)
But det (
is the characteristic polynomial of
which is the same
as the characteristic polynomial of by Theorem 4.22(d) . Thus, Equation (5) implies
that the algebraic multiplicity of is at least p, its geometric multiplicity.
Theorem 4 . 2 1
The Diagonalization Theorem
Let be an n X n matrix whose distinct eigenvalues are 1 , A 2 ,
statements are equivalent:
The following
a. is diagonalizable.
b. The union B of the bases of the eigenspaces of (as in Theorem 4.24) contains
n vectors.
c. The algebraic multiplicity of each eigenvalue equals its geometric multiplicity.
308
Chapter 4 Eigenvalues and Eigenvectors
a)=? (ebm) I4.f2A3.isIfdiagonalof thieszable eie,gtenvect
hen it ohasrs corlirneearspondly intdependent
eigenvec­
ttohrsen, bycont(Theor
o
t
h
e
ei
g
enval
ueare A;ly,
a
i
n
s
at
l
e
as
t
vect
o
r
s
.
(
W
e
al
r
e
ady
know
t
h
at
t
h
es
e
vect
o
r
s
ar
e
l
i
n
ng that micontghtaprinesventat letashemt vectfromorbeis. nButg a, basby Theor
is for emis4.th2at4,
tinhdependent
eyis amilingearht lnoty; tihnesdependent
ponlanyitt.h) iThus,
(multbip)l=?icit(yc)ofLetA; bethe geometBy setLemma
ricinmul4.tihence,
p26,licid;ty ioft contA; beforaidn;s==exactdi1,m...ly andvectNowloerst t.ashesualmegebrthaatic
property (b) holds. Then we also have
iplicities ofy, the
eiButgenvalIt foluloeswsofthAati·sd·j1·ust tdh2e degr=· · ·eesiofndcekth=tehchare suamcteofristth·iec· pol·algyebrnomiaiwhic amullcofhtA-namel
implies that
(6)
Uscanindeduce
g Lemmathat4.each26 agaisummand
n, we knowin Equatthat ion-(6)d;is2:zer0 ofor; that=is1,, ...= d; from
whi1, ...ch we
fo
r
=
ity has andd1 thde2 geometridck mul= tip1 licity d; are
equal(c)for==?eachvect(a) Ioeifrtgsh,envalwhie alcguhebrearA;aeicoflinmulA,earttliyhpenliincdependent
Thus,etmhes4.e2ar3.e
linearly independent eigenvectors of A, and A is ,dibyagonalTheorizeablm e4., 2by4.Theor
n
Proof
�
B;
n;
n;
n;
B
B
n
!R n ;
:s m;
m;.
m1 + m 2 +
+ mk
n
Exa m p l e 4 . 2 8
n
EA ;
, k.
i
n,
+ mk
+
EA ;
+
+
m1 + m 2 +
+ m k,
m;
i
B
n
m;
+
+
n.
m;
· · ·
, k,
m + m2 +
+
, k.
· · ·
(a) The matrix A = [ 2� - �5 4� i from Example 4. 1 8 has two distinct eigenvalues,
A2 1but= Ageomet
2 = 1 andric mulA3 t=ipl2.icSiitynce1, Athies einotgenvaldiagonalue A1iz=ablAe2, by= 1thhase DialaggonalebraiiczatmuliontipTheo­licity
rem. (See also Example-4.25.)
(b) The matrix A = [ �l 0� - �1 ] fromExample 4. 1 9also hastwodistincteigenvaltipluicesit,yA2,1 =andA2 t=he0eiandgenvalA3 =ue--2.2Thehas eialggenval
uc eand0 hasgeomet
algebrriacicmulandtigeomet
ricThusmul­,
ebr
a
i
p
l
i
c
i
t
y
1.
tfihnidis matngsriinxExampl
diagonale 4.i2z6.abl) e, by the Diagonalization Theorem. (This agrees with our
tionWeof tconcl
he powerude sthofis asectmatiornixwi. th an application of diagonalization to the computa­
Compute A10 if A = [ � � ].
In Example 4.2 1, we found that this matrix ]has eigenvalues]. A1 = - 1
and A2 = 2, with corresponding eigenvectors = [ _ � and = [ � It follows
-
is
Exa m p l e 4 . 2 9
i
Solution
v1
v2
Section 4.4 Similarity and Diagonalization
309
P-(fro1AmPany= Done, wherof ae number of theorems in this section) that A1 is diagonalizable and
l J and D = [ - 0 2O J
2
Solvi1.ng for A, we have A = PDP- 1 and, by Theorem 4.22(f), An = PDnP - 1 for all
Since
[ ( -01 r 2On J
we have
An = PDnp- 1 = [ _ � � J [ ( -01 r �n J [ _ � � r l
[ _ � � J [ ( -Ol )n 20n J [iI - iJ
2( - 1 )" + 2" H r' + 2· l
[ 2( - 1 )•: + 2•+• r - 1r: + 2·+•
Sisent ce=we10wertoefionlndy asked for A10, this is more than we needed. But now we can simply
10 ( - 1 ) 1 + 2 10 2 1
�
)
(
+
1
2
2
AlO = [ 2( - 1 ) 1 + 2 1 :3 + 2 " ] � [68234 68343 ]
3
n 2:
n
(- !)'
I
Exercises 4 . 4
A
A = [! � l = [� �J
2. A = [ - 42 6l J , [ 3 - l J
1
3.A = [ : 02 ! J .B � [- : 403 � ]
4.A � [ : - 21 - : J B � [ : 011 � ]
In Exercises 1 -4, show that and B are not similar matrices.
1.
B
B=
-5
7
P- 1AP = D.
AA
[ - 21 - � J [ � - � J [ � � J = [� �J
1 - 11
1
0
0
[! 1 -I
1 1 �m 0 J
0
0
� [: 0 -� l
In Exercises 5-7, a diagonalization of the matrix is given
in the form
List the eigenvalues of and bases
for the corresponding eigenspaces.
5.
6.
l
6
-2
3
-3
Chapter 4 Eigenvalues and Eigenvectors
310
0
3
[ - ! -�I23 03 �m _ : ]
02
� [ � 0 - �l
A
A = -3
8.A = [ � � ]
A = [ �]
0
IO.A = [ � 03 � ] A = [ : il
2
0
12.A = [ � 20 : J 13. A = H 0 il
02 00
03 02
14.A = 0 3 ] 15. A � 0 - 2 ]
ll 0 0 ; l� 0 0 j
[ -- 43 :r
[ - � �r
[ 4 - r6 [ � �r
"
�
r
�
[� n
[i 0
0
[ � - 2 ff
[: 0 J
A
A = [� �]
24.A = [ � �]
0
26.A = [ � � ]
A = [i 0 �]
I
8
3
4
3
-3
7.
1
-1
-3
In Exercises 8-15, determine whether is diagonalizable
and, if so, find an invertible
matrix P and a diagonal
matrix D such that P 1 P D.
-
9.
-1
11.
1
In Exercises 1 6-23, use the method of Example 4.29 to
compute the indicated power of the matrix.
1 7.
1 6.
18.
1 9.
-1
21.
20.
22.
1
1
-1
-1
23.
In Exercises 24-29, find all (real) values ofkfor which is
diagonalizable.
25.
27.
1
Pr
o
ve
Theor
e
m
4.
2
(
c
)
.
l
31. PrProoveve Theor
e
m
4.
2
2(
b
)
.
Theoreemm 4.4.222(2(ce))..
PrProoveve Theor
Theor
e
m
4.
2
2(
f
)
.
PrIfAoveandTheorB areemin4.ver22(tibgl)e. matrices, show that AB and
BAProvearethsiatmiiflaAr. and Bare similar matrices, then
tr(A) Sect= tri(oBn) .3.2.] Find a way to use Exercise 45
from
A
B
= B.
38.A = [ � - � l B = [ � � ]
39.A = [ : -- 23 ],B = [ -l 4l ]
2
40.A = [ : - 20 : J B � [ : 22 -- 4S i
2
0
2
[
3
41. A = [ : 0 � ] ,B = : 45 - � l
PrProoveve tthhatat iiffAA iiss disimagonal
ilar toizB,ablteh,ensoAisTAisT.similar to BT.
matrix. Prove that ifA is diago­
nalLetProiveAzablbetheat,ansoifinAisverAis taibdileagonal
eimatgenvalrix isucale letdhena A is of thiezablforemmatA rix wi(tShuchonlya one
eivectLetgenvalAorands iufesandB. bePronlovey tihfABatmatA=andriBA.cesB, eachhavewithtehsamedisteiingctences. PruoesveofthAatandthe Balarge­e
brtLetheaiAscamulme.andtiBplbeicitsiiems ioflarthmate eirgienval
30.
32.
33.
34.
35.
36.
37.
[Hint:
In general, it is difficult to show that two matrices are simi­
lar. However, if two similar matrices are diagonalizable, the
task becomes easier. In Exercises 38-41, show that and
are similar by showing that they are similar to the same
diagonal
matrix. Then find an invertible matrix P such that
P - 1AP
-6
1
-1
-1
42.
43.
44.
-
45.
46.
47.
A,
i.
scalar matrix.)
nXn
= AI.
n
Section 4.5 Iterative Methods for Computing Eigenvalues
48.
ove that it ivects notorposs sible toifinnIRd6tshurchee ltihnatearly
metLetthe sraicme.andmulBtibeplicsiiShow
tmieislaofrtmaththate,reiiifcgesBenval. Prouvees tofhatthtandhene geo­everB arye iPrndependent
and
eivectgenvect
o
r
of
Bi
s
of
t
h
e
for
m
for
s
o
me
ei
g
en­
Ihf e eiisgdiensagonal
izable, whatandare the dimensions of
o
r
of
t
p
aces
Preverovey eithgatenvalif uies ofa diaigonal
ierzableormattrhixensuchistihdatem­ 52.LetA = [ : �] .
s
ei
t
h
potLetentbe(thaatniilsp,otent matrix (that is, for some Prove that is diagonalizable if
and is not diagonalizable if
Pr
o
ve
t
h
at
i
f
i
s
di
a
gonal
i
z
abl
e
,
t
h
en
mus
t
beSupposthe zere tohatmatrisixa. matrix with characteristic
Find two examples totdemons
trateorthmayat ifnot be
h
en
may
diagonalizable.
( (
polynomial
A
50.
51.
v A.]
A
A
m > 1).
(a)
A
= P - 1AP,
p - 1v
[Hint:
49.
311
A
A 2 = A).
A
0
1,
v1 , v2 , v3
Av1 = v1 , Av2 = v2 , Av3 = v3 .
(b) A
E2 ?
E_ 1 , E 1 ,
A
(a)
Am = 0
4bc > 0
(a - d) 2 +
(a - d) 2 +
A
4 bc < 0.
A
(b)
A 6X6
cA (,\) = ( 1 + ,\) 1 - ,\) 2 2 - ,\) 3 •
(a - d) 2 + 4bc = 0,
A
Iterative M e t h o d s t o r C o m p ut i n g E i g e n v a l u e s
Inematician
1824, theNielsNorwegian
math­
Henri
k
Abel
(1802-1829)
proved thatpolynomial
a general
fiequation
fth-degreeis not(quintic)
that
is,inthere
isofnoitformula
for its
roots
terms
s
coefficients
that uses onlsubtraction,
y the operations
ofi­
addition,
mul
t
i
p
l
cati
on,Indivision,
andwrittaki
ning 1830
nth
roots.
a
paper
ten
and
published
posthumously
in
1846,
the
French
mathematician
Evariste
Galois (1811-1832)
gave
alished
moreconditions
complete
theory
that
estab­
under
which
an
arbitrary
polynomial
equation
can
bewassolinstrumental
ved by radicals.in establishing
Galois's work
the branchhis approach
of algebratocalled
equations is now knownpolynomial
as
solvable by radicals;
group
theory;
Galois
theory.
Atto stohlivsepoithnetchar, theaonlcteyrismettic hequatod weion.haveHowever
for comput
iengartehesevereigenvalal pruoesbleofmsa wimatthritxhiiss
,
t
h
er
metit depends
hod thatonrenderthe comput
it impraatctioincalofina aldetl butermsimnantall exampl
esis. Thea verfiyrsttiprme-oblconsemuismithnatg
,
whi
c
h
pra polocesynomi
s for alalrequat
ge matiorn,ices.andThetherseecondare noprforoblmemulaiss tforhatstohlveicharng polactyenomiristicaequat
ioinonsis
l
equat
ofquadrdegrateiec forhigmherulathandan its(panalolynomi
al.sThusof degr, weeesare forcanded to can be solvedeigusenvaling uthese
o
gues)
t praalctariecalquiprteoblseensms.itivUnfor
tuundoff
nately,ermetrorhandods arfore tapprhereoforxime unratineglitahbleer. oots of a
polin mosyInomi
e
t
o
r
o
cteristoirc fipolrstyandnomithaenl alustogetinghterhisandeigenvect
take aodir tffoefirenntd
apprthe coronach,streead,spapprondiweobypas
xingmeiatgsienval
ntgheancharue.eigIaenvect
one such method that is based onn athsiismsplecte iioten,rawetivewitelchniexplque.ore several variations on
2, 3,
4
4
approximate
Thethat power
metgenvalhodueapplthatiesistloaranger in absmatolutreixvalthuate hasthanaall of the other eigenvalues.
i
s
,
an
ei
Onandthuee.otthherenhand,is athmate domirix nwiantth
eieiForggenval
envalexampluuese, es,inifcea matrixandhas eihasgenvalno 2domiues n2ant eigenval
meta sheodquenceproceedsof vectiteorrastitvhelatyconver
to produceges toa tshequence
ofpondiscalnargseithgatenvec­con­
vertor gThees tthopower
and
e
cor
r
e
s
sume metthathtod.he matrix is
diagonaliezable. The following theorForemsiismtplheicbasity,isweforwithl easpower
The Power Method
nXn
4 = 1 -41 > l -31
- 4, - 3, 3,
4
v1 ,
,\ 1
dominant eigenvector.
dominant eigenvalue A 1 -
- 4, - 3, 1,
I l l.
131
3,
-4
A
312
Chapter 4 Eigenvalues and Eigenvectors
Theorem 4 . 2 8
an o vectdioargonalsuichzabltheatmattherisxequence
with domiof vectnantoreis xgenval
u
e
Then
t
h
er
e
exiLetsts abenonzer
defi
n
ed
by
k
,
approaches a dominant eigenvector of
We may assume that the eigenvalues2: of have2: · ·been· 2: labeled so that
be),ththeeycorforersmpondia basngiseifogrenvectConsors. Siequent
nce ly, we...can, vwrn aritee linearaslya
ilLetinndependent
(
w
hy?
ear combination of these eigenvectors-say,
Nowx1
and,
gener
a
l
y
,
fo
r
k
2:
As we saw in Example
nXn
A
A1.
x0
x1 = AXo , x2 = Ax1 x3 = Ax2 , . . . , xk = Axk _ 1 ,
•
•
•
A.
Proof
_.
I A 1 I > I Az I
v1, v2 , . . . , v"
A
I A3 I
I A" I
v1, v2 ,
!R n .
x0
Xo = C 1 V 1 + C2V2 + . . . + C nVn
= Ax0, x2 = Ax1 = A (Ax0) = A 2 x0, x3 = Ax2 = A (A 2 x0) = A 3 x0,
xk = A k Xo
4.2 1 ,
1
A kXo = C 1 A �v1 + C2 A ;v2 + . . . + cn A �vn
(
= A i c 1 v1 + c 2
(�:)\2 + · · · + c" ( ��yv")
(1)
wherThee wefacthavethusated thise tfacthe domi
that nant eigenvalue means that each of the fractions
is les than in absolute value. Thus,
A1 * 0 .
A1
A 2 / A1, A 3 / A1, . . . , A n / A1,
1
all go to zero as k � It folxk lows that� ask�
Now,
and
x
i
s
appr
o
achi
n
g
a
mul
t
i
p
l
e
of
(
t
h
at
i
s
,
an
s
i
n
ce
v1
k
eiongenvect
(This is thine trheequidirreedcticondi
tdomihne antinioteiiraglcorrenvect
vectesoproondir Int gmusto t have a nonzero component
on oftitohne
Appr
Theoroeximmate the dominant eigenvector of [ � � ] using the method of
We wil take [ � ] as the initial vector. Then
[ � � ] [� ] [ � ]
[� �] [�] [�]
We continue in this fashion to obtain the values ofxk in Table
oo.
A1 * 0
= A k Xo
v1 * 0,
x0 :
v1.)
Exa m p l e 4 . 3 0
(2)
oo
nonzero
A1) provided c 1 * 0.
c1
A=
4.28.
Solution
A � c 1 v1
x0 =
x 1 = Ax0 =
x2 = Ax1 =
4. 1 .
Section 4.5 Iterative Methods for Computing Eigenvalues
313
Ta b l e 4 . 1
xk [ � ] 0.[ � ]0 1.[ �5]0 0.[ :8J3 [1.�1�0] [0.�9�5] [1.!0�2] [0.!:99] [1.1701710 1]
rlkk 1.050 3.00 1.67 2.20 1. 9 1 2.05 1.98 2.0 1
k
0
2
1
3
4
5
6
7
8
y
7
6
5
4
3
2
o
x
0-/-t----2 --t- 3---1 --t-----t
---r---r-4
5
6
7
Xo
Figure 4 . 1 4
�
e 4.n14antsheiowsgenvect
whatoisr happeni
ngdigeomet
rnic1.all(yW. Wehy?knowSee Exerthatcitshee46.eig)ensTherpacee­
foforretFi,hietgiurdomi
wi
l
have
m
ens
i
o
in sTheas thfioughrst fewthieteirtaetreastexskarareeconver
showngalinogngonwiththe
the diresctaiolinsnetthheyroughdeterthmeinore.igIitnappear
lweinesewhosek, wee dineedrectionlon yvectobsorerivse[t�h].atTotheconfiratiormr tofhatthtehifisrisst tthoethdomie secondnant eicomponent
genvector
k line in the body of Table 4. 1 gives
ofthesxkegetvalsuveres, yandcloyouse tocan1 asseke iclnecrarelasesy th.atTherk isseicond
n
deed
appr
o
achi
n
g
1.
We
deduce
t
h
at
a
dominant eigenvector of is [ � ].
Oncenantweeighaveenvalfound
a domiapprnoantacheigisenvect
ore,rvhowe thcanat ifweanfixnkd itsheapprcororxiesmpondiatelynga
domi
u
e?
One
t
o
obs
dominant eigenvector of for the dominant eigenvalue A1, then
Iaskt folilnocrwseastheats. tTablhe reat4.io1 lgik ofvesthtehefivalrst ucomponent
ofyouxk+can1 tosteheatthofat xthk eywilarappre approachoach­A1
es
of
l
,
and
k
ing 2, which is the dominant eigenvalue.
IR 2 .
A
A
314
Chapter 4 Eigenvalues and Eigenvectors
e ivers aydrlaarwback
srsof. Totheavoiiterd­
attheissTherdragetwback,
gwe cane vermulytoquitthipcelklymetyeachandhodicanteofraExampl
causte byessoiegmenifisccantalThearrotcomponent
undoff
er
r
o
hatwirleduces
the magni
­
tdomiude ofnantits eicomponent
s
.
Si
n
ce
s
c
al
a
r
mul
t
i
p
l
e
s
of
t
h
e
i
t
e
r
a
t
e
s
s
t
i
l
conver
g
e
t
o
a
oach byis accept
wayseachto accom­
pla ish itvect. Oneor)gienvect
s. Anto noreasorm,ietalhr imetiszeappreachhod-and
dithveidoneianblgweeit. Ther
bywil eusare(e-ii.vare.s, titooousmake
iterabyte
di
v
i
d
e
each
largest component
nowtthhee maxicomponent
1. Thimums metabsolwihodthuittsehecalvalmaxiluee,d wemumwilabsolrThusepluacet,eivalf byue,denotso the(1ats ththeecomponent
of witihs
is notWehinilgutsotrdo,ate tshinisceapproach1. wiHence,th the calculations from Example For there
4.30:
xk
xk
unit
xk
II xd l
scaling.
mk
xk Yk = / m k) xk .
xk
m0 =
xk
4.30.
x0,
We then compute [ � ] as before, but now we scale with 1 to get
m =2
x1 =
Now the calculations change. We take
and scale to get
C�5 )[ � ·5 ] [ �.67 ]
The next few calculations are summarized in Table 4.2.
], a
You
can
now
see
cl
e
ar
l
y
t
h
at
t
h
e
s
e
quence
of
vect
o
r
s
i
s
conver
g
i
n
g
t
o
[
�
sdomipondinantng domieigenvectnantoeir.gMorenvaleoverue ,\,1 the sequence of scalars converges to the corre­
=
Y2 =
Yk
mk
= 2.
Ta b l e 4 . 2
k
xk
Yk
mk
[ � ] [ � ] [ � ·5 ] [ �·67 ] [ 1.1.6837 ] [ �·9 1 ] [ 1.1.9951 ] [ �·98 ] [ 1.1.9998 ]
[ � ] [ � · 5 ] [ �. 67 ] [ � . 83 ] [ �. 9 1 ] [ � · 95 ] [ �. 9 8 ] [ � · 99 ] [ � .99 ]
1.5
1
1.8
1.99
1.95
0
1
2
2
3
2
5
4
3
2
6
7
2
8
Section 4.5 Iterative Methods for Computing Eigenvalues
315
This method, called the
is summarized below.
Levaltue be a diagonalizable matrix with a corresponding dominant eigen­
Let x0 =thy0e folbeloanywiningisttiaelpsvectforkor i=n whos... :e largest component is
Repeat
(((cba))) SetyComput
Let mk k=bee( lxt/mhke=component
of xk with the largest absolute value.
.
)
x
k
k
k converor. ges to the dominant eigenvalue and Yk converForgmoses tot achoidomicesnantof x0ei,gmenvect
4
eiUsgeenvectthe power
or of method to approximate the domi5 nant- eigenvalue and a dominant
= [ -� 0l
Taking as our initial vector
power method,
The Power Method
A
1.
2.
nX n
,\ 1 .
!R n
1.
1, 2,
AYk - 1·
,\ 1
Exa m p l e 4 . 3 1
A
-2
12
-2
6
- 12
1
Solution
we compute the entries in Table 4.3.
0.
5
0
l
You can see that the vectors Yk are approaching [ - 0.50 and the scalars mk are
tapprhe domioachinnantg eigThienvals suueggesof ts that they are, respectively, a dominant eigenvector and
1
16.
A.
the domimetnhantod
eigenvectIf tohre in(iit.iea.l, vectif o=r x00 inhasthae zerprooofcomponent
of Theoremin the dirthecteniotnheofpower
R e m a rks
•
v1
4.28),
c
Ta b l e 4 . 3
xk [: l [ = : l [ l [ - .00 l [ . 5 l [ - 80..0053 l [ - 8..00 1 - [ - 800.00 ]
8
Yk [ : l [ = � �; l [ - 0.�40 l [ - 0.�s5o2 l [ - �s0.50o l [ - 0.�s50o l [ - 0.� 5s0o l [ - �s0.50o l
mk
. 5 .05
k
0
1
1
I
1
6
2
3
4
5
6
7
- 9 33
- 1 9.33
1 1 .67
8 62
1 7.3 1
9
8 12
16 2
- 8.20
16
8 4
1 6.0 1
8 1
1 6.00
8
6
- 1 9.33
1 7. 3 1
16 2
16
1 6.01
1 6.00
316
Chapter 4 Eigenvalues and Eigenvectors
ee tsoubsa domi
,undoff
it is quierterloikr elwiylthpratodurduceinang thxek
calwiwiltchulnotaanonzer
ticonver
on ofothgcomponent
equentninantitthereeiadigteenvect
sre, ctatisoonomerof. However
poiv1. Thent ropower
metherodrowirs actl thuenallysthelartpt!o)
convergThee to power
a multimetple ofhodv1.s(tTilhiwors is kones wheninstatnceherwher
e
r
o
undoff
e is acertain condidomitionns.antDeteigenval
ue,
or
even
when
t
h
e
mat
r
i
x
i
s
not
di
a
gonal
i
z
abl
e
,
under
a
i
l
s
may
be foundForinsomosmetmatmoderricesn tthexte power
books onmetnumer
ical analgesyrsaispi. d(SlyeetoExera domicisesnant eigen­
John
h
od
conver
Baron
Rayl
e
i
g
h,
was
a
British
physictoistthewhofieldsmadeof major
contribu­
vectof Theoror, whiemle for otrehvealers sthwhy.ekconverSincegencel ,\2 /,\mayk 1 2:bel qui,\3 /,\te s1lo2':w.· A· ·car2:elful,\n /,\look1 , atif lt,\h2e/,\proof1 is
tions
acoustics
and
optics.
Inexplanation
1871, he gaveof whythe thefirstsky is close to zero, then (,\2 /,\k 1 ) , , ( ,\n /,\ 1 ) wil all approach zero rapidly. Equation
correct
t
h
en
s
h
ows
t
h
at
x
=
A
x0
wi
l
appr
o
ach
,\
i
c
r
a
pi
d
l
y
t
o
o.
v
k
1
1
blue,
and
in
1895,
he
discovered
the
As
an
i
l
u
s
t
r
a
t
i
o
n,
cons
i
d
er
Exampl
e
The
ei
g
enval
u
es
are
,
and
s
o
inert
gas
argon,
for
whi
c
h
discovery
by
t
h
e
sevent
h
i
t
e
r
a
t
i
o
n
we
s
h
oul
d
have
,\
16
=
Si
n
ce
,\
=
4
/
1
2
heRaylreceieighvedwasthepresident
Nobel Prize
inRoyal
1904. close to four- decimal-place accuracy. This is exactly what we saw.
of
the
Ther
e
i
s
an
al
t
e
r
n
at
i
v
e
wa
y
t
o
es
t
i
m
at
e
t
h
e
domi
n
ant
ei
g
enval
u
e
,\
of
a
ma­
1
Soci
e
ty
from
1905
to
1908
and
t
r
i
x
A
i
n
conj
u
nct
i
o
n
wi
t
h
t
h
e
power
met
h
od.
Fi
r
s
t
,
obs
e
r
v
e
t
h
at
i
f
Ax
=
,\
x
,
t
h
en
1
became
chancellor
ofHeCambridge
Uni
v
ersi
t
y
in
1908.
used1873 paper
(
)
(
(
)
)
,\
x
·
x
A
A
x
x
·
x
·
x
1
1
Rayl
e
i
g
h
quotients
in
an
=
=
= Ai
x·x
x·x
x·x
onbookvibrating systems and later in his
tsThehymmet
e itexprerarteeicss matxiok,ntrhiec(essxu),ccest=he(sRaylAivx)e Rayle·ix)gh/equot(ixgh· x)quotientis calimetentleshdoda(xisk)aboutshoultdwapprice asoachfasAst,\as1we. tIhncomput
fae sccalt, ifornge
factor method. (See Exercises 17-20.)
Thewhatpower
metwehoddo canif wehelwantp us apprthe otoxihmerateiegtenval
he ues? Foreitugnatenvalelyu, ethoferae matare rsiexver, butal
s
h
oul
d
variTheations of the power methodustheats tcanhe obsbe applervatieid.on that, if ,\ is an eigenvalue of A,
tThushen ,,\if-,\1 iiss tanhe eidomigenvalnantue ofeigAenval- ueforof A,anythsecaleiagrenval(ExeruescofiseA -in,\Sectif wiiolnbe
A22 -- ,\A11,, and,\3 -fr,\om1, .th. i.s,valAn u-e we,\1. canWe ficannd t,\h2en. Repeat
applyinthgethpower
metcesshwiodl talo lcomput
e
i
s
pr
o
o
w
us
t
o
,\comput
e all of the eigenvalues.
Use th]e shifted power method to compute the second eigenvalue of the matrix A
[ � � from Example
method toIn Example we foundA -that =,\1 - To fi]nd ,\2, we apply the power
[� �
We take x0 = [ � ], but other choices wil also work. The calculations are summarized
in Table
repeated
•
2 1 -24.)
William Strutt ( 1 842- 1 9 1 9),
•
4.28
.
(2)
•
1 6 4,
4.3 1 .
0.25 7 = 0.00006,
0.25.
/
•
2,
•
The Theory of Sound.
R
Rayleigh quotient.
R
The Sh illed Power Melhod and lhe Inverse Power Melhod
dominant
shifted power method
a
al
a
22
4.3).
0,
=
Exa m p l e 4 . 3 2
4.30.
4.30,
Solulion
= 2.
21
4.4.
_
Section 4.5 Iterative Methods for Computing Eigenvalues
311
Ta b l e 4 . 4
[ � ] [ - � ] [ - �·5 ] [ - �·5 ] [ _ �·5 ]
Yk [ � ] [ -2�.5 ] [ - �.5 ] [ - �.5 ] [ - �.5 ]
teratiuonse of. Ther:._+e­
foreOur, A2 -choice of-x0 shaso produced the ei2g-envalue isafthteersonlecondy tweiogienval
k
1
0
2
3
4
-3
-3
-3
xk
1
mk
A 1 = 3,
A2 = A 1 - 3 =
-3
3 = -1
A
m preigoenvalpertyu(eb) of Theor
eforme, if wethapplat ify thiseipower
nvertibmetle wihtodh eitogenvall, uites
tdomihRecalennantl eifrl gohasenval
Ther
e
u
e
wi
l
be
t
h
e
(
i
n
magni
t
u
de)
ei
g
enval
u
e
ofmethod,To except
use thisthat in step 2(a) we computweefollow the sYkame- l · (sItnepsprasactiince,thwee power
don't
actforuallusy icomput
e
expl
i
c
i
t
l
y
;
i
n
s
t
e
ad,
we
s
o
l
v
e
t
h
e
equi
v
al
e
nt
equat
i
o
n
ng Gaussian elimination. This turns out to be faster.)
A-
A,
A.
xk
Exa m p l e 4 . 3 3
4. 1 8
1 /A.
A
A-
recip rocal of the smallest
inverse power method,
xk = A - 1
i
A-
Axk = Yk - i
Use the] inverse power method to compute the second eigenvalue of the matrix
[ � � from Example
We start, as in Example with [ � ]. To solve
we use row reduction:
A=
4.30.
Xo = y0 =
4.30,
Solution
Ax1 = y0,
Thus, [ � ] ,so [ � ] · Then we get from
[2 I J [ I J
0·5 ], and, by scaling, we get [ ] Continuing, we get the
Hence,
[
valsmualeslesstheiowngenvalinuTable ofe is thwhere recieptrhoecalvalofues m(kwarhiechconver
ging toThis agrThusees ,withthe
i
s
al
s
o
our previous finding in Example
x1 =
y1 =
x2
[A I Yi ] =
x2 =
- 0.5
4.5,
A
Ax2 = y1:
1 o 0.5
1 1 o
�
0 1
0 1 - 0.5
1
y2 =
.
-1
4.32.
-1
- 1).
- 1.
Chapter 4 Eigenvalues and Eigenvectors
318
Ta b l e 4 . 5
k
[ � ] [ �] [ - �:� ] [ - 0.l .SS ] [ - o0..s83 ] [ - o1..s1 ] [ - o0..9sS ] [ - o1..0s2 ] [ - o0..9s9 ] [ - o1..s0 1 ]
[ �] [ � ] [ _�O].S [ - �l ..S33 ] [--0.�.863] [ --�1..41S ] [ -- 0.�.9SS2 ] [ -- �1..0429 ] [ -- 0.�.9S9l ] [ -- �1..0SO1 ]
0
1
2
3
7
6
5
4
8
9
mostiversoned.atiIltecanof tbehe usvareidanttosfiofndthane power
metmathiodonifors one theiatgenval
combiune,esprtohvie tdwedo
jweTheusthavement
appr
o
xi
n othuerewor,\ ofds, tifhaatscalis clarosesits gitoven,
the If ,\ aisclanoseeiapprgenvaloxiumeatofion tando thwiatl eifignenval
d,\, tthheenueie.gIenval
- is inver(SeetibExerle ifciseis4S.not) Ifan
eiisgclenvalose tuoe,\of, thenand1/(1 /,\(,\- - wiils anbe eia gdomienvalnuante ofeigenval
of tude thanInthfacte next, if
ieigsenvalucle,osseo t(oas,\not, theend i1n/(t,\he-thirdwiRemar
l be k follbiowiggerng uiExampl
ne magni
e 4.3 1) the conver­
gence wil be very rapid.
Use the shifted inverse power method to approximate th6e eigenvalue of
]
s
[ - �2 -122 - 1210
that is closest to S.
Shifting, we have
- SI= [ =:- 2 - �2 -� �s l
Now we apply the inverse power method with
T h e Sh illed Inverse Power Method
any
shifted inverse power method
A
a)
a)
A
a very
Exa m p l e 4 . 3 4
a*
A al
(A - aI ) - 1 .
a)
much
A
a
a
(A - aI ) - 1 .
A= -
Solulion
A
We solve
[A
for
6
s
s
- SI [ -- 42 - 2 - 12s :]--� [ : �0 �1 =- 0.�::39� ]
(A - SI )x1 = y0
x1 :
I y0 ] =
7
a.
a
Section 4.5 Iterative Methods for Computing Eigenvalues
319
Ta b l e 4 . 6
050
050
050
041
061
049
047
[: J [ -- 0.0.3898 ] [ -- 0.0.6395 : [ -- 0.0.89 : [ -- 0.0.985 ] [ -- 0.0.998 ] [ -- 0.0.9590 ] [ -- 0.1.0500 ]
053
050
050
069
059
051
050
[: ] [ 0.1.005 [ 0.1.0501 ] [ 0.1.0500 ] [ 0.1.0500 l [ 0.1.0500 ] [ 0.1.0500 ] [ 0.1.0500 - 0.88 - 0.69 - 0.89 - 0.95 - 0.98 - 0.99 - 1.00
This gives
-[ - 0.0.6881 ] , - 0.88, and
1- -0.88 [ -- 0.0.8681 l [ 0.1 69 ]
- 0.39
- 0.39 0.45
Wethatcontthe ieingueenvalin tuheisoffasAhiclonosteostobtto a5inisthappre valoxiuesmatineTablly 5 e 4.16/, from whi5 ch1/we( - deduce
1)
which, in fact, is exact.
4
and itservar5, iweantswirel prdiessentcussonlanoty oneherapprmethoodachbasto ethdeoncomput
a­
tfactionoTheofrizeiatpower
gioenvaln ofmetua esmat.hIodnrixChapt
t
h
e
For a morical emetcomplhodset. e treatment ofthis topic, you can consult
almost any textbook on .numer
�
I
n
t
h
i
s
sect
i
o
n,
we
have
di
s
c
us
s
e
d
sever
a
l
var
i
a
t
i
o
ns
on
t
h
e
power
met
h
od
for
ap­
pr
o
xi
m
at
i
n
g
t
h
e
ei
g
enval
u
es
of
a
mat
r
i
x
.
Al
l
of
t
h
es
e
met
h
ods
ar
e
i
t
e
r
a
t
i
v
e,
and
t
h
e
WeRussian
owe this
theorem
to
the
s
p
eed
wi
t
h
whi
c
h
t
h
ey
conver
g
e
depends
on
t
h
e
choi
c
e
of
i
n
i
t
i
a
l
vect
o
r
.
I
f
onl
y
we
had
mathematician
sidedinmakeformaatjiuodin"ciabout
theceloofcatthioeninofitiathl vecte eigoenval
uperes ofhapsa gispveeden matup rthixe,
S.statedGerschgorin
(1901-1933),
whove tshoenmewe"incoul
o
us
choi
r
and
i
t
i
n
1931.
I
t
di
d
not
recei
muchresurrected
attention unti
l 1949,Taussky­
when it convergence of the iterative process.
was
by
Olga
ion ofuesthofe eia g(renvaleal oruescomplof anyex)
Todd in a note she published in the matFor
rn X nixmat.tunatrixelaly,l tlhieereinsiids eathwaye unitoonesofsttiamnteatcis erthcultathaetrhldieocateiskgsenval
in the complex plane.
LetA
[
a
;
]bea(
r
e
alorcompl
e
x)nX
nmat
r
i
x
,
a
ndl
e
tr
;
d
enot
e
j
tish,ers; um of tl ahue· IabsTheolute values of the off-diaisgonalthe cientrculrieasr idin stkhe ithinrotwheofcomplA; theatx
plane wijt1'hicenter a;; and radius r;. That is,
k
0
1
2
xk
Yk
5
4
3
44
4
6
7
4
4
mk
=
m1 =
X1 =
+
m7 =
+
= 4,
QR
Gerschgori n 's Theorem
American Mathematica/ Monthly.
Gerschgorin's Disk Theorem
Definition
= 2:
=
ith Gerschgorin disk
D;
320
Chapter 4 Eigenvalues and Eigenvectors
Olga
Taussky-Todd
(1906-1995)
was born
in Olmiitz
indoctorate
the Austro-Hungarian
Empirefrom the
(now
Olmuac
in
the
Czech
Republic).
She
recei
v
ed
her
i
n
number
theory
Uni
versity ofinVienna
inwhere
1930. Duri
nvesti
g World
WartheII,problsheeworked
for thein theNational
Physi
cal
Laboratory
London,
she
i
n
g
ated
m
of
flutter
wi
n
gs
of
super­
sonic
aircraft.on theAlthough
the problem
involmatrix.
ved diffTaussky-Todd
erential equations,
the stabiGerschgori
lity of an aircraft
depended
eigenvalues
of
a
related
remembered
n's
Theorem
from
her
graduate
studies
in
Vienna
and
was
able
to
use
i
t
to
simplify
the
otherwise
laborious
computations
needed
to determine
thein 1947,
eigenvalues
relevant
to thesheflutter
problem.
Taussky-Todd
moved
to
the
United
States
and
ten
years
later
became
theover
first
woman
appointed
to
the
California
Insti
t
ute
of
Technol
o
gy.
In
her
career,
she
produced
publications
and receivnow
ed numerous
the200 branch
of mathematics
known asawards.
matrixShetheory.was instrumental in the development of
Exa m p l e 4 . 3 5
Sketch the Gerschgorin disks and the eigenvalues fo-r the following matrices:
(a)A = [ � - � J
(b)A = [ � � ]
adii 1and2,
respectively(a. )TheThechartwoactGererissctihgorc poliynnomidisksaarl ofe centA ise,\r2ed at,\2-and8, s-o3thwie teihgrenval
ues are
,\ = ( - 1 v12 - 4( - 8) /2 2.37, - 3.37
diFi(bsg)kurs.Thee 4.t1w5oshGerowsschgorthat itnhedieiskgsenvalare centues earreedcontat 1aandined3wiwiththinrtahdiei two3 Ger= s3chgorand i2,n
respectively. The characteristic polynomial ofA is ,\2 - 4,\ 9, so the eigenvalues are
(4 V( - 4)2 - 4(9) /2 = 2 iVs 2 2.23i, 2 - 2.23i
Figure 4. 1 6 plots the location of the eigenvalues relative to the Gerschgorin disks.
Solulion
+
±
=
+
A=
±
±
Im
4
-4
Figure 4 . 1 5
=
+
I- I
Section 4.5 Iterative Methods for Computing Eigenvalues
321
Im
4
-4
Figure 4 . 1 6
GerAsschgorExamplin diesks. Thesunextggesttsh, eortheemeigverenvalifieusesthofat tahimats is rsiox. are contained within its
4.35
Theorem 4 . 2 9
Gerschgorin's Disk Theorem
wiLetthinbea anGerschgor(rienaldiorskcompl
. ex) matrix. Then every eigenvalue of is contained
respondihenceng einonzer
genvecto.or(Why?Let) Thenbe
the entrLety tofhe itbehwirantohwteiofhgewhienvallargchesuites absof oluwitetvalh corue-and
nXn
A
A
Proof
_....
Ax = Ax,
A
A
x
x;
x.
2: a x = Ax;
1 �1 ij J
n
Rearranging, we have
becaus
Appendie x weTakiobtanign absolute values and using properties of absolute value (see
X; * 0.
C),
IA
-
a;; I
a;1x1
2:
i
*
J
=
I fi1 aij xJ I J2:* i la;1x1 1 J2:* i lau l l x1 I 2: la;1 I r;
l x; I
X;
l x; I
:S
l x; I
:S
J*i
=
becausThies establishes forjthat thei.eigenvalue is contained within the Gerschgorin disk
centered at with radius
l x1 1 :S l x; I
a ;;
*
r;.
A
322
Chapter 4 Eigenvalues and Eigenvectors
Ther
e
i
s
a
cor
r
e
s
p
ondi
n
g
vers
i
o
n
of
t
h
e
precedi
n
g
t
h
eorem
for
Gers
c
hgor
i
n
disks whosIt cane rbeadisiharowne thethsatumifofofthethoffe Ger-diasgonal
entn dirisekssinartehediistjhoint fromofthe other
c
hgor
i
haenr, ifexacta sinlgly e dieisgkenvalis diusjesoinartefrcontom tahienedothwiertdihisnks,thtehunien iotnmusof tthcontese aindiexactsks. Ilny
pardionesktiseic, ulgtenval
utheatofinthExampl
e matrixe.4.Exampl
e i4.s 3not5(acont) il uasitnraedtesinthaisGer. schgorin disk; that is,
Not
e
3
5(
a
)
,
0
tful0hiatswhennotthe anmatappleirgiixenval
ied tosuliaenrvergofertmatibHence,
lerbyicesTheor, wibecaustheoutme4.tanyh1e6.GersfurtThihschgorerobscomput
eirnvatdiisokantsiicanosn,parwebetidetccanulaerrdeduce
lmy iusnede­
directly from the entries of the matrix.
R e m a rks
•
•
k
•
k
A.
A
Exa m p l e 4 . 3 6
column A.
k
Consider the matrix [ 2� 06 �8 l Gerschgorin's Theorem tel s us that the eigenvalandues2, ofrespectareivcontely. aSeeinedFiwigurthein4.tlh7(reae).diBecaus
sks cente tehreedfiatrst2,di6,skandi s di8swijoitnht rfradiomi 1,th1,e
otremher4.t2w9.o,Becaus
it mustecontthe acharin exactactelryisonetic poleigyenvalnomiuae,l ofby thhase secondreal coeffi
Remarcikentafst,eirfTheo­
i
t
has
complxex rootHences (i.e.t,heiergeenvalis a uunies qofue realtheieygenval
mustuoccur
ineenconj1 andugate3,paiandrs.th(SeeeuniAp­on
pendi
e
bet
w
o (possibly complex) eigenvalues whose real parts
lofietbethOne wotteenhhere ot5twandhoerdihand,
10.sks conttheaifinrssttwRemar
kdiafsktsercentTheoreredemat4.2,296,tandel s us8 withtath rtahdie isame, 1,
tandhree eiregsenval
u
es
of
are
cont
a
i
n
ed
i
n
Figrealure )4.eil 7(genvalb) . Thesue. eCombi
disks narinegmutthesuale lryesdiusltjos,inwet, sodeduce
each con­that
taihasns athsirnpegleectereiv(alaendleiy.gSeehence
(Compute the actualenvaleigenvalues, oneues ofin eachto verofitfhyethinist.e)rvals [ 1 , 3] , [5, 7], and [7.5, 8.5] .
A=
·
A
A
A),
D.)
t,
�
A
A
A
Im
Im
4
4
-4
Figure 4 . 1 1
(a)
-4
(b)
. '�
1
Section 4.5 Iterative Methods for Computing Eigenvalues
Exercises 4 . 5
In Exercises 1 -4, a matrix A is given along with an iterate
x5 , produced as in Example 4.30.
(a) Use these data to approximate a dominant eigenvector
whose first component is and a corresponding dominant
eigenvalue. (Use three-decimal-place accuracy.)
(b) Compare your approximate eigenvalue in part (a) with
the actual dominant eigenvalue.
1
1. A =
2. A =
3. A =
4. A =
[ l 42 ] ' Xs [ 111094443 ]
[ _ � - 41 ] , Xs [ - 78113904]
[ � l1 ] ' Xs [ 144 ]
[ 1.5 0.3.05 ] ' 5 [ 60.625 ]
6. A =
7. A =
=
89
x =
2.0
239.500
[ _ � -1 03 ] ,xs = [ -11.3.060167 ]
[ � - � l X1 0 [ � : ! � � ]
1
�
X
,
�
[-� 4-12 g [ �:��3.�4115
1- 1 - 31 1 , X10 [ - 2.1.290714 1
=
=
0
2
1 0.000
In Exercises 9-14, use the power method to approximate
the dominant eigenvalue and eigenvector of A. Use the given
initial vector x0 , the specified number of iterations k, and
three-decimal-place accuracy.
10. A =
12. A =
13. A =
=
=
9. A =
11. A =
=
5
In Exercises 5-8, a matrix A is given along with an iterate
xk > produced using the power method, as in Example 4.31.
(a) Approximate the dominant eigenvalue and eigenvector
by computing the corresponding m k and Yk· (b) Verify that
you have approximated an eigenvalue and an eigenvector
ofA by comparing Ayk with m kYk ·
5. A =
323
[ 14 l � l [ � l
[ - 6 - �l [ �l 6
5
8
Xo
Xo
=
k=
=
=
- 0.5 ' Xo
=
0
=
5
In Exercises 15 and 1 6, use the power method to approxi­
mate the dominant eigenvalue and eigenvector ofA to
two-decimal-place accuracy. Choose any initial vector you
like (but keep the first Remark after Example 4.31 in mind!)
and apply the method until the digit in the second decimal
place of the iterates stops changing.
16. A =
60 - 6 1
[ -6 6 1
12
2
-2
2
Rayleigh quotients are described on page 316. In Exercises 1 7-20, to see how the Rayleigh quotient method ap­
proximates the dominant eigenvalue more rapidly than the
ordinary power method, compute the successive Rayleigh
quotients R(x;) for i =
for the matrix A in the given
exercise.
18. Exercise 2
17. Exercise
20. Exercise
19. Exercise
1, ... , k
1113
114
The matrices in Exercises 21 -24 either are not diagonaliz­
able or do not have a dominant eigenvalue (or both). Apply
the power method anyway with the given initial vector x0,
performing eight iterations in each case. Compute the exact
eigenvalues and eigenvectors and explain what is happening.
[� � ],x0 [ � ] [ _� � ],x0 [ � ]
� [ � � +• � m
� [� � n � � m
21. A =
23. A
k=5
=
[ � � l Xo [ �l k 6
[ 3.1. 55 1.5 ] [ l ] ' k 6
9[ 4 14
8 -4
: n � � uk � 6
24. A
=
22. A =
=
Chapter 4 Eigenvalues and Eigenvectors
324
In Exercises 25-28, the power method does not converge to
the dominant eigenvalue and eigenvector. Verify this, using
the given initial vector x0 • Compute the exact eigenvalues
and eigenvectors and explain what is happening.
[ = � �l [ � ]
[ _� �l [ � ]
�
_
n ; n [: J
- [: �: � ] · � - [: ]
25. A =
Xo =
26. A =
Xo =
27. A
42. p(x) = x 2 - x - 3, a = 2
43. p(x) = x 3 - 2x 2 + 1 , a = 0
44. p(x) = x 3 - 5x 2 + x + l, a = 5
45. A
A
x. a * A
a
A,
1 /(A - a)
be anor eigIenvalf ue ofandwitishnotcorranespeiondigenvalngue of
eiLetgsenvect
is an eiogrenval(Wuhye ofmust
wibetihnhverowcortrtiehbsatlpeondi?) ng eigenvect
Igensf haspacea domiis one-nantdeiimgenvalensiounale . , prove that the ei­
46. A
_
_
x.
(A - al ) - 1
A - al
A1
EA,
� In Exercises 47-50, draw the Gerschgorin disks for the given
matrix.
2& A
1
-i
4
2i 1 +
48.
47.
In Exercises 29-32, apply the shifted power method to
-2i
0
approximate the second eigenvalue of the matrix A in the
2
4 3i
-2
given exercise. Use the given initial vector x0 , k iterations,
0
0
l
+
i
and three-decimal-place accuracy.
49.
2;
5 + 6i
1 + i
-i
30.
10
29.
9
1
2i
-2i
-5 - Si
31.
32.
I2 0
In Exercises 33-36, apply the inverse power method to
4 !4
approximate, for the matrix A in the given exercise, the ei­
50.
I 6
6
genvalue that is smallest in magnitude. Use the given initial
0 8I
vector x0 , k iterations, and three-decimal-place accuracy.
51.
strictly diagonally dominant
34.
10
33.
9
36.
k-5
ExerExercciissee
ExerExercciissee
tinhethsuatmroofw.th(SeeeabsSectoluitoenvaluesUseof thGerse remaichgorniinng's entDisrkies
Theor
e
m
t
o
pr
o
ve
t
h
at
a
s
t
r
i
c
t
l
y
di
a
gonal
l
y
domi
n
ant
matafterriTheor
x musetmbe invertible. See the third Remark
Itfhe sius msan of thematabsolrixu,tleetvalues denot
of theertohwse maxiof mtumhat ofis,
c� ) (See Section Prove that
iLetf ,\ is beananeigeienvalgenvalueuofe of athsentochastic matrix
Apply
Exer(see Sectcise i52onto Prove that
Prove that the eigenvalues of [ � � � �1 are
alcllosreeald ,inandtervloalcatone teachhe reofal tlihnese.e eigenvalues within a
2.5.)
14
In Exercises 37-40, use the shifted inverse power method
to approximate, for the matrix A in the given exercise, the
eigenvalue closest to a.
38.
12, a = 0
37.
9, a = 0
39.
7, a = 5
40.
13, a = - 2
Exercise 32 in Section 4.3 demonstrates that every poly­
nomial is (plus or minus) the characteristic poly nomial of
its own companion matrix. Therefore, the roots of a poly­
nomial p are the eigenvalues of C (p). Hence, we can use
the methods of this section to approximate the roots of any
poly nomial when exact results are not readily available. In
Exercises 41 -44, apply the shifted inverse power method to
the companion matrix C ( p) ofp to approximate the root of
p closest to a to three decimal places.
41. p (x) = x 2 + 2x - 2, a = 0
�i
[
l
[ ; �]
Aabssquaroluteevalmatueriofx ieachs diagonal entry is greater thifanthe
Exercise
Exercise
Exmi" � - [ _ :J
Exercise
7.
[�
�
ExerExercciissee 14
ExerExercciissee 13
35.
u !]
nXn
52. A
ll A ll = �'!,�
1
53.
A
3.7).
A r.]
54.
[Hint:
4.29.]
11A 11
A;
l a iJ I .
7 .2.)
A,
I A I ::; II A II .
I A I ::; 1 . [Hint:
A=
A
! 0 3 2
0 0 � 7
'
Section 4.6 Applications and the Perron-Frobenius Theorem
325
A p p l icati o n s a n d the Perro n - Fro b e n i u s T h e o re m
genvaleruses. and eigenvectors.
WeIn thbegiis snectbyiorn,eviwesitiwingl sexplomeoapplre seivercatiaolnsapplfroimcatprionseviofouseichapt
sSectitiisoitnohne(sttroachasnsinittritooicduced
)n matmatrrMariixcesofkasaovMarsochaiciaktnovesdandchaiwithmade
tn, hthem.enseverInhasparal obsatiscteuleradyvarat, iwesotnsatobseaboutvectervoedtrhethtThatratan­if
eiis,gtenvalhere uise.aWevectaroer nowsuchintahposat ition toThiprosveistequihis factvale. nt to saying that has as an
If Pis the transition matrix of a Markov chain, then is an eigenvalue of
Recal
l
t
h
at
ever
y
t
r
a
ns
i
t
i
o
n
mat
r
i
x
i
s
s
t
o
chas
t
i
c
;
hence,
each
of
i
t
s
col
u
mns
sciusmse toin SectTherionefore, iTakif isnag rtoransw vectposoesr, conswe haveisting of l s, then (See Exer­
whiExercchisiempliiens Sectthation is an eiandgenvecthaveor ofthe samewitheicorgenvalrespuondies, snog eiisgenval
ueeigen­By
al
s
o
an
value of
is truue.e Foris most transitthioatnismat, if rices, theneigenvaluWee needsat­
itshfieeIfolsn lfactowi,nmuch
andg twothmordefie einegienval
are posie­,
tive, and a sq] uare matrixtiios ns:callAedmatrix i]sifcalsolmeed power ofifitalisl ofpositistientve. rFories exampl
[ � � is positive but B [ � � is not. However, B is regular, since B2
[ ! � ] is positive.
Leta. be an transition matrix with eigenvalue
b. If Pis regular and then
As
i
n
Theor
e
m
t
h
e
t
r
i
c
k
t
o
pr
o
vi
n
g
t
h
i
s
t
h
eor
e
m
i
s
t
o
us
e
t
h
e
fact
t
h
at
has(a) tLethe sabemeaneigeienvalgenvectuesoasr of corresponding to and let be the component of
withethktthhecomponent
largest absols ofutethvale equat
ue ioThenn we havefor
Comparing
···
Markov Chains
3.7
P
P
Theorem 4 . 3 0
x.
Px = x.
x
P
nXn
1
P.
1
Proof
13
1.
3.7.)
jr
19
P.
jP = j .
n
j
pYf = (j Pf = f
pT
pT
4.3, P
I A I ::=::: 1
1 dominant;
1
every
A * 1,
I A I < 1.
1.
A
positive
regular
Theorem 4 . 3 1
=
=
A=
1
nXn
P
I A I ::=::: 1
A.
A * 1,
4.30,
Proof
x
IAI < 1 .
pT
P.
pT
m.
A
lx;I ::=::: lxk l = m
PTx = Ax,
Plk X1 + P2 k X2 +
xk
i = 1, 2, . . . , n.
+ Pnk Xn = Axk
x
326
Chapter 4 Eigenvalues and Eigenvectors
(Remember that the rows ofpT are the columns ofP.) Taking absolute values, we obtain
I A l m = I A I l xk l = I Axk l = l p lk x 1 + P 2 k X2 + · · · + Pnk xn l
:S l p lk x l I + I P 2 k X2 I + · · · + I Pnk Xn I
= P1k l x, I + P2 k l x2 I + · · · + Pnk l xn l
:S Plk m + P 2 k m + · · · + Pnk m
=
+ P2 k + · · · + Pnk)m = m
(1)
(p,k
Thecomesfirsfrtoimnequalthe factity folthatlowsthefrroomwstofhepTrT isaunglm etoInequalThusit,y in andAfttheerladistviequal
ibyty
d
i
n
g
we
have
as
des
i
r
e
d.
ove tPhe(aequind tvhalereentforime PplTi)catis iaon:posIfitive matrtihxen. If Fithrenst, weall ofshtowhe
ti(nhbequal
at) Weit isitwitireuls eiprnwhenEquat
ions are actually equalities. In particular,
Equivalently,
itive, p;k in Equatfor ion must beAlszero, o, and this canforhappen only if
TherNow,esforinforcee, eachPis possummand
thermoraree, weposgetitivequal
itlyarine tnegat
he Triivae;ngline Iotnhequaler woritydisn,
ithefp;andkX; 'sonlallyhaveif alltofhetsamehe suFurmmands
e
or
al
sign. This implies that
IAI :s 1,
m,
IR,
IAlm :s m.
1.
IAI = 1 ,
( 1)
A = 1.
IAI = 1,
(2)
>0
lx;I = m
i = 1 , 2,
IR
i = 1, 2,
(2)
m - lx;I 2: 0
i = 1, 2,
. . . , n.
. . . , n.
wher
e.m Thus, in either case, the eigenspace
l s, as inspTheor
r
of pButTecorj,iussreaisnrpogondiwthvectenprgootroofofofisTheor
an(
j
)
em This handl
we seeesththate jcasT ekwher
pT {e P is{,posand,itivkcompar
­
ing Icomponent
s
,
we
fi
n
d
t
h
at
e.
kt prandovedofkPit+hatsarpose eiitgivenvale-suay,esPofTher.pItkfolandeforlowspek,+thk(atresppect+ I musivelyt,
albysoTheorbef Pipossermeitgulive.ar(W, twehhy?enhave)soSimenjucespower
which implies that since is impossible if
ChaptWeercanInnowExamplexplaein someweofsatwhethbehavi
at for othr eoftrMaransiktiovonchaimatnrisxthat we observed in
p [ ]
and ]initial state vector [ ] , the state vectors xk converge to the vector
[ , a steady state vector for P (i.e., We are going to prove that for regular
4.30.
n
A EA =
A = 1.
�
. . . , n.
4. 18,
3.
A 1
A
A = 1,
4.30,
A=0
Ak = Ak + l = 1 .
IAI = 1 .
=A
1,
A A - 1) = 0,
3.64,
=
0.4
0.6
=
Xo =
0.6
0.4
0.7 0.2
0.3 0.8
x=
Px = x) .
Section 4.6 Applications and the Perron-Frobenius Theorem
321
Marthe sktovate chaivectnosr,s txhkissataliwsfaysy xkhappens.
Indeed,
wetigwiatel whatprovehappens
much morto teh.eRecalpowerl tshat
k
P
x0.
Let
'
s
i
n
ves
as P becomes large.
The transition matrix P [ 0.0.73 0.0.82 ] has characteristic equation
0 det(P 1 0.70.3 0.80.2 I 2 - l .5A + 0.5
- 0.5)
so3i1,ts weeigenval
uesin aradvance
e thandat would0.5be. (Nanoteiegtenval
hat, thuankse andtothTheor
emseig4.enval30 andue
4.woul
knew
e
ot
h
er
eigensdpbeaceslesartehan in absolute value. However, we stil needed to compute The
span( [ � ] ) and E0.5 span( [ � ] )
So, taking [ 3 ], we know that [ 0 0.O 5 ] From the method
used in Example 4.29 in Section 4.4, we have
[ 3 ][0 0(0.5 ] [ 3 ] - !
Now, ask� (0.5)k � 0, so
and � [ � � ] [ � � ] [ � ] [ 0.0.46 0.0.46 ]
(Observe that the columns of thias ]"limit matrix" are identical and each is a steady
state vector for P.) Now let [ b be any initial probability vector (i.e., a + b
Then
� [ 0.0.46 0.0.46] [ ab] [0.0.46aa ++ 0.0.46bb] [0.0.46 ]
Not only does this explain what we saw in Example 3.64,0.4it also tel s us that the state
veoto;s wil oonmge to the steody stote vedoc x [ 0.6] foe any choke of x,!4
e is notorhalinwgaysspoccur
ecial about
Exampl
etr4.ans37.itioThen matnextrices.theorBeforemeswehowscanthpratesentthis
ttyhpee tTherhofeorbehavi
s
wi
t
h
r
e
gul
a
r
em, we need the following lemma.
LeteigenvalP beuaeregular nhas aln gtrebransaiitciomuln mattiprliicxi.tyIfP is diagonalizable, then the dominant
pk
=
Exa m p l e 4 . 3 1
=
-A
- AI) =
=
A1 = 1
-A
1
1
=A
A2 =
A2 .)
E1 =
Q=
2
= (A - l)(A
=
1
-1
Q - 1 PQ =
2
pk = QDk Q - i =
_
1
l lk
)k
-1
oo,
pk
= D.
2
1
-1
1 -l
=
-1
_
= 1).
Xo =
k
Xk = p ·v�
·-u
�
x,
Lem m a 4 . 3 2
=
=
A1 = 1
X
1.
328
Chapter 4 Eigenvalues and Eigenvectors
eigenvalricuesmuloftPipandlicity 1 araseantheeisame.
FruoemoftPher.prSionofceofPiTheor
em 4.i3z1abl(be),,
Aso1 i=s P1rhas,Theby geomet
g
enval
s
di
a
gonal
Thereeform.e, the eigenvalue A1 = 1 has algebraic
multiplicityExer1, bycitshee41DiiangonalSectiiozatn i4.on4.Theor
Let P bemata rriexgulL awhosr e colturamnsnsitiarone matidentriixc.alThen, eachasequalk ---+ to thek sapprameovectachesor anx.
This vector xis a steady state probability vector for P.
seorimplemifyisthtreupre, however
oof1 , we wi, wil tconshoutidtheris onlassyumptthe icason.e wher1 e P is diagonaliz­
ableWe. ThediToathgonal
ize Pas Q- PQ =Dor, equivalently, P = QDQ- , where
pT
Proof
Theorem 4 . 3 3
n
See and J. L. Snell (Newby J.York:
G.
Kemeny
Springer-Verlag, 1976).
Finite Markov Chains
n
X n
X n
oo,
p
Proof
0
From Theor
e
ms
4.
3
0
and
4.
3
1,
we
know
t
h
at
each
ei
g
enval
u
e
ei
t
h
er
i
s
1
or
sat
i
s
fi
e
s
k
ask---+matrix-sappray, D*-each
oaches1 1or0of whosfor e=di1a, gonal
... entIt folriesloiwss 1or0.that DThusap­,
prpko=achesQD1. Hence,
akQdi-a1 gonal
approaches L = QD*Q- . We write
lim = L
Obs
e
r
v
e
t
h
at
Wethe notion
are takiofngasome
l
i
b
erties
wi
t
h
limibet. Nevertheless,
PL
=
P
l
i
m
l
i
m
=
l
i
m
=
L
these
steps
should
i
n
tui
t
i
v
el
y
clear.properties
Rigorousoflimits,
proofs folwhilowchfrom
Ther
e
for
e
,
each
col
u
mn
of
Li
s
an
ei
g
envect
o
r
of
P
corr
e
s
p
ondi
n
g
t
o
A
=
1.
To
s
e
e
1
themay
you
t
h
at
each
of
t
h
es
e
col
u
mns
i
s
a
vect
o
r
(
i
.
e
.
,
L
i
s
a
s
t
o
chas
t
i
c
mat
r
i
x
)
,
we
have encountered
ingeta side­
calcu­ need only observe that, if j is the row vector with 1 s, then
lustracked
course.
Rather
than
discussion
of matrix
j
L
=
j
l
i
m
l
i
m
j
P
=
l
i
m
j
=
j
limits, wewiwithla omit
the proofs.
snowinceipmkplisieasstthoatchasL itsicstmatochasrixt,ibyc. Exercise 14 in Section 3.7. Exercise 13 in Section 3.7
umn oforLs ofis
jPuforstWeLe;mi,nneedwherg a basonle isyisofsthhowe itwithhattshtatvndarh1ecorcoldruebassmnspiondis vectofnLgortar.oLete i1dvent1,1.vi2cWr,al.... iThete, vnibeth eicolgenvect
for scalars c1, c2, , cw Then, by Theorem 4. 1 9,
By---+Lemma0 as k4.---+32, for 1 for1. It fol1,loswso, bythatTheorem 4.3 l (b), 1 for 1. Hence,
= lim = c 1 v1
I A; I <
oo,
A7
A;
i
k---+ 'X!
, n.
pk
pk =
+
pk i
pp k
probability
k---+ 'X!
e;
k
k---+ oc·
•
.
Aj -=F
oo,
k---+ oc·
A =
!R n ,
.
AJ
pk =
n
j -=F
I Aj l <
j -=F
Le ;
k---+ oc,
P k e;
j -=F
Section 4.6 Applications and the Perron-Frobenius Theorem
329
Isnhownotherthworat thdes,colcoluumnsmn ofofLLiarseanproeibabigenvectlity vector cororrse, sspoondiLe; ins gthteo A1 mulButtweiplehavex of
val1lwhosof thee colcomponent
umns ofsLsarume itdoentiSicaln,ceeachthisequalis truetoforthieachs vectcolorux.mn of L, it implies that
SiofntceheLMaris akstovochaschaitinc. matThatrixis,,weLijcanreprinesteentrpsretthiet pras othbabie lity of being in
Theas
facttshtaetenextthathaviexampl
thengcolstueamnsritleudstoffrraotLmes.arsetaitdeentical says that the
Recall the rat in a box from Example The transition matrix was
= 1.
i
unique
1.
long range tran­
Remark
sition matrix
i,
Exa m p l e 4 . 3 8
j, if the transitions were to continue indefinitely.
starting state does not matter,
3.65.
We determined that the steady state probability vector was
Hence, the powers of appr[ � oach � i [
]
L
frmentom whiandch we canofseeitstthimatethine eachrat wiofl the other twspoendcompartofmentits tsi.me in compart­
discussioofn tofhereingulitiaalr sMartatek. ovThechaipronofs byis preasoivilynadapt
g thatetdhteostcoveready
sthtaeteWecasvecteconcloforsxtuaitdese ivectnourdependent
o
r
s
whos
e
component
s
s
u
m
t
o
an
ar
b
i
t
r
a
r
y
cons
t
a
nt
s
a
y
,
I
n
the exercises, you are asked to prove some other properties of regular Markov chains.
gular em traThen,
nsitionformatanyrixi,nwiititahl xprtohbabie steladyity vectstateorprx0obabi, thelisteyquence
vector
forofLetiterbeaasteaisnrxeTheor
k approaches x.
Let
P
�
=
1
�
� � �
8
8
-
0.250 0.250 0.250
0.375 0.375 0.375
0.375 0.375 0.375
eventually
3 7 .5%
25%
s.
Theorem 4 . 3 4
P
P,
Proof
nXn
4.33.
330
Chapter 4 Eigenvalues and Eigenvectors
kx0, we must show that lim Pkx0 = x.kNow,
wher
e
=
Si
n
ce
x
=
P
X
k
n
byTherTheoreforee,m the long range transition matrix is L = [ x x x] and lim p = L.
lim pkx0 = ( lim p )x = Lx0
x1 + x2 +
· · ·
4.33,
1.
+
···
k
k-'>GO
k-tGO
k-toc-
k--+oo
0
== X1X XXz2X XnX)nxX= X
WeSectiroentu3.rn7.tIontExampl
he Lesleie3.model
67 in thofat popul
sectioan,tiweon sgrawowtthhat, forwhitchhe weLeslfiiersmatt explrixored in
[�s L �]
iterates of the population vectors began to approach a multiple of the vector
+
(xl +
+ '''+
+ ···+
Pooulalion G rowlh
r �
populiattiisonstaeventble, suinalcely tended
rfolInatliotoowiherngworyear.dMors,artehegieovervthenr,ebyoncee agetclhiasssteasteofisthreisached,
he ratioups forin tthhee
[�5 00.25
l . 5x
s arofethstiisl popul
in theatriaotniowhen it1.has5 =reached itObss steeadyrvestthaatte.1.5 repre­
sandentWesththe ecancomponent
recognithezsetetadyhat sxitasteangroeiwtgenvect
oirs aof L correeispgondienvalnugetooftL,heandeigen­an
valeigenvect
ue =or1.nowcor5. Thus,
h
r
a
t
e
thibeens eigenvalreached.ue reWeprecansentcomput
s the e thesesidizesreofctlyt,hwie ageth­
cloutas havies whenng totihteerresasttpeeondiadyas wesntagditetodhasbefor
e.
wth rate and the corresponding ratios between the age clas es
forFindthtehLese stWeeliadye matneedstartixetogrLfioabove.
L. The characteristic polnydnomiall posal ofitivLeieis genvalues and corresponding eigenvectors of
det(L = 00.5 0.25 0
0.375
18:6: 1
IF
4
27 : 9 :
growth rate
,\
Exa m p l e 4 . 3 9
18:6: 1 .
positive
relative
Solution
- ,\
- AI)
4
-A
3
- ,\
- ,\ 3 + 2 ,\ +
Section 4.6 Applications and the Perron-Frobenius Theorem
331
sinog,wewemushavet solve 2,\ 0. 5 0 or, equivalently,
0. Factor­
(2,\
) 0
1
x Since- t1.he1,setcondhe onlfacty posor hasitiveonlroyotthofe rtohotis sequation i s - 0.1.159.
andThe(See corAppendi
row reductrespioondin: ng eigenvect1.o5rs are in the null space of - l .50I, whi18ch we find by
O
- 1.5II O J [ �-5 - 0.1.255 - 01.5
01 0 00 J
Thu,, if [ :: ] i"n eigenvectoc '°""ponding to 1.5, it "ti'fi" I
and That is,
- ,\ 3 +
8,\ 3 - 16,\ - 3 =
+ 37 =
- 3)(4,\ 2 + 6,\ +
=
D.)
( - 3 - Vs)/4
L
4
=
[L
( - 3 + Vs)/4
A=�=
=
3
=
3
-6
x�
A�
x, � Bx,
x2 = 6x3 •
Hence,
t
h
e
s
t
e
ady
s
t
a
t
e
gr
o
wt
h
r
a
t
e
i
s
1.
,
and
when
t
h
i
s
r
a
t
e
has
been
r
e
ached,
t
h
e
age
5
clas es are in the ratio 18: 1, as we saw before.
ddatwee forhavethedonesteadyif stahadte grhadowtmorh raetet:htanhe
unioneqIposnueExampl
positivietieiveeg4.enval
ei3g9,envaltuheerorueewasnone?
of onlButWey onewhatwercandiwoul
e also appars werentleyposfortituivnate, ewhithatchtalherloewedwasusa
corto rreelaspteondithesnegcomponent
eigenvectors altol tofhewhossize eofcomponent
atandiona icors notresaccipondidentngaeil; gthenvect
at is, or witLesh poslitehimatteivpopul
re component
ix hasatioexactn. Wesl.ycanoneprposoveitivtheateigthenvalis situue­
Recall that the form of a Leslie matrix is
0
0
0
0
00 0 0 00 00
000
0
Since thoe(oenttherriews ise,rethpreepopul
sent sautrivoinvalwoulprodbabirapilidtliyesdi, wee outwi)l. Weassuwimel althsato asthseyumearethalatl
nonzer
atand,leasagait onen, tofhethpopul
e birtahtiparon awoulmetedrsdie iouts nonzer
o
(
o
t
h
er
w
i
s
e,
t
h
er
e
woul
d
be
no
bi
r
t
h
s
now prove the assertion we made above as). Wia thteorh theesm.e standing assumptions, we can
a uniques. positive eigenvalue and a corresponding eigen­
vectEveroyrLeswitlhieposmatitrivixe hascomponent
6:
L.
L
every
b 1 b2 b 3
51
52
L=
53
(3)
Sn - I
sj
b;
Theorem 4 . 3 5
bn - 1 bn
332
Chapter 4 Eigenvalues and Eigenvectors
Proof
Let L be as in Equation The characteristic polynomial of Lis
(3).
= ( - l ) nj(A)
rs(Yuorotouvsivareofal prasokbabiedSitnloicetipresatovelareastehtipossoneinitiExerofve,tthcheiesebicoeffirth parcTheientametseiofgenval
ers uchange
iess posof Litsivarigeneandtexactheraleforllyofonce.e tthhee
x D), therefore, has exactly one positive
rByootDescar
.ByLetdirusetctecals'cals lRulictuleatofion,Sigwens can(Appendi
check that an eigenvector corresponding to is
16.)
f(A).
b;
(A)
f
f(A)
sj
A 1.
A1
s1 /A 1
S 1 S 2 I Af
S 1 S 2S3 /A i
pos(Youitiarve.e asked to prove this in Exercise Clearly, all of the components of are
biruthe
parofaImetnLifactsers, morande is trtuhe.atarWieis,posteverh tithiyvee,otaddiihterttui(orrnnalesaloutrorequicompl
thratementtheex)uniteihgatqenvalue posueitiveofeiLgenval
satproisoffieiss
(
I
t
i
s
beyond
t
h
e
s
c
ope
of
t
h
i
s
book
t
o
pr
o
ve
t
h
i
s
r
e
s
u
l
t
,
but
a
par
t
i
a
l
i.n) edThiins explExeraciinses whyforweregetaderconver
s who garencee famitoliaasrtwieadyth tshteatale gvectebraorofwhencomplweexinum­
bertouthelspopul
ation vectors: It is just the power method working for us! terate
x1
18.)
b;
two consecutive
b;+ i
A1
dominant;
I A I < A1.
A
27
Ieingtenvalhe prueeviofousinttweroesapplt wasicposatioinstiv,eMarandkovdomichainantns .andMorLeseoverlie mat, therrieceswas, wea corrsawetshpatond­the
heoremof
guartinhge oneseiantgenvect
eeswethhaveoatr twihbeenitshwiposlconsbeitivtiehdeercomponent
casing.e Thefor afislr.asrItgt veretuclrsanisosnoutofofmattthhiatsritcaheseorr,emarineclmukidiabls nforeg tmany
matFiricres.st, we need some terminology and notation. Let's agree to refer to a vectposoitrivase
ifweall wioflitwrs component
sf are positfoivre.alForl andtwo (Similamatr defiricnesitions wil applandy
i
t
e
i
for to be theandmatsorixon.of)tThus
he absol, a posuteivaltiveuesvectofotrhe entsatirsifieessof Let us define
The Perro n -Frobenius Theorem
Oskar Perron
(1880-1975)whowasdida
German
mathematician
work
in manyanalysis,
fields of mathemat­
ics,equations,
including
differentiandal
al
g
ebra,
geometry,
number
theory.inPerron'
s Theorem
was
published
1907
in
on continued fractions. a paper
positive
B = [b ii ] ,
A > B, A :s B,
IA I = [ la;j l l
A 2: B a ii 2: b ij
i
j.
mX n
x
A = [a ii ]
x > 0.
A.
Section 4.6 Applications and the Perron-Frobenius Theorem
Theorem 4 . 3 6
333
Perron's Theorem
prLeta. operbe>tie0as:positive matrix. Then has a real eigenvalue with the following
b.c. If hasis anya corotrheserpondieigenvalng posueioftive eitghenvect
o
r.
en
Iofntauitively,posweitcanive matsee whyrix thThee fircorrst tweospsonditatementng mats shrioulx trdansbefotrrume.atConsion mapsider tthhee casfirset
quadr
e pltoaneactpronoperthelyiminagesto itsweelf,getsin,cethaleyl neces
component
sconver
are posgeittiovware. Ifdwesomere­
peatray iendlathntyealoffilrostwthquadr
s
a
r
i
l
y
a
nt
(
F
i
g
ur
e
A
di
r
e
ct
i
o
n
vect
o
r
for
t
h
i
s
ray
wi
l
be
a
pos
i
t
i
v
e
vectleavesor thewhiraychfixmused. tInbeotmapped
ins,to some posiwitivteh mulandtiple ofbotithseposlf (siatyiv, e. since
h
er
wor
d
oalvectl o>rs thus2, we foneedr someonlyscconsalar iderWhen tvecthis happens
,
tChapt
hen erForwes2omewilnonzer
for
o
r
s
I
n
vectors iinve vect(tohres on this unit
isnpthero ae",gtenerherealwiizedl beelseleiaptsoimaxihatd:'mSo,mapsumasvalthruaeengessetof ofoversalulchtunihethtnonnegat
Denote this number by and the corresponding unitatvecto2r by x1.(See Figure
nXn
A
A1
A1
A
A
A,
2X2
4. 18).
x,
Proof
7,
I A I :::; A1.
A.
A
A(kx)
A(kx)
A
k
Ax = A 1 x,
x, Ax Ax
x
A
y
y
Figure 4 . 1 8
y
Figure 4 . 1 9
x
A1),
A1
A.
O;
A1
y
A1
Ax
A
unit
x.
n
!R
unit sphere)
Ax.
4. 19.)
y
334
Chapter 4 Eigenvalues and Eigenvectors
obtaWein now show that Ax1 = A1x1. If not, then Ax1 > A1x1, and, applying A again, we
wher
Exerthereceiwistehle beinequalsButometithAyen2is>yprA1=eses(rulv/ched,l Athx1siatnl ce)AyAAx1isisAposa2yuni.itThiivte.vects (contSeeorExerrtahdiatcctsissaettihsfie efactands AytSecth>at A1iA1onywas, so
tA1hex1maxi; thatmisum, A1valis anue wieigtenval
h thisupre ofoperA.ty. Consequently, it must be the case that Ax1 =
Now
A
i
s
pos
i
t
i
v
e
and
x1
i
s
pos
i
t
i
v
e,
s
o
A1
x
1
=
Ax1
>
Thi
s
means
t
h
at
A1
>
andTox1 >provewhi(c)c,hsucompl
etAesisthanye protoofherof(r(eaal) andor compl
(b) . ex) eigenvalue of A with cor­
ppos
e
responding eigenvector z. Then Az = Az, and, taking absolute values, we have
e uniinequalt vectityorfoul oiwsn thfreomdirtehcteioTrniaofngllzel Iinsequalalso iposty. (itSieeve Exerand csiasetisfies
SiAuwhernceeIAlzthlul e>. Bymitdhtdlheemaxi
mality ofAJrom the first part ofthis proof, we must have IAI ::::: A1.
I A1 forric, anymuleitigpenval
AWe IwinA1lfact. notIt ,imorspralosveeoitsththeresucase.e factIet ttuhsrat.nsA1outhasthalatgA1ebrisadomi
ic, andnanthence, so IAgeomet
licity ue
tain nonnegat
iveematmatrrixi­.
cesA squar. PerFrorbenieon'mats uTheor
sridix Ad esimos calincanledbe gener
The raelsiuzifedlt, srufrebjquiomectrepostsoasittoeivmechnie topercceralmcondi
t
i
o
n
on
t
h
the same permutation of the columns, A can be writ en in bloutckatforionmofasthe rows and
[ � �]
wher
matriex andsuch tharate square. Equivalently, A is reducible if there is some permutation
A [ � �]
(See page For example, the matrix
A=
is reducible, since interchanging rows and and then columns and produces
40
36.)
3.7
2:
0.
0,
0
(4)
40.)
0,
2:
<
i=
1.
1912.
reducible
B
P
D
P PT =
1 87.)
0
2
2
0
0
2
4
1
6
1
0
1
7
0
0
3
5 5
3 0
2
7 2
3
1
7 2: 1 3 0
�----�-J � ?.
0 0i2 1
0 0!6 2
0 0! 1 7
__
____
____
?.
3
1
2
3
Section 4.6 Applications and the Perron-Frobenius Theorem
335
(This is just where
PAPT,
0
0
0
P= 1
0
0
0
1
0
0
�
Theorem 4 . 3 1
A
Ak > 0
irreducible.
primitive.
Letwith thbee anfolliorwieducing prbloepernonnegat
ive n n matrix. Then has a real eigenvalue
t
i
e
s:
b.c.a. If hasis aanycorroteshperondieignenvalg posuietivofe eigenvect
or. If is primitive, then this
t
h
en
iInf equal
strict.ue of such that ,\ then is a (complex) root of the
d. equat
,\ isioaninty,\eiisgenval
e. has algebraic,\mul� tiplicity
rested readerive matcanricfiesndoramatprorofix analof thyesisPer. Theron-eiFgenval
robeniuues Theor
eemn calinlemany
tTheextsionntenonnegat
i
s
oft
dartihlye
of
and
a
cor
r
e
s
p
ondi
n
g
ei
g
envect
o
r
(
w
hi
c
h
i
s
neces
s
unique) is called the
of
The e, after the first two arteermthse, eachnumbernews itnertmheisseobtquenceained by summing the two ter...ms,
wher
plpreetcediely defing int.edIf weby tdenot
he equate thieonsnth Fibonacci number
and, forbyn then this sequence is com­
The Perron-Frobenius Theorem
X
A1 > 0
A1
A
A,
n
-
A1
Matrix Analysis
0
0
0
0
0
A
A
See C. R. Johnson (Cambridge,
by R. A. Horn
and
England:
Press, 1985).Cambridge University
0
0
0
CheckA stqhuaris!)e matrix that is not reducible is called If for some
ttranshenitionismatcalrleixd, by definitiForon. exampl
e
,
ever
y
r
e
gul
a
r
Mar
k
ov
chai
n
has
a
pr
i
m
i
t
i
v
e
ir educible. (Do you see why? TrIyt sihs owinotnharg thdetocontshowraposthatitivevere ofythprisi.m) itive matrix is
k,
__..
1
0
0
0
0
= 0.
A
A
A
IAI ::=::: A1.
IAI =
1,
A1
A
1.
A1
Perron root
A,
probability
Perron eigenvector A.
linear Recurrence Relalions
0, 1, 1, 2, 3, 5, 8, 13, 21,
Fibonacci numbers
Jo = 0, f1 = 1,
Jn = fn - 1 + fn - 2
fn ,
2: 2,
ThiFibonacci
s last equatnumbers
ion is, anbutexampl
e ofwial lconsinearidrerecurlinrearencerercurelatrieoncen. Werelwiatilonsretsuormewhat
n to the
fi
r
s
t
we
more general y.
336
Chapter 4 Eigenvalues and Eigenvectors
Leonardo
of Pisa"son(1170-1250),
pictured
left,a number
is betterofknown
by hisbooks,
nickname,
Fibonacci,
whi
c
h
means
of
Bonaccio:'
He
wrote
i
m
portant
many
ofappears
which have
survived,
including
and
The
Fibonacci
sequence
assurrounded
the solutionontoallasides
problem
inwall. How many pairs
certainof rabbits
man putcana pairbe produced
of rabbitsfrom
in a place
by
a
thatthe
pair
in
a
year
if
i
t
is
supposed
that
every
month
each
pair
begets
a
new
pair
which
from
month onbybecomes
ve?" The name
was given to the terms
ofsecond
this sequence
the Frenchproducti
mathematician
Edouard Lucas (1842-1891).
Liber abaci
Liber quadratorum.
Liber abaci: "A
Fibonacci numbers
be a sequence ofnumbers that is defined
as follows: Let
where wherare e scalars. are
Forscalaalrls. k,
If ionstihne equatareiorenfeinr ed tios ascaltlheed a
equat
of the recurrence. The
Thus, the Fibonacci numbers satisfy a linear recurrence relation of order
the entiht, ttherenmthine raecurrecurrencerencerelraetliaotniohasn, weordrerequik. re the
rmordbuter tnoofo defitineritmniaelbeforcondi
- k)IThetfh, itnenumber
tfiirosnst teisrmtheofortdheersofequence
the recurbercalenceledrelatiWeon. could
I
t
i
s
not
neces
s
a
r
y
t
h
at
t
h
e
start at It isorposanywher
e
el
s
e.
curbyralencelowirnelgatanionsextbyra,alisloowilatnedg
tcoeffi
he coefciefintci,ewhintsscibhtlomaye tbeo havealso evenbe a funct
rmorathereiogener
tn.hanAnasclexampl
allinaearrs andreewoul
d be the recurrence
We wil not consider such recurrences here.
rsConseecurquence.irdeerncethreelsaetiquence
on defined by tforhe initial Wrcondiite toutionsthe first five termsandof tthhies
late the nextWetharreeegitevrmenst. hWee fihaverst two terms. We use the recurrence relation to calcu­
D e fi n it i o n
(xn ) = (x0, x1, x2 , . . . )
a0, a 1 , . . . , a k - l
1 . x0 = a0, x 1 = a 1 , . . . , xk - l = a k _ 1 ,
2.
c 1 , c2 ,
n 2: Xn = c 1 xn - l + c2xn _ 2 + · · · + ckxn - k'
ck * 0,
(2)
( 1)
.
•
.
, ck
linear recurrence relation of order k.
initial conditions
2.
Remarks
(n
•
•
•
•
Exa m p l e 4 . 4 0
x0.
x1
C;
functions
(xn )
xn = 5xn - l - 6xn - l
n 2: 2.
x 1 = 1, x2 = 5
Solulion
X3 = 5 X2 - 6X1 = 5 • 5 - 6 • 1 = 19
X4 = 5X3 - 6X2 = 5 . 19 - 6 . 5 = 65
X5 = 5X4 - 6X3 = 5 · 65 - 6 · 19 = 2 1 1
1, 5 , 19, 65, 2 1 1,
so the sequence begins
....
Section 4.6 Applications and the Perron-Frobenius Theorem
331
pltoeapplCleyarttlhyhe,enirfetwecurhe rwerapprenceeoiachrnetleatrusieosneteddthinteri,mesay,eswoul. Itthdwoule belOOdrtahbethterenirmtceediofifoweusth,ecoulsisnecequence
wed findwoulanin Exam­
d have
forcurmreulncea forrelaXtnioasn.aWefunctwiilonil ofustraWete threefprerotceso fisnwidinthgtshuechseaquence
formulfraoasm Exampltehe re­
To begin, we rewrite the recurrence relation as a matrix equation. Let
A [� -�J
and introduce vectors xn [ :: J for 2. Thus, x2 [:: J [ � l X3 [:: J
[ � l x4 [::J [ �: l and so on. Now observe that, for 2, we have
AXn- [ 5 - 6 J [Xn-2Xn-I J [ 5Xn-IXn-- 16xn-2 J [Xn-Xn 1 J Xn
NotLesliiceematthatritches.is iAss thine stahmeose tcasypeesof, weequatcaniowrn weite encountered with Markov chains and
We Thenowcharuse tahcteetreichnistic equat
que ofioExampl
n ofA2 ise to compute the powers ofA.
A
6
5,\
hat thieoneigfolenvallowsuesthatareof,\t1he r3eandcurre,\nce2 r2.el(aNtiootn.iceIftwehatwrtheitfore thme
ofrfreocurtmherwhiecharncechaasctweeXrnfiisn-tdic5tequat
the same!) The correspn-ondin6g ein-genspacesit isarappare ent that the coefficients are exactly
E3 span( [ � ]) and E2 span( [ � ])
SettingP [ � � J , we knowthatP- 1AP D [ � �J. ThenA PDP - 1 and
Ak PDk 1 [ 3 2 J [ 3k 2OkJ [ 3 2 J - 1
[ 3 2 J [ 3k 2OkJ [ - 23 J
- 2-(32k(+3kI)) 33((22kk)+ 1) J
It now follows that
4.40,
98
explicit
solving
n.
4.40.
=
=
1
=
n
x
=
=
2:
=
n
i= 1
=
=
0
2:
=
4.29
+ =0
=
x
I
+ x
z
= 0,
=
=
=
=
=
=
p-
=
1 1
0
1 1
0
=
=
1 1
1
-1
+
+
=
338
Chapter 4 Eigenvalues and Eigenvectors
�
frinonm which we readtooffvertihfye sthoatluttihoins xforn mul3na-gi2vnes. (tThoe check
ourermwors thkat, wewecoulcalcdulplatuedg
s
a
me
t
usinObsg theervreecurthatrenceXn isrealalitnioearn. Trcombiy it! )nation of powers of the eigenvalues. This is nec­
essariicliyt]t.hUse casinge asthilsongobseras tvhateieion,genvalwe canues aresavediourstinsctelv[esas Theor
eworm 4.k.38(Oncea) wiwel makehave
explcomput
s
o
me
ed the eigenvalues ,\1 3 and ,\ we can immediately write
where c1 and are to be determined. Using the initial conditions, we have
when n and
when n We now solve the system
for cThi1 ands is thtoe obtmetahinod1we wiandl use in practThusice. ,Wexn now3n -il u2snt,rasatebeforits use. e to find an
explicit formula for the Fibonacci numbers.
Solve the FiWrbonacci
r
e
cur
r
e
ncef0
,
f
andf
f
for
n
f
O
1
2
n
n
n
l
f
f
f
we
s
e
e
t
h
at
t
h
e
char
a
ct
e
r
i
s
­
i
t
i
n
g
t
h
e
r
e
cur
r
e
nce
as
2
n
l
n
n
tic equation is ,\ ,\ so the eigenvalues are
and
Itthefolforlomws from the discussion above that the solution to the recurrence relation has
for sUsomeingstchaleairnsitci1aandl conditions, we find
fo
and
in 1 is and
Hence, an explicit
forSolmviulnga forfor theandnth Fibweonacciobtanumber
=
= 1, 2, . . . , 5
=
c2
2 = 2,
1 = x 1 = c 1 3 1 + c2 2 1 = 3c 1 + 2c2
=1
5 = X2 = C 1 3 2 + C2 2 2 = 9c 1 + 4c2
= 2.
c2
Exa m p l e 4 . 4 1
c =1
=
Solulion
X
= 1,
=
+
= 0,
2 - - 1 = 0,
A1 =
Jacques
Binet (1786-1856)
made
contributions
to
matrix
theory,
number theory,
physics, theandruleas­
tronomy.
He
discovered
forBinet'matrix
multiplication
in 1812.
s
formula
for
the
Fibonacci
numbers
isishedactualit inly due
tohow­
Euler,
who
publ
1765;
ever, it washisforgotten
unti
l Binet
published
version
in
1843.
LikeheCauchy,
Binet
was
ay royal
ist,
and
lost
his
uni
v
ersi
t
posi­
tion
when
Charles
abdicated
in
1830.
He
recei
v
ed
many
honors
for hisinwork,
including
his elec­des
tion,
1843,
to
the
Academie
Sciences.
3c 1 + 2c2 = 1
9c 1 + 4c2 = 5
c2 = - 1 .
=
----
1 + Vs
2
A2 =
0=
= C 1 A � + C2 A � = C 1 + C2
2: 2.
1 - Vs
2
---
c2 .
c1
c = 1 / Vs
c2 ,
l
_
Jn =
Vs
c2
- 1/ Vs.
( 1 +2 vs) n Vs ( 1 -2 vs) n
-
l
_
(5)
4
Section 4.6 Applications and the Perron-Frobenius Theorem
Fornumber
mula is ayetremarthe Fikablbonacci
e formnumber
ula, becauss aree alitl iisntdefiegersned! Triny tplerumggis ofngtihnea few
valuisesknown
for toassee how the terms cancel out to leave the integer valuesfn- Formula
lineduwores arkes alforl dianystinctse.condWhenorthderereliisneara repeatrecured­
reiegnceenvalThereluamete,tiotnhhodewhostewechnie ashaveqsueocimusjautsetdtouteibegenval
ince thizeesdibotagonalh sitiuzatatiioons.n method we
used may no longer work. The next tmodiheorefimed,susmmar
LetLeta,\ -X,\n1b=and=ax0.n,\-2l be bxthne-eiz begenvala recuruesrofencethereaslatsioocinattheatd ichars sataisctfieerdisbyticaequatsequenceion ,\(2xn-) .
a.b. IIff AA11 = AA22, =A,thentXhnen=Xcn1A=7 c1Acn2A� cfor2n,\some
n for ssocmealarsscalc1aandrs c1cand2. c2•
In either case, c1 and c2 can be determined using the initial conditions.
Axn_ 1, wher(a) Gener
e alizing our disXcussion above, we can write the recurrence as xn =
xn = [Xn-n 1 ] and A = [ � �]
Sifonr ceExerA chasise distinct eigenvalues, it can be diagonalized. The rest of the details are left
(bxb)n-Wez orwi, equil shvowtalenthlatxy, n = c1,\n c2n,\n satisfies the recurrence relation Xn = axn- l
if ,\2 - a,\ - b = 0. Since
sXubs-tiaxtution intbxo Equat= i(ocnAn yicelnd,\s n) - a (c An- I c ( ),\n- l)
n n- I - n-2 -1 b (c1An-2z c2( - 12),\n-2) 2 - l
= C1 (An --a,\2)n,\- 1 - b,\n-2) C2(n,\n - l )An- 1
== cc1,\Ann-2-2((,\02)- a,\c n-,\bn)-2(0c)2n,\nc-2,\(,\n2-2-(aa,\,\ - 2bb) ) c2An-2(a,\ 2b)
1= c An-2(a,\ 2b2 ) 2
2
Buta/2,, susiniceng,\thies aquadrdoublateicroforotmof,\ula2. -Consa,\ e-quentb = ly0,, wea,\ mus+ 2bt have= a2/a22 4b2b == 0-and4b/2,\ =
2b = 0, so
tional
n
(5)
Theorem 4 . 3 8
339
(5)
Vs
irra­
Vs
Binet's formula.
+
*
+
+
Proof
53.
+
+
(6)
( 6)
+
+
+
n
+
n - Z)
- b(n
n
- a (n
+
+
+
+
+
+
+
+
+
+
340
Chapter 4 Eigenvalues and Eigenvectors
is a Suppos
unique esothluetiinonitiforal condic1 andtions (arSeee Exerrandcise 54.) Then, in either (a) or (b) there
Solve the rTheecurrcharenceactreelartisiotinc equation is and- Xn n- l -whi9xnc-hl hasfor 3 as a
n
n
n
doubl
e
r
o
ot
.
By
Theor
e
m
4.
3
8(
b
)
,
we
mus
t
have
X
3
(
c
n
)
3
.
3
2
n
Since c1 and ( c2)3, we find that so
Therelattioechnins. Wequesstaoutte, lwiintedhoutin Theor
38 canalberesextulte. nded to higher order recurrence
proof,etmhe4.gener
LettXhnat isamsat- iiXsfin-edl byama-lsXequence
Xn-m bee athreecurasrsoenceciaterdelachartionaofcteorrisdteric
n-l ( n ) . Suppos
polynomial
where
factThenorxs nashas the form
X (
x0 =
c2 .
Exa m p l e 4 . 4 2
x0 = 1, x1 = 6,
A2
Solulion
1 = x0 =
Theorem 4 . 3 9
m
x1 = s .
6 = x1 = c 1 +
=
+
+
(,\ - A 1 ) m 1 (A - A 2 ) m2
•
•
· · ·
•
= 6x
n 2 2.
A=
6,\ + 9 = 0,
= c1 + c2 n = c 1 +
c2 = 1,
+ a0
x
(,\ - ,\ k ) m',
m1 + m 2 +
· · ·
+ m k = m.
n - C 11 1t n1 + C 1 2 n 1t n1 + C 13 n 2 fl1n + . . . + C 1 m 1 n m 1 - I fl nJ ) + . . .
+ (ck1 AJ: + ck2 nA J: + ck3 n 2 AJ: + · · · + ck m, n m, - I A J: )
_
\
\
\
\
Ientn calialcequat
ulus, iyouon ofletarhne forthatmif ( wher) is aediffiseraentconsiabltaentfunct, thenionthsate gener
isfyinagl sola diufftieorn­
is specitfiheed,unithqen,ue
bysisolusutibsontittouttiwher
nhegdiffe ereiisntnaitahconsleequatgenertantioa. nlIfstoanhlatuitnsioaittn,iiasfilweecondis fithnedtiitnohintatial conditiHence,
on is
Suppose we have differentiable functions oft-say, , Xn-that satisfy a
{
We can write this system in matrix form as where
[x{�(tt)) l , and
( t)
Now we can use matrix methods to help us find the solution.
svs1ems of Linear Differenlial Equalions
x = Ce kt,
x=x t
x' = kx,
C
t=0
k
x ( O ) = x0
C = x0•
x = x e kt
o
n
system of differential equations
x1, x2 ,
x = a 11 x 1 + a 1 2x2 + · · · + a 1n xn
x� = a z1 X 1 + a z 2X2 + · · · + a zn Xn
x' = Ax,
x' ( t) =
x�
x�
•
•
•
Section 4.6 Applications and the Perron-Frobenius Theorem
341
a usefulions:observation. Suppose we want to solve the following sys­
temFiofrsdit,ffweeremakential equat
2
Each equation can be solved separately, as above, to give
wher
a e andcoefficarieentconsmattraintx s. Notice that, in matrix form, our equation has
X{ = X 1
x; = 5X2
X1 = C1 e 21
Xz = C2 e s t
C1
diagonal
C2
A=
tion. cThiients
smatanduggesrtihxet,sieifthgposenval
at,sforiblueanes. 2arandbitraryoccursysteim,n thwee exponent
should stiaarltsby diandagonalofizitnhge tshoelucoeffi
Solve the following system of differential equations:
5
Exa m p l e 4 . 4 3
[� �]
x' = Ax
e 21
e 51
Here the coefficient matrix is [31 22 ] , and we find[ 2th]at the eigenval[ -u1 es]
arreespective4lyand. Therefor-e,1, wiisthdicoragonalrespizondiablen, gandeigenvect
,
o
r
s
and
1
3
the matrix that does the job is
A=
Solution
,\ 1 =
,\ 2 =
A
P = [v1 v2 ] =
We know that
P - 1AP =
[� - � ]
[40 - OJ1 = D
v1 =
P
v2 =
Let to get(so that or, equiandvsaluebsnttliytu, te these results into the original equation
This is just the system
whose general solution is
or
x = Py
x ' = Ax
x ' = Py' )
Py' = APy
y' = P - 1APy = Dy
4
Y1 = C 1 e 1
Y2 = C 2 e - t
342
Chapter 4 Eigenvalues and Eigenvectors
To find we just compute
x,
x = Py =
[�
(Check that these values satisfy the
gisoven system.) and
Observe that we could also expres the solution in Example as
[] [ ]
This itzeablchnie.qTheue genernext athlieorzeseeasm,ilwhosy to e proofsyisstelemsft aswheran eexertheccoeffi
cmmar
ient matizesrixthies
disitaugonal
i
s
e
,
s
u
ation.
Let be an diagonalizable matrix and let
be such that
x1 = 2C 1e 4 t - C2 e - t
x2 = 3C 1e 4 1 + C2 e - t.
4.43
Remark
2
-1
x = C I e41 + C 2 e -1 l = C 1 e41v1 + C2 e -1v2
3
n
Theorem 4 . 4 0
A
n
X n
X n
P = [ v1 v2
•
•
•
v" ]
Then the general solution to the system is
Theecosnextysteexampl
ereinasvolonablves ea tbio oaslosgiumecal tmodel
ingrwhiowtchhrtawteoofspeachecies slpivecie iens de­the
same
m.
I
t
i
s
h
at
t
h
e
pends
ations.sim(plOfe courby igsnore, thinergethareese.ot)her factors that govern
growtIf hon, butthandewesizwieslofkeepdenotourpopul
model
wo populis ofattihoensforatmtime then and
are their rates of growtehthate tsiimzese ofOurthe tmodel
0
x' = Ax
x = C 1 e A 1 t v1 + C 2 e Air v2 +
·
·
·
+ C " e A"1 V"
both
x�(t)
Exa m p l e 4 . 4 4
x 1 (t)
x 2 ( t)
t.
x{(t) = ax 1 (t) + bx 2 (t)
x;(t) = cx 1 (t) + dx2 (t)
a, b, c,
d
t,
x{(t)
where the coefficients and depend on the conditions.
Raccoons
andr, andsquisprace.els iLetnhabithte traccoon
he same andecosysqstuiemr eandl populcompetationse wiatthtieachme otyearhers forbe
food,
wat
e
giravteenisby and butreswhen
pectivesqlyui. Irnetlhs earabse prenceesentof, tsquihe compet
r els, tihtieornaccoon
grheorwtac­h
s
l
o
ws
t
wth rat.eIntothe absence of raccoonsThe, thsquie grorwtelhpopul
atitohne sisqsuiimr ielal rpopul
ly affeacttioend
bycoon
tis hegrroaccoons
r
a
t
e
of
e populisation growth rate for squir els when
the ecosystem wiandth rtahccoons
Suppostheeytharate isnhiartiainl gy
r ( t)
s ( t),
r'( t) = 2.5r ( t),
r' ( t) = 2.5r ( t) - s ( t) .
s'( t) = 2.5s ( t) ,
s'( t) = - 0 . 25r ( t) + 2.Ss(t).
t
Section 4.6 Applications and the Perron-Frobenius Theorem
343
raccoons
and squir els in the ecosystem. Determine what happens to
tthheresee tarweo popul
a
t
i
o
ns.
Our system is where
[ ] and [ -I.O J
The- ei] genvalues of] are and with corresponding eigenvectors
[ � and [ � By Theorem the general solution to our system is
60
Solution
60
x' = Ax,
2.5
r (t)
Ax = x(t) =
s (t)
-0.25
A2 = 2,
A A1 = 3
.
4.40,
v2 =
[ ]
2.5
v1 =
[]
2
-2
x(t) = C 1 e 3 tv1 + C2 e 2 tv2 = C1 e 3 r
+ C2 e 2 r
1
1
(7)
.tionThe iwenitihaveal population vector is [ : ���] [ :� ], so, set ing in Equa­
Solving this equation, we find and Hence,
(7),
C1 = 15
x(t) = 15e3 r
t=0
C2 = 45.
[ - � ] + 45e 2t [ � ]
Ficcoongurepopulshatowsion
tdifrhoeemsgroutwhiaphsafchtofewer tahfilesitnedletwmoro functe thanions,yearand. you(Canandcanyouseedetcleerarmlyintehat the rawhen
it dies out?)
.+
We
now
cons
i
d
er
a
s
i
m
i
l
a
r
exampl
e
,
i
n
whi
c
h
one
s
p
eci
e
s
i
s
a
s
o
ur
c
e
of
food
forl
tbehedrotahsteric. alSuchly overa model
i
s
cal
l
e
d
a
Once
agai
n
,
our
model
wi
simplified in order to il ustrate its main features.
r(t) = - 30e 3 1 + 90e 2 1
�
=
x( 0) =
s(t) = 15e 3 1 + 45e 2 1.
1
exactly
predator-prey model.
Raccoon and squirrel populations
Figure 4 . 2 0
4.20
344
Chapter 4 Eigenvalues and Eigenvectors
Exa m p l e 4 . 4 5
Robionlynsours andce worof food.ms cohabi
t
an
ecos
y
s
t
e
m.
The
r
o
bi
n
s
eat
t
h
e
wor
m
s
,
whi
c
h
ar
e
t
h
ei
r
Theivelyr, oandbinthande equatwormionspopulgoveratnioinnsg atthetigrmoewtyearh ofsthareetwdenoto populed bya­
and
r
e
s
p
ect
tions are
Ithf eintiwtioalpopul
y roabitionnss andover timwore. ms occupy the ecosystem, determine the behavior of
fiandrst thinginwethnote twicoe equat
aboutiotns.his exampl
e eilsyt, hwee prcanesenceget rofidthofe texthemra
conswithtaantsims,plThee change
For
t
u
nat
and ofvariablSubses. Itfitweutilnegt into Equations andwe have then
r ( t)
t
w(t),
r'(t) = w (t) - 12
w '(t) = - r (t) + 10
6
(8)
20
Solulion
- 12
r'(t) = x'(t)
10,
w'(t) = y'(t).
r(t) = x(t) + 10
w(t) = y(t) + 12,
(8),
x'(t) = y (t)
(9)
y'(t) = -x(t)
which is] easier to work with. Equations have the form where
[ . Our new initial conditions are
and
so Proceedi[ -ng:].as in the last example, we find the eigenvalues and eigenvectors of
2 ewhix rootch hass, whinocrhealareroAot1 s. WhatandsAh2ould weThedo?
TheWe havecharanoctechoiristiccepolbutynomito usaeltihseAcompl
=
=
correspemonding oureigenvect
Theor
solutioorns hasare althseoforcomplm ex-namely, v1 [�] and v2 [ -�l By
x' = Ax,
(9)
O 1
-1 0
x(O) = r (O) - 10 = 6 - 10 = - 4
A =
y (O) = w (O) - 12 = 2 0 - 12 = 8
x(O) =
A.
+ 1,
i
=
4.40,
From
x( O ) =
- i.
=
[ - : ] ,we get
whose solution is C1 =
- 2 - 4i
and C2 =
So the solution to system is
(9)
- 2 + 4i.
of thisesxonumber
lution? Robis! Fearnsleands lyworpromceedis innhabig, wet a applreal yworEullde­r's
yetformWhat
ourulasoarluetiowen itnovolmakeves compl
cos sin
e ;1 =
t+i
t
Section 4.6 Applications and the Perron-Frobenius Theorem
345
IN.Si\NC.T.
T\<?sERS AAE
�n ·tum n .
CALV I N A N D H O B B E S © 1 988 Watte rson. R e p r i nted with p e r m i s s i o n o f U N IVERSAL P R E S S SYN D I CATE. A l l r i g hts reserved
(Appendix to get cos ( - t) + i sin ( - t) cos t - i sin t. Substituting, we have
x(t) ( - 2 - 4i) (cost + isin t) l�J + ( - 2 + 4i) (cost - isin t) � �J
(
)
)
+
4s
i
n
t
+
i
2s
i
n
t
4cost
]
[ ( -(42cost
)
(
)
cost
+
2s
i
n
t
+
i
2cost
+
4s
i
n
t
+24ssinint)t)++i (i2(4coscostt -+42ssinint)t) J
+ l ( -(42cost
cos
t
+
[ - 4coscost t++4 sisnint t]
Thiterms sgiofvesourx(tor) igi-nal4 cost+
variables,swein tconcland y(udet) thatcost+ 4 sin t. Putting everything in
r(
t
)
x(
t
)
+
10
4
cost
+
s
i
n
t
+
10
w( t) y( t) + 12 cost + 4 sin t + 12
and
e - it =
C)
=
=
_
=
8
8
=
=8
8
=
=
=
= 8
8
25
20
15
10
5
15
0
2
4
6
8
10
Robin and worm populations
Figure 4 . 2 1
12
14
30
16
Figure 4 . 2 2
346
Chapter 4 Eigenvalues and Eigenvectors
Sotheourtwosolpopulutionatiisornsealosafctielraaltel!perTheiodigrcaalphsly. ofAs theandrobin populin Fiagturioen increshasesow, tthhate
wortheimr number
populastisotnarstttaortdecls to idecrne aseaswele, lbut. Asasthethpre erodatbinors's onldisyappear
food,stohure worce dimmpopul
inishesa­,
tandion tbegihe cyclns etorerpeatecovers it.seAslf. Thiits foods oscisluappltiony iisntcryepiasescal ,ofsoexampl
does tehseinrowhibinchpopulthe eiatgien­on,
valuPlesoart iencompl
enx.s, worms, and time on separate axes, as in Figure clearly
g
r
o
bi
reveals the cyclic nature of the two populations.
secta diioffnerbyentlioaokiblenfunctg at whation oftwe, havethendonethe generfromaal sdiolffuetrioentn ofpoithnet
ofordviiWenearw.yconclIfdiffuerdeentthiailissequat
ionwe have beenis consid,erwhering havee is tahscale forarm. The syste,mssoofif
lweinearsimdiplffyeplreontwedial equat
i
o
ns
ahead witwher
hout ethiniskianvectg, weor.miButghtwhatbe teonmptearedthtocouldeduced thisthmean?
at the
sOnolutthioenrwoul
d
be
i
g
ht
h
and
s
i
d
e,
we
have
t
h
e
r
a
i
s
e
d
t
o
t
h
e
power
of
a
Thi
s
appearLets'stsotabert nonsby consensied,eryetingyouthewiexprl seees tihoatn therIeniscala cwayulus,toyoumakelearsennstheatoftiht.e func­
tion has a power series expansion
2! 3!
that converges for every real number By analogy, let us define
2! 3!
Theit converrightg-eshandfor anysideriesaljumatst defirixnedSoi nnowterms ofispower
siofx, callanded thitecan be shownofthat
a
mat
r
But how can we compute or For diagonal matrices, it is easy.
Compute for [ ].
From the definition, we have
2! 3!
] [ ] [
]
[
r(t)
w ( t)
4.2 1
4.22,
x = x(t)
x' = ax x = c e a1
x = ce A 1,
c
x' = Ax
c
number e
eA.
ex
matrix.
xz x 3
ex = 1 + x + - + - + . . .
x.
Az A3
eA = I + A + - + - + · · ·
eA
Exa m p l e 4 . 4 6
eD t
D=
A.
e A 1?
A,
eA
exponential
4 0
0 -1
Solution
e
Dt
= I + Dt +
(Dt) 2
--
+
(Dt) 3
--
+· · ·
(4t) 3
(4t) 2 0
0
+ _i,2
+
· 0 ( - t) 2
31
0
-t
1 + (4t) + t C4t) 2 + t, C4t) 3 + · · ·
o
1 + C - t) + ti C - t) 2 + t, C - t) 3 + · · ·
0
et
e� t
I
[; ]
The matrix exponential is also nice if is diagonalizable.
A
A.
Section 4.6 Applications and the Perron-Frobenius Theorem
Exa m p l e 4 . 4 1
Compute eAforA = [� � ].
In Example 4.43, we found the eigenvalues of-A],to be ,\1 = and ,\2 =
with corresponding2eigenvectors = [ � ] and v2 = [ � respectively. Hence, with
P = k V2 ] k=-[1 3 -l ],wehaveP- 1AP = D = [ 0 0 ].sinceA=PDP - 1, we
have A = PD P , so
eA = I A A2!1 A3!3
= PIP- 1 PDP- 1 -2! PD 2P- 1 -3! PD 3P- 1
= P(I D -D2!z -D3!3 )
0- 1 ] [ 23 ]
e
[ 2e3e44 - 3e3e-- 11 2e3e44 - 2e2e-- 11 ]
are nowial"insoalupostionitiofon to=showwasthatnotoursobolfardoff(andaftseeremiall!ngly foolish) guess at
an "Weexponent
Let A abel solanution todithaegonalsysteimzable=matrixis wi=th eA1eigcenval, wherueesc,\is1,an,\2,arbit,raryThenconsttahnte
gener
vector. If an initial condition is specified, then c =
Let P diagonalize A. Then A PDP - 1, and, as in Example
1
Hence,
we
need
t
o
check
t
h
at
i
s
s
a
t
i
s
fi
e
d
by
Pe01
P
c. Now, everything
is constant except for e01, so
( 1 0)
If
4
Solution
v1
[vi
4
1
+-+-+
+
+
+
+
·
·
1
1
+
+
+
·
-1
+
·
·
·
+
·
·
·
p- i
- 1 -I
1
_!_
5
+
+
x'
Theorem 4 . 4 1
341
n
Proof
X n
Ax
•
x' Ax x
x ( O)
.
Aw
x ( O).
=
x' = Ax
.
4.47,
x=
- 1,
348
Chapter 4 Eigenvalues and Eigenvectors
then
Taking derivatives, we have
[I
e Dt =
0
e A2 t
0
0
0
0
). , ]
0
0
d ( A,t
e )
dt
0
� n·, JJ
� [ ! �r·' J,]
0
A z e A,t
0
0
A1
0
e A,t
0
An
0
0
Substituting this result into Equation we obtain
as reThequirlead.st statement follows easily from the fact that if
then
since (Why?)
, Theorateimon ofmatis rtirxueexponent
even if ialsisfornotnondidiagonalagonalizablizable, ebutmatwericwies lrenotquirpres othvee
tInhisfact. Comput
in more advanced linear
algebrIdeala telyxt, tsh. is shortofdiagrmates iroixn, hasa tospiercvtedhattomayil usbetrafound
t
e
t
h
e
power
of
mat
h
emat
i
c
s
t
o
gener
imporatlaintze andtoolsthine valmanyue ofapplcreicatativioensthiofnkilinng.earMatalgrebrix exponent
a, both thieorals etutircnaloutandtoapplbe veried.y
Wechaiconcl
er as weofbegan
itt-ionbygrloookiwthngarate dynami
csalofsystems. Markov
ns andudethtehiLess chaptliEache model
popul
a
exampl
e
can be described by a matrix equation of the form
( 10),
x' = PDe Dtp - 1 c = PDP - 1 Pe Dtp - 1 c = (PDP - 1 )(Pe Dtp - 1 )c = Ae A tc = Ax
x = x ( t) = e A tc,
x(O) = e A · Oc = e0c = le = c
e0 = I.
see A. J. Insel, and
byL.ForE.S.example,
H.Spence
Friedberg,
(Engle1979).
wood Cliffs,
NJ: Prentice-Hall,
Linear Algebra
4.41
A
Jordan normal form
Discrele Linear ovnamical svs1ems
discrete linear
dynamical systems.
Section 4.6 Applications and the Perron-Frobenius Theorem
349
wher
ate oftohre ofsystthemeseatsy"tsitmemse" isandrelateids atosqthuare eie gmatenvalrix­.
Asuesweande thaveheiegvectenvect
seen,or othrrseeofcorlong-thdes ttmatehremsrtibehavi
x oxiThematpower
methuodes explandoeiitsgenvect
the iteroartsi,veandnatuthree
ofPersruochn-Fdynami
c
al
s
y
s
t
e
ms
t
o
appr
e
ei
g
enval
inforcmieatntiomatn aboutrix thisenonnegat
long-termivbehavi
or
ofaWhen
discrertoebeniliins earaus2Theor
dynami2 matemcalgirisxvy,esswetsepmecicanwhosaldesizedeccoeffi
e.
reibalelythane evolinfiuntiitoencolofleactdynami
cal syisotensm.
geomet
r
i
c
al
l
y
.
The
equat
i
o
n
i
s
r
i
o
n
of
equat
Beginning with an initial vector we have:
xk
A
k
A.
A
A
X
xk + I = Axk
XcJ,
X1 = A Xo
x2 = Ax1
x3 = Ax2
wiNotThel eisetdthentatify each vect} oisr calin laetdraajectory withofittshheade systseom.th(atForwegrcanaphiplcoalt purit aspaospoies,nwet.)
Letin the traj[e0.0ct5orie0.0s 8wi] .thForthethfole dynami
c
al
s
y
s
t
e
m
pl
o
t
t
h
e
fi
r
s
t
fi
v
e
poi
n
t
s
lowing initial vectors:
(a) [ Os ]
1.0 25 ]
2.0 5 ]
(
a
)
We
comput
e
[
[
0.0 625 ] , [ 0.0 3 125 ] . These are plot ed in Figure 4.23, and the points are
[connect
marked ed to higandhlight itnheFitgraurjeect4.o2ry3.. Similar calculations produce the trajectories
{ Xo , x1, x2 ,
.
•
trajectory
.
xk = A k x0 .
Exa m p l e 4 . 4 8
xk + 1 = Axk ,
A =
Xo =
x 1 = A Xo =
Solution
, x2 = Ax 1 =
X4 = Ax3 =
( b), (c),
(d)
y
X4
-2
(a)
Xj
X3 X 2
2
Xo
4
6
x
-2
-4
(b)
Fioure 4 . 2 3
, x3 = Ax2 =
350
Chapter 4 Eigenvalues and Eigenvectors
everysttraandjectwhyory conver
in thInisExampl
case. Wee can under
t] his i]s gsoesfrtoom TheTheororeimgin is calTheled anmatrix A in
Exampl, respeectivelhasy. (Ceiheckgenvectthios.r)sAccor
[ � anddingl[ �y, focorrr anyespiondinitianl vectg to oitrs eigenvalues and
4.48,
0.
attractor
4. 19.
4.48
�
0.5
0.8
we have
Because
zero askem gets ltahrgate,becausappreo0.aches8 is thefordomiany­
choinantceie gofx0envalbot•hIuneaddiof A,tandion, wiwel knowapprapprofroachachomaTheor
mul
t
i
p
l
e
of
t
h
e
cor
r
e
s
p
ondi
n
g
ei
g
envect
o
r
cient of corresponding to [ � ]) . In other words,
[al�l ]trasajelctongorieass except th(toshee coeffi
y-axis, as Figure shows.that begin on the x-axis (where wil approach the
(0.5) k
(0.8) k
xk
c2 * 0
Xo
c2 = O)
4.23
Exa m p l e 4 . 4 9
Discuss the behavior of the dynamical system xk+i Axk corresponding to the
]
matrix A [
The eigenvalues of A are and 0.8 with corresponding eigenvectors [ � ]
andwe have[ - � ],respectively. (Checkthis.) Henceforaninitialvector Xo [ � ] [ � l
xk Ak ( )k [ 11 ] (0.8)k [ - 11 ]
Once agaithen ttrhaejeorctoigriynwiislanapprattroaachctort,hbecaus
ehxrkoughapprtohachese originfowir tanyh dichoirectcioenofvectoIrf
e
l
i
n
e
t
[ - � ]. Several such trajectories are shown in Figure The vectors where
arineg tonrajethcteolriyneintthhriosughcasethfole lorowsigitnhiwis ltihnediinrteocttihoenorvectiginor. [ � ], and the correspond­
=
0.65 -0.15
=
.
-0.15 0.65
0.5
Solution
�
0
xk
4.28
= c,
=
Xo = c 1
0.5
+ c2
-
+ c2
0
c2 * 0,
4.24.
Xo ·
Xo
c2 = 0
Section 4.6 Applications and the Perron-Frobenius Theorem
351
y
Figure 4 . 2 4
Exa m p l e 4 . 5 0
folDislocwiussngthmate behavi
rices: or of the dynamical systems corresponding to the
(a) [ � �] (b) [ �.5 �-S J
(], a) The eigenvalues of are 5 and with correspondi] ng ei-genvect], ors [ � ]
and [ � respectively. Hence for an initial vector [ � [ � we have
xk + 1 = Axk
A=
A=
A
Solution
3
Xo = c 1
+ c2
AsBecausbecomes
l
a
r
g
e,
s
o
do
bot
h
and
Hence,
t
e
nds
away
fr
o
m
t
h
e
or
i
g
i
n
.
5
],
e
t
h
e
domi
n
ant
ei
g
enval
u
e
of
has
cor
r
e
s
p
ondi
n
g
ei
g
envect
o
r
al
l
t
r
a
j
e
c­
5
[
�
tories for which 0 wil eventually end up in the first or the third quadrant. Tr-lajec­].
tSeeorieFis gwiurteh 4.25(a0).start and stay on the line - whose direction vector is [ 1
(b) In this-example, the eigenvalues are 1.5 and 0.5 with corresponding eigenvectors
[ � ] and [ � ], respectively. Hence,
k
k
3k_
xk
c 1 -=fa
c1 =
y =
x
352
Chapter 4 Eigenvalues and Eigenvectors
y
y
20
- 20
(a)
Figure 4 . 2 5
If c1 = 0, then xk = c2(0.5)k [ - � ] [ � ] ask ---+ But if c 0, then
---+
cxi.
1 -=fa
and such trnjeoto;ies osymptotkally apprnooh the line See Egme 4.25(b) 4
etu4.de50(becaus
a), all epoints th1atforstabotrt outh eineargenvalthueesor; iginis become
increasinglIny
lExampl
argIeninExampl
magni
a
cal
l
e
d
e
4.
i
s
cal
l
e
d
a
becaus
e
t
h
e
or
i
g
i
n
at
t
r
a
ct
s
poi
n
t
s
i
n
s
o
me
5
0(
b
direThectionsnextandexampl
repelsepoishnowsts inwhatothercandirhappen
ections.whenIn thitshcase eieg, envalues1 andof a real 2 1. 2
matrix are complex (and hence conjugates of one another) .
y � x.
), 0
Exa m p l e 4 . 5 1
l,t l >
saddle point
0
repeller.
l -\1 1 <
l -\2 1 >
X
Plcorotretshpeonditrajnectg otorythbegie folnloniwingngwimatth rices=: [ ! ] for the dynamical systems xk+ 1 = Axk
(a) A = [ 0.0.55 - 0.0.55 ] (b) A = [ 0.0.62 - 1.1.42 ]
ure 4.e2as6((a)b)andappear(b),srteospfoectl oivwelany. Notellipettihcatal
(ora)biits. a trajTheectortryajsepctiroalriinegs arinetoshtownhe oriingiFin,gwher
Xo
Solulion
Section 4.6 Applications and the Perron-Frobenius Theorem
y
y
-4
-1
(a)
Figure 4 . 2 6
Theorem 4 . 4 2
353
0
(b)
ExamplThee 4.folSllo(wia)n. g theorem explains the spiral behavior of the trajectory in
LetbothAzer=o,[tabhen-Aab Jcan. Thebe facteigenvaloreduases of A are = a ± bi, and if a and b are not
A = [ ab - abJ = [ r Or J [ cs�ms()() -cos()sin () J
where r = I A I = Va2 b2 and() is the principal argument of a bi.
TheA = eit(g2enval
u
es
of
A
ar
e
(
)
a
±
=
2
a
±
2\/b2\/=l
=
a
±
b
i
=
a
±
bi
t
l
l
by Exercise 35(b) inaSect-iobn 4. 1 . Fia/rgure 4.-2b/r7 displaysra O bic, r,s()and()-.sIint fol() lows that
A = [ b a J = r [ b/r a/rJ [ r J [ sm� () cos() J
Geomet
r
i
c
al
l
y
,
Theor
e
m
4.
4
2
i
m
pl
i
e
s
t
h
at
when
A
=
[: �b J
thce�sl()inear- sitnra()nsfothrrmoughationthe angle A() folilsowedthe bycompos
i
t
i
o
n
of
a
r
o
t
a
t
i
o
n
=
r
O
a scaluiesng are= [ = 0.r5J ±wiOth.Sifac-so
[tosrmr() (Figurcos()e 4.J 28). In Example 4.S l (a), the eigenval
r = I A I = \./2/2 1, and hence the trajectories all spiral inward toward
The next theorem shows that, in general, whena a-realb 2 2 matrix has complex
eigenvalues, it is similar to a matrix of the form [ b a J . For a complex vector
x = [ :J [ ::�:J = [ : J [ �}
,\
0
+
+
Proof
�)
+
=
Im
0
-=F 0,
Remark
a + bi
r
e
Figure 4 . 2 1
T(x) = x
S
=
b
R
,\
0.707 <
X
+
0
0.
354
Chapter 4 Eigenvalues and Eigenvectors
y
ln /
Sca i g
Ax = SRx
Rx
/
Rotation
rotation followed by a scaling
Figure 4 . 2 8
A
we define the real part, Re x, and the imaginary part, Im x, of x to be
Rex [ :] [ :::] and Im x [ �] [ :::]
andandLet corrbe easpreondial ng eimatgenvectrix wiorthx.aThencompltheexmateigenvalrix ue A[ Re x Im x](wheris inevertible
=
=
Theorem 4 . 4 3
A
2X2
P=
= a - bi
b i= 0)
Let x Axso that Rex and Im x From x Ax, we have
Equating real and imaginary parts, we obtandain
Now[ so
] [ ]
to s,htowhenthitatwoulandd folarelowlinthearatly indepen­for
dentsomeTo. I(fnshonzerowandthoatcompl
weriseinnoteverx) slticinbalearlae,rilyt iisbecaus
nenough
dependent
e neither nor Thus,
x
Now, because is real, x Ax implies that
Ax
Ax
Butso is an eigenvector corresponding to the other eigenvalue A
= u + vi
=u
= v.
A =
Au + Avi = Ax = = (a - bi) (u + vi)
= au + avi - bui + bv = (au + bv) + ( - bu + av)i
Proof
Au = au + bv
Av = - bu + av
P = [u l vJ ,
a -b
a -b
= [ul vl
= [au + bv l - bu + av] = [Au l Av] = A [ u l v l
P
b a
b a
= AP
P
u v
u
v
v = ku
k,
u v is 0.
= u + vi = u + kui = ( 1 + ki)u
A
A =
Ax = Ax = Ax = =
x = u - vi
= a + bi.
x = ( 1 + ki)u = ( 1 - ki)u
Section 4.6 Applications and the Perron-Frobenius Theorem
355
becaus
muleigenvect
tipleesoofriss coraandrerealstphvectondiereforonr.geHence,
artoedimulstitnhtctiepleieiesggenvect
ofenvaloneuoesanotrsmusxhandert.beThiliofsniearsAimlaryposienbotdependent
sibhlenonzer
becausbyoe
Theorems.)4.20. (This theorem is valid over the complex numbers as well as the real
number
inverThitibslecont. It nowradicfoltiolonwsimtplhaties that and are linearly independent and hence P is
Theor0.2 em- 1.4.24]3 serves to explain Example 4.5 l (b). The eigenvalues of
A = [ 0.6 1.4 are 0.8 0.6i. For ,.\ = 0.8 - 0.6i, a corresponding eigenvector is
Frhaveom Theorem 4.43, it follows that for P = - 01 ] and C = [0.0.68 - 0.0.68 ] , we
A = PCP- 1 and P- 1AP = C
For the given dynamical system 1 = Axk, we perform a change of variable. Let
Then
so
NowChast
calthe sorysitgeimn byhesameei
=TheorCyemkg4.enval
sim42.pluyesasA(
rotates wthhy?e poi) andnts i0.n 8ever0.y6triaje=ct1.Thus,t
ory in a cihedynami
rcle about­
cttroarnsyfoofrmthateiodynami
c=al Axsyst=emPCPin Exampl
e t4.ra5nsl (fob)r,mweatioitn­
1
(
)
ercanatiTobeveltdetyhapploughtermyitnofheeasalinttrhearaejecompos
n
T
x
x
.
The
ofvariaofblevar(xitaobly)e (,yfolbacklowedto x)by. Wethe
rwiotlatencount
ion deteerrmtihnieds idbyeaC,agaifolnlowedinittihobyen applofthae irchange
ecatveriosne tchange
os"grinaphiSectnigonquadr6.3. aInticExerequatcisieons74 iofn
SectSectiioonn 5.5.55,and,you wimorl sehgener
a
l
y
,
as
"
c
hange
of
bas
i
as itToappearsummars to ibeze frthoen:mowIFifgaturhreateal4.t2h2e6(trb2a)jmat.ectorriyx AinhasExamplcomple 4.ex5ei1 (gbenval) is inudeedes,.\ =an ellipsbie,,
thenis athe trajectories of stphieradynami
cald isfy,.\stem 1 i=s aAxk spiral inwardandif l,.\ie on a1
l
out
w
ar
closed orbit if ,.\ = 1 is an
u
x
u
u
v
±:
xk +
�
l
Yk +i
X
(O
1 1
spiral attractor) ,
(O
xk + I
1 1 > (O
orbital center) .
±:
l
a ±:
spiral repeller) ,
1 1<
Vi gnette
Ranking Sp o r t s Te a m s and S e arch ing
t h e I nternet
Inrankanythecompet
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league,inigt iwis notns andneceslossaersilyaloanestroveraightlofooksrwarthde proces
slittoy
pl
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pos
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i
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i
tanothat honeer tteeamam maymay haveaccumulfewaerteviacltaorgrieesnumber
of vithcemtoriagaies againstnssttroweakng tetamseams. Whi, whiclhe
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eams dispoibetnttesrs?cHow
shtoulakend weintocompar
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Despite thaceeseandcomplmuch-exitiaents,itchipeatraenkid fenatguofreatinhltehteesmediand asp. Fororts exampl
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itenniouss plannual
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and
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and
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er
ustheedprtooblpraeyeromducebys arusesuialnchsgotrhraaenkinkedidneasgsi,nfrbuttoemrnweatthiioscannalchaptlgaiy. Therenr.somee arienmany
sight incopyrto howightteodapprschemesoach
let's revi. sWiit Exampl
elos es arFiveeretecornnidsedplianyerthseplforaymoneof
anota digTohraerphesitnainbla whiirsohund-tchheabasrodibiricenctitdoeea,durnament
n
s
and
edgerifrx omtherefore
to indihascates that1 plifaplyerayerdefeatdefesatpls playerayer
Theandcorhasrespondi0notghaderwacency
mat
ise.
3.69.
j
aiJ =
j
A
5
i j
3
j.
i
0 1 0
0 0 1
A= 1 0 0 1 0
0 0 0 0 1
0 0 1 0 0
2
4
i
aiJ =
We iwoulndicatd elsiktehatto plasasyerociate isa rraanked
nking morwie thhigplhlayyerthaninplsauyerch a wayFor tthhatis
r; > r1
356
i
r;
i
j.
purpose, let's require that theand'sthbeenprorogbabianizleittiehse (rtahnkiat nisgs, in a for all and
r1 + r2 + r3 + r4 + r5 = 1 )
0 :s r; :s 1
i,
ranking vector
r
;
d be pre,oplporayertionaldefeatto thede plsuamyerofs
tFurhe trhaandnkiermnorgsseo,oflweetth'swant
eiplnsaisyert ths atdefeatplayered byi's plraankiyerng Forshoulexampl
he consthteantfoloflowiprnoporg sytsitoenalm:ity. Writing out similar equations for the other
plwherayeres prisotduces
2, 4,
i.
5,
1
a
r 1 = a(r2 + r4 + r5 )
r2 = a(r3 + r4 + rs )
r3 = a(r 1 + r4 )
Observe that we can write this system in matrix form as
or
'1
r2
f3 =
f4
f5
ll'
0
0
1
0
0
1
0
0
0
0
0
1
0 1 0
0 0 1
1 0 0
'1
r2
f3
f4
f5
r = aAr
Equiis anvaleiegntenvect
ly, weosrecore thratespthondie rannkig tnogtvecthe mator rixmust satisfy In other words,
Fur
t
h
er
m
or
e
,
i
s
a
pr
i
m
i
t
i
v
e
nonnegat
i
v
e
mat
r
i
x
,
s
o
t
h
e
Per
r
o
nF
r
o
beni
u
s
Theo­
vectremoguarr turannts eoutes tthoatbethere is a ranking vector In this example, the ranking
r
r
A
1
Ar = -r.
a
A!
unique
r.
0.29
0.27
r = 0.22
0.08
0.14
so weBywoulmodidfryainkng tthhee plmatayerrixs in tiht eisorposdersible to take into account many of the com­
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Thereforor eof, the rIannkithnisgexampl
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unordered.
ordered
i
i
i
i.
356
r4 = a(r 1 + r2 + r3 )
A r.
0 0
0
T
A = 0 1
1
1
1
0
0
1
0
0
0
0
0
0
0
1
0
0
transpose
1
AT = a
0 14
0.08
= 0.22
0.27
0.29
5, 4, 3, 1, 2.
4.5).
358
I
Section 4.6 Applications and the Perron-Frobenius Theorem
Exercises 4 . 6
17.
Markov Chains
Which of the stochastic matrices in Exercises 1-6 are regular?
1.
3.
5.
[� �]
[i � ]
O[ 0.J5 0 � ' ]
0.4 0 0.5
2.
4.
1
6.
[ � !]
0
[! 0 �]
[ �".5' 00 � ]
10.
P - 1 LP
18.
[l [
[i iJ
02[ 0.6 0.0.31 00.4
]
0.
2
0.
2
0.
6
eovstchaieadynsitsatunie prqoue.bability vectUseorTheo­of a
rremPregulove4.art3h3MaratorthkTheor
em 4.34.]
3
2
I
6
8. P =
4
In Exercises 1 1 -14, calculate the positive eigenvalue and a
corresponding positive eigenvector of the Leslie matrix L.
1
11. L =
12. L =
15.
16.
[ 0.5
[ 0.O 5 � ]
7
5
L [�s 00.5 � ] [i 0 � ]
If a Lesisltiheematsigrniixfihascancea uniforqtuehe pospopulitivaetieiongenvalif ue
what
VermatirfiyxtLihatntEquat
he charioanct(e3r)isistic polynomial of the Leslie
()
det(L Use matalohngemattheiclaalstincolductumn.ion] and expand
14. L �
2
3
A1 < l ? A1 = 1 ?
S1
A1,
A1 > 1 ?
cL A = ( - l ) " (A " - b 1 A " - 1 - b 2s 1 A " - 2 - b 3s 1 s2 A " - 3
- · · · - bns 1 s2 · · · Sn - I )
[Hint:
- Al)
S1S2
S1S2 · · · sn - 1
L. [Hint:
L
A1
S1 / A 1
S1S2 / A i
S1S2S3 / A i
l
[Hint:
�
1
1
Population Growth
13.
s;
l
7. P =
9. p =
If all ofthe survival rates are nonzero, let
0
0
0
00 0 0 00
00 0
Comput
tSectic polionynomi4.e 3.] al ofand use itRefeto firndtotExerhe charciseact32eriins­
Verify that an eigenvector of corresponding to is
P=
In Exercises 7-9, P is the transition matrix of a regular
Markov chain. Find the long range transition matrix L of P.
359
SectionCombi
4.3 andneExerExercciissee4617iabove
n Sectiwionth4.Exer4.] cise 32 in
[Hint:
GAs In Exercises 1 9-21, compute the steady state growth rate of
the population with the Leslie matrix L from the given
exercise. Then use Exercise 18 to help find the corresponding
distribution of the age classes.
19.
20.
21.
GAs 22.
ExerExercciissee 4039 iinn SectSectiioonn 3.3.77
ExerManycisspe eci44eisnofSectsealionhave3.7 suffered from commercial
huntand meating. They
havefur trbeenade, ikin lparedtforicultahrei, rresduced
kin, blsuobberme ,
.
The
pointaoftioextnsiarenctideclon. iToday,
tshhe
grssteoalecksatpopul
esduet thartteoiatoovernss totofisealtshheinpopul
n
e
of
fi
habiby fitsahter, enty owners
anglement. Someing,smarepolalsilnuhaveetiodebrn,beendiiss,tuanddeclrbanceculareldiofng
endangereTabld specie 4.e7s;giotvheserthspeecibiretsharande carsuerfulvivlyal rates
managed.
forclasthees. nor[Thetherdatnafuraresealbas,edidvonided iE.ntoYor2-kyearandageR.
HarNorttlheery,n"PFurup Product
i
o
n
Fol
l
o
wi
n
g
Har
v
es
t
of
Femal
e
Seals(1981)
;' , pp. 84-90.]
A.
Aquatic Science, 38
J.
Canadian Journal ofFisheries and
Chapter 4 Eigenvalues and Eigenvectors
360
(b)
sShowents that = if and only if ,\1 = Let(This repre­
g(,\) = ,\ ,\2 ,\3
"
,\
Show
t
h
at
,\
i
s
an
ei
g
enval
u
e
of
i
f
and
onl
y
i
f
g(As,\s)umi= n1.g] that there is a unique positive eigen­
lvalpopul
atiuoen,\aits1i,odecrsnhowiseiasnthcrinategasanding. if andifonlandyonlif thyeifpopu­
the
r 1
1.
zero population growth.) [Hint:
bnS1S2 · . . Sn - 1
b l bzS1 b 3S1S
- + - + --2 + . . . + ---L
(c)
A sustainable harvesting policy is a procedure that allows a
certain fraction of a population (represented by a population
distribution vector x) to be harvested so that the population
returns to x after one time interval (where a time interval
is the length of one age class). If h is the fraction of each
age class that is harvested, then we can express the harvest­
ing procedure mathematically as follows: If we start with
a population vector x, after one time interval we have Lx;
harvesting removes hLx, leaving
Ta b l e 4 . 1
Birth Rate
0.00
0.02
0.70
1 .53
1 .67
1 .65
1 .56
1 .45
1 .22
0.91
0.70
0.22
0.00
Age (years)
0-2
2-4
4-6
6-8
8- 10
10-12
12-14
14- 1 6
16- 1 8
18-20
20-22
22-24
24-26
r< 1
r> 1
Survival Rate
0.91
0.88
0.85
0.80
0.74
0.67
0.59
0.49
0.38
0.27
0.17
0.15
0.00
Lx - hLx
=
( 1 - h)Lx
=
mattIhf ath,\1riixs theanduni1/,\qiues 1t.hpose siutisvteaieinablgenvale haruveesoft araLestiol,ipre ove
Fiandnd cartheibsuoustaininExerable charisevestinraSecttio forionthe wood­
Cons
t
r
u
ct
t
h
e
Les
l
i
e
mat
r
i
x
for
t
h
es
e
dat
a
and
l
comput
eigenvalor.ue and a corre­
Useduceing tthheedatcaraibinouExerhercdisaccor
e idninSectg toioyourn answer
sInpondithe lnoegngtposheruposin,tivwhatietieivegenvect
r
each age clas and whatperwicentl thaegegrofowtsealh sratwielbe?be in Findtorothipargeinsutals(talae)ivel.nVerablafeitfyeharrtonehvatesttthimreapopul
etioinfoterravtthalioe.nserealtuinrns to its
Exerseal popul
cise ati(oCnsonswhenervatoverionifistsshhaveing hashadretduced
o harvteshet
lable foodof stasurvpplatiyon.to)the point where the seals are
The =
of a population is defined as avaiiLetn danger
be,\1.aShowLesliethmatat ifri,\x iwis anyth aotuniherque(reposal oritivcom­e eigen­
val
u
e
wher
rth raatteison.and the are the
plr(ecx)os eigenvalsinu()e)ofand tshuenbst1i,\tu1 te i,\t 1in. to the Wrequatiteio,\=n
survExplivealthraeaintewhysarfore ththecanebipopul
bers iborntenrptroetaesdinasgltehfeemalavereaoverge g(Theor,\) =em andas inthparentta(bke) ofExer
ciparse t ofUsbote hDesidMoies. vThere's
number
of
daught
e
t
h
e
r
e
al
her lifetime.
Triangle Inequality should prove useful.]
(a)
L
Sustainability requires that
(1 - h)Lx
24.
cAs
x
L
h
=1-
25. (a)
44
44
(b)
3.7.
3.7,
(b)
26.
Exercise 23 shows that the long-run behavior of a popula­
tion can be determined directly from the entries of its Leslie
matrix.
23.
net reproduction rate
� 27.
r b 1 + b 2s 1 + b 3 s 1 s2 + · · · + b n s 1 s2 · · · s n - l
b;
sj
(a)
r
22.
L
() + i
1,
L,
:s
[Hint:
23.
Section 4.6 Applications and the Perron-Frobenius Theorem
In Exercises 28-31, find the Perron root and the correspond­
ing Perron eigenvector ofA.
2[ 1 01 ]
[ : � i]
28. A =
30. A �
29. A =
31. A �
(
1[ 2 03 ]
[: 0 �l
39.
[Hint:
i
( a) l cA I = l c l I A I
(c) I A x l :s I A I l x l
41.
42.
33. A �
A
In Exercise 40, the absolute value of a matrix A = [ a j is
defined to be the matrix IA I = [ l a u l
40. A
B nXn
x
IJ�r,
c
It can be shown that a nonnegative n X n matrix is irreduc­
ible if and only if I + A) n - t > 0. In Exercises 32-35, use
this criterion to determine whether the matrix A is irreduc­
ible. If A is reducible, find a permutation of its rows and
columns that puts A into the block form
[! � � �l
(b) I A + B l ::; I A I + I B I
(d) I A B I ::; I A I I B I
a a
A = " ,2 ·
a z1 a z 2
a2 1 =
Prove that a matrix 0.[ ] red 'ble
iLetf and beonlaynonnegat
if 0ivore, ir educible matrix such
tAh1atand- beistihneverPertirbolne androot and- Per) ron2eigenvect
Let or
of Prove that 0 A 1. Apply Exercise
iDeduce
n Sectiofrno4.m3 (anda) thTheor
e
m
4.
1
8(
b
)
.
]
at
2X2
a1 2 =
A
I
A.
( a)
IS
A
v1
(I
< 1<
A -1
UC!
0.
[Hint:
22
00 01 0 00 0 00 01 00 00 0
10 00 1 01 01 01 00 01 00 01
0
0
0
1
1
0
0
0
0
showed.
1128,, for fo2r 12
tI(hfAatgrisaphtihs eiirsadjreduciacencyble imatf andif trhixeronlofe yiasigraf pataphishconnect
2
,
for
everWhicyhpaiofrthofevergratphsices.i)n Section 4.0 havebetanween
l
1, 1,
for 2
iprr iemduciitivbeladje adjacency
acencymatmatrixri?x? Which have a
Let Showbe atbihatpartiitsenotgraphprimwiitthivadje. acency matrix
0,
5,
4
for
2
Show tUsehat iExerf A iscanise ei80genval
u
e
of
s
o
i
s
A.
0,
1,
for 2
i
n
Sect
i
o
n
3.
7
and
par
t
i
t
i
o
n
anthiseipargenvecttitioniorngforofA soUstheatthitisisparcompat
i
b
l
e
wi
t
h
titioning to find
an
ei
g
envect
o
r
for
A.
]
Atex.grLetaph isbeakcalledregular graiph.f k edges meet at each verEV
- The recurrence relation in Exercise 45. Show that your
s
o
l
u
t
i
o
n
agr
e
es
wi
t
h
t
h
e
ans
w
er
t
o
Exer
c
i
s
e
45.
Show
t
h
at
t
h
e
adj
a
cenc
y
mat
r
i
x
of
has A emk4.as30.an] eigenvalue. Adapt
Theor
tCompl
hat if tehtee rthecure prroenceof ofreTheor
lationem 4.38(a) by showinhasg
35. A =
34. A =
36. (a)
A
A
G,
G
connected
( a)
(b)
G
A.
A
A,
[Hint:
(b)
v1 > Av, .
Linear Recurrence Relations
In Exercises 43-46, write out the first six terms of the
sequence defined by the recurrence relation with the given
initial conditions.
43. Xo = Xn = 2xn - I
n
44. a , =
a n = a n _ , /2 n 2
45. y0 = O, y 1 = yn = Yn - I - Yn - z n 2
46. bo = b , = b n = 2bn - I + b n - 2 n 2
(b)
37.
Showuestarhate alifl leiss prthiankmitivine,absthenoluthteevalothuere. eigen­
valAdapt
Theor
e
m
4.
3
1.
]
iExpln ligahtinofthExere resucilstessof36-38your andexploSectratiioonn i4.n5Sect. ion 4.0
]
].
Leta scalaandr. Provebethe follomatwinrgicmates, raixvectinequalor inities: and
(b)
The Perron-Frobenius Theorem
361
In Exercises 47-52, solve the recurrence relation with the
given initial conditions.
47. Xo = X 1 = Xn = 3Xn - I + Xn - 2 n 2
48. Xo = X 1 = Xn = 4Xn - I - 3Xn - 2 n 2
A.
38.
k-regular
( a)
52.
G
=
A
[Hint:
G
53.
Xn = axn - + bxn - 2
i
Chapter 4 Eigenvalues and Eigenvectors
362
diofstthinectforeimgenvalues ,\2, then the solution wil be
e 4.40 works in
generShowaShow
l.] thatthforat tanyhe metchoihodce ofofiExampl
nitiandal condic2 cantiobens
rand
t
h
e
s
c
al
a
r
s
found,
as
s
t
a
t
e
d
i
n
Theor
e
m
4.
3
8(
a
)
and
(
b
)
.
iInfitthiaeleicondigenvalltiounses areand ,\20,are dist1,inscthowandththate
( )(,\7 - ,\�).
asThesociFiabtonacci
ed matrriexcurequatr]enceion l l wherhase the
and [ 1 0 ]
tWiionthtf0o pro0veandfthat1 1, use mat] hematical induc­
[
Usforinalgl par2t 1.(a), prove that ( - 1 )"
This is Gicalolvanni
ed Domenico Cas­
afsfointreialr(1625-1712)
tlhe 2ast1.ro[nomer
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tation.heofCasLouibecame
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,\1 *
v
[Hint:
54. (a)
x0 =
x1 = s,
(b)
Xn =
55.
A 1 - Az
f,, = fn - I + fn - 2
xn = AXn _ 1,
(b)
=
Therectaarngleae'ofs artheeasiqs uar65 seqiuars 64e suniquartse! Wher
units,ebutdidtthhee
exthaveratsoquardo wie cometh thefrFiobm?onacci seWhat
does
t
h
i
s
Youof 1 have2 tialessupplandyoneofthkireendkiofn1ds of1ttiquence?
lieles:, astwsoh]kiownndsin
Figure 4.30.
56.
X
X
D CJ
=
ways toe,cover a
tthhatthesofe tdii5.leffs.eForrentexampl
1FigurLeter4.ect3bea1 nglshthowseewinumber
Fi(Dnoesd make any sense? If so, what is it?)
ence rteiolanstio, snofortlve t"h'e
UsrSetecurinupgreancesandecondretl2atasiorotndherienirnpareicurtital(rbcondi
against the data in part (a) . ) . Check your answer
Figure 4 . 3 0
A " = Jn + I fn
Jn fn - 1
n
fn + ifn - 1 - fn2 =
n
(c)
(b)
[Hint:
A=
n- 1
Cassini's Identity,
__.v"
......
Figure 4 . 2 9
x1 =
Xo =
Xn = ILtfn
(a)
_......
c1
,\1
�
v
�
1.._...
...- ..
X n
�
(a)
(b)
(c)
tn
t1, • • • , t5•
t0
t3 =
t1
X
X
\
v
v
......
�
\'
\
\
The five ways to tile a 1 X 3 rectangle
Figure 4 . 3 1
\
'
(a)
57.
Youto coverhaveaa2supplyreofcta1ngle2. Letdominbeoesthwie tnumber
h whichof
diFiffgurereent4.3ways2 showsto coverthat the re3.ctangle. For example,
X
X n
dn
d3 =
Section 4.6 Applications and the Perron-Frobenius Theorem
363
x' == Xx - 3z, yx((OO)) == 32
(
)
=
3x
=
z
O
Em
Aa petscireintdiissthpl. Iancesitiatlwy,othsterraeinars eof400bactofXeriaand, X and500 ofY.in
/ l �
nottThehe number
ftewedo bacton seacheofritahotcompet
he sterra. iInfesxfoat=rtifoodme andanddayss,p=tacehe grbutoarwtdoeh
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i
o
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ar
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v
en
by
t
h
e
s
y
s
t
e
m
[8 BJ ITIJ rates of the two popul
x' == - 0.l .22xx The
three ways to cover a 2 X 3 rectangle with 1 X 2
Detatioensrmbyinesowhatlvinghappens
tmo thofesdiefftweroentpopu­ial
dominoes
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t
h
e
s
y
s
t
e
iroensth.e effect ofchanging the initial populations
Fi
n
d
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o
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In Example find eigenvectors and corre- arsizees ofofXthe popul
and
1
0
ofY.
I
f
x
=
ar
e
t
h
e
and
=
a
t
i
o
ns
at
t
i
m
e
mont
h
s
,
t
h
e
gr
o
wt
h
sponding to, Aver1 =ify and A2 = 2 . With rates of the two populations are given by the system
xk = ] fo:mula (2) in Section That is,
0.
S
x
x'
=
=
show that, for some scalar c1,
Det
e
r
m
i
n
e
what
happens
t
o
t
h
es
e
t
w
o
popul
a
t
i
o
ns
.
x
hm A1 = C1V1
=
=
x'
=
x
x = [;]
x(O). x = =
x' = 2xx yx((OO)) = 0
] ,b = [ - 3010 ] ,x(O) = [ 2030 ]
l
=
_�
[
x' == 2x- x- y, xy((OO)) ==
[
]
1
- ' - [ 40O J ' x(o) - [ 3010 ]
=
x;x� == XX11 - XXzz,, XXz1 ((O0)) == 0
Let
x
=
be
a
t
w
i
c
ed
i
ff
e
r
e
nt
i
a
bl
e
funct
i
o
n
and
cons
i
d
er
t
h
e
YiY� == Y1Y1 - YzYz,, YzY1 ((0O)) == 1
x"
0
X1 == X - yx((OO)) == 0
Show= x althlatowstheEquatchangeionof vartioablbeeswri=t ex'n asanda sys­
tem of two linear differential equations in and
= x z (O) =
64.
65.
+
y'
z'
+
2y +
Z,
z,
4
x (t)
Figure 4 . 3 2
(a)
(a)
(b)
(c)
d1,
d0
.
•
d5.
•
d1
58.
(b)
dw
d2
4.41,
I fk
Ltk- 1
v1
v2
llili_
In Exercises 59-64, find the general solution to the given
system of differential equations. Then find the specific
solution that satisfies the initial conditions. (Consider
all functions to be functions oft.)
=
59. = + 3y,
5
y'
+ 2y,
60.
1
1
y'
+ 2y,
61.
+
1
EV
1
+
y'
z'
y
+
+ y,
b.
Y,
Z,
Z,
-1
y y (t)
a
symbiotic
x (t)
t
y'
svstems of L i n e a r D i f f e r e n t i a l E q u a t i o n s
63.
=a
4.5.
. kk
k->oo
62.
y'
15
1 - Vs
t
0.2y
+ I.Sy
66.
1 + Vs
Y,
b.
y y (t)
+ 0.4y
0.4x - 0.2y
In Exercises 67 and 68, species X preys on species Y. The
sizes of the populations are represented by x x (t) and
y y (t) . The growth rate of each population is governed
by the system of differential equations
A + b, where
and b is a constant vector. Determine what happens
to the two populations for the given A and b and initial
conditions
(First show that there are constants a and b
such that the substitutions
u + a and y v + b convert
the system into an equivalent one with no constant terms.)
67. A
1
1
b
-1 -1
69.
x ( t)
second order differential equation
+ ax' + bx =
( a)
y
z
1
1)
(
y
68. A
(1 1)
z.
Chapter 4 Eigenvalues and Eigenvectors
364
the
sShowShowytshteatmtthhinaterparethiest achar(achange
) ias ct,\erofistivarc equatiableisotnhatofconver
ts
the nth _ into acsiyentstematm ofrix liisntearhe compani
differentoianl equat
ions whosoftehe
coeffi
mat
r
i
x
pol[Theynominotaatilp(on ,\) denot,\" es th_e,\kt-h derivative of See
Exercompaniciseson matriinx.Sect] ion for the definition of a
(b)
70.
2 + a,\ + b = 0.
order differential equation
n
(
x ) + an 1x(n I ) +
+ a 1 x' + a o = 0
n
C( p)
I
n
=
+ an 1
+ + a1A + a0•
x (k)
x.
26-32
4.3
·
·
·
·
·
·
In Exercises 71 and 72, use Exercise 69 to find the general
solution of the given equation.
71. x " - 5x' + 6x = 0
72. x " + 4x' + 3x = 0
In Exercises 73-76, solve the system of differential equations
in the given exercise using Theorem 4.41.
74.
60
59
73.
76.
64
75.
63
ExerExercciissee
ExerExercciissee
D i s c r e t e linear D v n a m i c a l svstems
In Exercises 77-84, consider the dynamical system
xk + 1 = Axk.
1
(a) Compute and plot Xo' x1 , x2 , x3 x0 = .
1
1
(b) Compute and plot Xo' x1 , x2 , x3 Xo =
0
(c) Using eigenvalues and eigenvectors, classify the origin as
an attractor, repeller, saddle point, or none of these.
(d) Sketch several typical trajectories of the system.
2
0.5 - 0.5
78. A =
77. A =
0 3
0
0.5
for [ ]
for [ ] .
[ l]
[
Chapter Review
adjalgebroinat iofc mula mattiprliixci,ty of an
charchareigenval
aactcteerriiusstte,iicc polequatynomiion, al,
cofact
Crdetaemerrmoir'nsexpans
Rulant,e, ion,
Kev Defi nitions and Concepts
276
294
266
274-275
263-265
292
292
]
79. A =
[ _� - � ]
[ -�]
[ ]
80. A =
0.1 0.9
0.5 0.5
- 1 .5
84. A =
1 .2 3.6
1 .5
-1
0.2 0.4
83. A =
- 0.2 0.8
81. A =
[
[ -� -�J
[ ]
[O ]
82. A =
]
In Exercises 85-88, the given matrix is of the form
a -b
A=
. In each case, A can be factored as the
b a
product of a scaling matrix and a rotation matrix. Find
the scalingfactor r and the angle e of rotation. Sketch the
first four points of the trajectory for the dynamical system
and classify the origin as a spi­
xk + 1 = Axk with x0 =
ral attractor, spiral repeller, or orbital center.
1
0 0.5
85. A =
86. A =
1
- 0.5 0
V3
1
- V3;2 - 1 /2
88. A =
87. A = V3
1
1/2
- V3;2
In Exercises 89-92, find an invertible matrix P and a maa -b
trix C of the form C =
such that A = PCP i .
b a
Sketch the first six points of the trajectory for the dynamical
system xk + 1 = Axk with Xo =
and classify the origin as
a spiral attractor, spiral repeller, or orbital center.
[�]
[ -�]
[ ]
[
89. A =
91. A =
[ ]
[ � -�]
0.1 - 0.2
0.1 0.3
]
[
[
]
[�]
90. A =
92. A =
-
[ _� � ]
[o - 1 ]
1 V3
'
GersGerscchgorhgoriinn'sdiDisks,k Theorem,
dieigaenval
gonaluize,able matrix,
eieiggenvect
or,
LaplacemetExpanshod (iaondn Theor
em,
power
i
t
s
ens
p
ace,
Fundament
al Theorem of Invertible prvaroperianttiess),of determinants,
Mat
r
i
c
es
,
geomet
r
i
c
mul
t
i
p
l
i
c
i
t
y
of
an
eigenvalue,
similar matrices,
254
254
256
296
294
319
303
3 1 1-319
269-274
301
321
266
]
Chapter Review
365
MarFork eachall sofquartheefolmatlowiricnesg A,statdetement( -A)s tr=ue-or<lefat A.lse: LetA = [ - : - : � .
-2]
(b) <lIfeAt (andBA).Bare matrices, then det(AB) =
ristic upolesyofnomiA. al of A.
IfarAe tandhe sameBarebut in dimatfferreintcesorwhosders,etcolhenumns (b) FiFiFinnnddd alathbasleofcharitshforeaeicteachgeenval
of
t
h
e
ei
g
ens
p
aces
of
A.
<l
e
t
B
=
<l
e
t
A.
Det
e
r
m
i
n
e
whet
h
er
A
i
s
di
a
gonal
i
z
abl
e
.
I
f
Adiisago­
( IfIfA iisstihneveronltiybleie,gtenval
hen detue(ofA -a sq=uar<leetmatAT.rix A,
not
di
a
gonal
i
z
abl
e
,
expl
a
i
n
why
not
.
I
f
A
i
s
nalimatzablrei,xfindsuanchitnhverat tPib-l1eAmatP =rix P and a diago­
tTwohen Aeigienvect
s the zeroros matcorrreisxp. onding to the same
nal
I
f
A
i
s
a
3
3
di
a
gonal
i
z
abl
e
mat
r
i
x
wi
t
h
ei
g
enval
u
es
eiIfganenvalue musmattribex haslineardilystdependent
.
2,
3,
and
fi
n
d
<l
e
t
A.
iItfmusan t be dimatagonalrix iiszabldiaegonal
. inizctableieg,envalthenuitesmus, thent If-A is a 2 2 matrix with eigenvalues A1 = t, =
and cor-responding eigenvectors v1 = [ � ], = [ - �l
haveSimilardimatstinrictceseihavegenvalthueessa. me eigenvectors.
Irfeduced
A and rBoarweecheltwoon form,matthrenicesAwiis tshimthilearsatmeo B. fiIfnAdAis asdi[ �aJgonal. izable matrix and all of its eigenvalues
n
s
a
t
i
s
f
y
pr
o
ve
t
h
at
A
appr
o
aches
t
h
e
zer
o
ma­
3
t
r
i
x
as
get
s
l
a
r
g
e.
LetA = [ 3 � ] .
A
B.
A
B,
P
Comput
e
<l
e
t
A
by
cofact
o
r
expans
i
o
n
al
o
ng
any
rComput
ow or cole <luemn.t A by first reducing A to triangular P- 1 AP= B.
= [! � ] ,B = [ � � ]
form.
If = 3, find
14.A = [ � � ] ,B = [ � � ]
Let<let AB and= -B�.beFind <let matC forricteshewiintdihc<lateet dAmat= 2riandx C:
B
� �]
i
:
J
.
[
[
1
�
2
�
T
C=
(
A
B)
(
b
)C=A
B
(
3
A
)
prIfAoveis tahsatkew-<letsAymmet
= ric matrix and2 is odd, Let AA has= [ei� genval�] . Fiunesd3alandl values of for which:
Find all values of for which 2 =
(b) AA hashas noan eiregalenvaleigenvalue wiutesh .algebraic multiplicity 2.
x
A IIffAa sq=uarA,e whatmatriarex Athashe postwosibequalle eigrenvalows, uwhyes ofmusA? t A
have
as
one
of
i
t
s
ei
g
enval
u
es
?
,A
x = [ � ] = [! �]
Ihf atxixis ans aleisgoenvect
oenvect
r ofAowir ofthAeigenval
ue 2I.= What
3, showis
t
an
ei
g
SA
]
t
h
e
cor
r
e
s
p
ondi
n
g
ei
g
enval
u
e?
1
H J [ �: - 32
vectIf A oisr sofimA,ilasrhowto BthwiatthP -P1-x iAs Pan=eiBand
genvectxiosr ofanB.eigen­
Review Questions
1.
(a)
9.
nXn
(c)
0
( a)
nXn
d)
(c)
(d)
i)
0
(e)
(f)
(g)
nXn
(h)
nXn
n
(i)
n
5.
6.
11.
X
4,
12.
1
A2
- 1,
I A I < 1,
n
5
7 9 11
In Questions 13-15, determine, with reasons, whether is
similar to If � give an invertible matrix such that
(h)
4.
X
D.
nXn
(a)
3.
10.
D
Vz
(j)
2.
0
13. A
a b c
d e f
g h
4X4
( a)
0.
k
3d 2e - 4f f
3a 2b - 4c c .
3g 2h - 4i
15. A �
nXn
-1
1 k
1
4 k2
n
( a)
0.
17.
18.
7.
19.
s. F
A�
k
16.
In Questions 7 and 8, show that is an eigenvector of and
find the corresponding eigenvalue.
- 60 - 45
15
18
- 40
�
20.
( c)
- 1.
3
0
2
+
A
O rthogona l ity
. . . that sprightly Scot of Scots, Douglas,
that runs a-horseback up a hill
perpendicular-
-Wil iam Shakespeare
Act II, Scene IV
Henry IV, Part I
5.0
I n t ro d u ctio n : S h a d ows o n a Wa l l
r, we wierl extandendthentheagainotnioinnofChaptortheogonal
priol jnow,ectiowen thhaveat wediencoun­
tonlIenretyhdiprsfiorchaptsjtectinieoChapt
r
Unt
sbscuspsaceed
n
ont
o
a
s
i
n
gl
e
vect
o
r
(
o
r
,
equi
v
al
e
nt
l
y
,
t
h
e
oned
i
m
ens
i
o
nal
s
u
spannedas forbyprtohjeatctvection oontr) .oIna plthaisneseictn ion,Fiwegurwiel seeshifowswe whatcan fihappens
nd the anal, forogousexamplfor­e,
mulwhen
ona tawwalo-dli.mAenssimioinallar sprcroecesen,s soccur
s when
a
tputhreee-r moni
dparimaenslteolri.oliLatnalghteobjrraiysnectthcriisesatchapt
diesaplsaehyedradow
on
u
ch
as
a
com­
,2 weat wiwhatl conswe ialderreadythesknow
e ideasaboutin fulprl generojectaiolinsty.. In
To
begi
n
,
l
e
t
'
s
t
a
ke
anot
h
er
l
o
ok
Section we showed that, in IR , the standard matrix of a projection onto the line
through the origin with direction vector = [ �:] is
)]
]=[
Hence, the projShowectionthofattPhecanvectbeorwrv iontt eno itnhitshleinequie is vjualsetntPv.form
P = [ cos()cos2si()n () cos()sin2si()n () ]
(What does() reShowpresetnthatherP ecan?) also be writ en in the form P = where is a
vector in the dirUsectiniognProfoblem find P and then find the projection of v = [ _! ]
onto the lines with the following unit direction vectors:
(a) " = [ �:�J (b) " = [fl (c) " = [ -n
rm P e=nt) . show that (a) = P (i.e., Pis symmetric)
and (b) P2 = PUs(i.ein.,gPithseidfoempot
1
3.
IR 3 .
5.1
3.6,
d
dl !(dl + d i ) d , d2!(d l + di
d, d2!(df + d i ) d i !(df + d i )
d, d2
di
Problem 1
Shadows on a wall are projections
Figure 5 . 1
�
uuT,
Problem 2
d.
Problem 3
Problem 4
366
2,
uuT,
pT
u
unit
Section 5.0 Introduction: Shadows on a Wall
361
projects vectorsExplis thaeincolwhy,umnifspaceis aof projection matrix, the line onto which it
consider projections onto planes through the
origNowFiing. urWeewe5.wi2wilshexpll owsmoveoronee severinwayto atloapprprandooceed.aches.
norv mal vectoforr sandomeifsvcalisara vector in tIhf enis a plpraneojgpth(vro) ughis a vectthe ororiignin insuchwithtath
P
Problem 5
2X2
P.
IR 3
-c
n
=p
n
IR 3 ,
<!J'
c.
p=
n
v
<!J'
IR 3
- en
Projection onto a plane
Figure 5 . 2
v
=p
Us
i
n
g
t
h
e
fact
t
h
at
i
s
or
t
h
ogonal
t
o
ever
y
vect
o
r
i
n
s
o
l
v
e
forUsteothfiendmetanhexprod ofesPrioonblforem itnotfiernmdsthofe vprandojection of
Problem 6
-c
n
Problem 1
n
c
<!/',
p
n.
6
onto the planes wi(a)th the yfollowing equat(b)ions: (c) 3y
finding theethpre oprjeoctjeictonioofn ofa vectv ontoor ontinotoa
plthaeneAnotis shofuerggesitapprs tpredoojbyachectFiiotgnsourthontee5.pro3.otWehbleedimcanreofctdecompos
irson. Accor
vectordsinforgly, letThisandworksbeonldiyreictf itohne
divectrectorisonforvectwiorstharthe eorprthoogonal
uni
t
vect
o
perty that and
x+ +z=0
x - 2z = 0
2x
<!/'.
sum
<!J'
u 1 • u2 = 0
ll u 1 ll = ll u2 ll = 1
IV
I
I
I
I
'"""
I
I
U2
: p = pI
.- -:::�
.L.:...-
Figure 5 . 3
+ P2
-
+z=0
<!J'
u1
u2
368
Chapter 5 Orthogonality
By Problem 2, the projections of onto andand are
gioves tIhteisprenough
ojectiontoofshowonttohat we-need to showis
trorhesattphectogonal-ivelyt. oTobotshhowistandhoratthuogonal
t
2. (Why?) and
Show
t
h
at
ive formunioft vectthe dotors.pr] oduct, together with the fact that
andUseItthfoleareallotwseorrnthfratogonal
projection ontootmhe Prsuobsblpeacem andof thespcomment
anned bysorprthecediogonalng iunit thtatvecttheormats randix of thies
Usortehogonal
the sameunitRepeat
andvectusoresPrinoblandtheemgivenasuspliinnadine.g ctat)heedforbelmoulw.a(FforirstP, vergivienfy tbyhatEquatandion are
(a) x z with [ - �j�] and [ � ]
2
(b) x - 2z with [ 1:5] and [ � ]
(c) 2x - z with [ - ] and [ 2 ]
ties (a) and (b) ofShow
PrShowobltthheatmat athpreomatjectrioixn Pmatofriax prgivoenjectbyionEquatontoioan pl( anesatiinsfies prcanoperbe­
expres ed as
for some 2 matShowrix that if PiShows the tmathat rEquat
iaonprojeicts ianonoutonteropra oplductane expans
itohn.en]
i
x
of
i
n
rank(InPt)his chapt
2. er, we wil look at the concepts of orthogonality and orthogonal pro­
jgener
ectionaliiznedgrandeatetrhdetat tahiley. Wehavewimany
l see tihmatporthteantideasapplinictratoduced
ions. in this section can be
v
v
( p1 + p2 )
u1
u2
p 1 = u 1 u fv
p 1 + p2
p2 = u2 ufv
!JP.
u1
!JP,
v
v
( p1 + p2 )
u1 (v - ( p1 + p2 )) = 0
u2 (v - (p1 + p2 )) = 0. [Hint:
xTy = x y,
u1
Problem 8
•
•
·
u2
8
!JP
IR 3
u1
u2
( 1)
7,
Problem 9
v
u1
+y+
u2 ,
= 0
= 0
3y +
u1
Uz =
u1 =
1/\/6
u1 =
l/ Vs
= 0
u1 =
u2 =
l/ v'3
l/ v'3
l/ v'3
( 1).
l/
- 1/ V2
0
/\/6
1 / \/6
- 1 / \/6
u2 =
1)
Problem 10
Problem 11
u2
4.
IR 3
p = AA T
3X
Problem 12
A. [Hint:
=
•
( 1)
IR 3 ,
O rth o g o n a l ilV i n IR n
tsIwntaondarthvectis sdeobasctrsiotion,s setwes wioflvectgenerorsofa. lIinzedoiteashenygnottsooiwor,owen kofwiwiorltthsh:eogonal
eFirthstat, anyittwyotofwprovectodipersotirtnsiectisnvectmakeofrrsothmine
!R n
{ e1, e2 , . . . , en } !R n
Section 5. 1 Orthogonality in
lffi 11
369
terhteiesets learade usortthoogonal
. iSecond,
eachhogonalvectbasor ienstandhe setortishonora unimt vectal basore. Thess-concept
e two prs tohp­at
t
h
e
not
o
n
of
or
t
we wil be able to fruitful y apply to a variety of applications.
O rth ogonal and Orth onormal Sels of Veclors
rths e set are ort,hogonalin -itshcalat lies,difan
if
all pairs of distinActsvectet ofovectrs inowhenever
i j for i, j
tThehe fisrtsatndarexampld base iils ustrates, th,ere arofe manyis anotorherthposogonalsibilsiteite,sas. is any subset of it. As
Show that is an orthogonal set in if
Definilion
{v1 , v2 ,
•
Exa m p l e 5 . 1
z
{v1 , v2 , v3 }
•
.
!R n
vd
*
vi · vj = 0
{e 1 , e2 ,
•
.
•
en }
orthogonal set
= 1, 2, . . . , k
!R n
IR 3
is true, sinceWe must show that every pair of vectors from this set is orthogonal. This
Solution
V1 • Vz = 2(0) + 1 ( 1 ) + ( - 1 ) ( 1) = 0
Vz • V3 = 0( 1) + 1 ( - 1) + ( 1 ) ( 1) = 0
y
x
V1 • V3 = 2( 1) + 1 ( - 1) + ( - 1) ( 1) = 0
An orthogonal set of vectors
Figure 5 . 4
Geomet
r
i
c
al
l
y
,
t
h
e
vect
o
r
s
i
n
Exampl
e
ar
e
mut
u
al
l
y
per
p
endi
c
ul
a
r
,
as
Figure shows.
4
of thesamairilynlinadvant
of work,iasngTheor
with eormthogonalshowsset. s of vectors is that
theyOneare neces
early iangesdependent
is an or.thogonal set of nonzero vectors in then these vectors
arIfe linearly in, dependent
If are scalars such that
then
or, equivalently,
t, all oftothe dot products in Equation
are zerSinoce, except Thusis ,anEquatorthiogonal
on rseeduces
5.1
5.4
5.1
Theorem 5 . 1
{v1 , v2 ,
Proof
.
•
c1,
!R n ,
vd
•
.
•
.
, ck
c 1v1 + + ckvk = 0,
(c 1 v1 + · · · + ckvk ) · vi = 0 · V; = 0
· · ·
c 1 (v1 · vi ) + · · · + ci (vi · vi ) + · · · + ck (vk · vi ) = 0
{v1 , v2 , . . . , vk }
V; · vi .
( 1)
c; (vi · v;) = 0
(1)
(1)
310
Chapter 5 Orthogonality
The fact tsheatt.
tNow,his is true for allbecause imbypliehypot
s thathesis. So we musistahavelinearly independent
tloyTheor
em . weForknowexamplthate, iwef a setcanofimvectmediorsatieslyordeduce
thogonalth,atit
itsheautthormatee vecticalThanks
loyrlsininearExampl
independent
dependent. diContrectrlays!t this approach
with the work needed to estaeblish tarheie rlilninearearlyinindependence
for a subspace W of is a basis of W that is
an orthogonal seAnt.
The vectors
V; V; * 0
•
V; * 0
i = 1, . . . , k
C; = 0 .
{v1 , v2 , . . . , vk }
5.1,
Remark
5.1
Definition
Exa m p l e 5 . 2
!R n
orthogonal basis
e arevectortohrogonal
and,formhence,
lifornearly byindependent
. Sinceal anyTheorthereme
lfroflinoearmnverExampl
ly tiinbdependent
s
i
n
a
bas
i
s
t
h
e
Fundament
le Matrices, it follows that is an orthogonal basis for 4
onlvecty tohre orttohmake
ogonal vectors anandorthogonalwere
gibasvenis forand youOneInwerExampl
ewayasktoededototfihinssudippossattohireredmember
theeatExplin oratthioen:crTheos product
ofoducttwo
vectin Chaptors erand Henceis orwethogonal
t
o
each
of
t
h
em.
(
S
Cros
s
Pr
may take
5.1
IR 3
5.2,
Remark
IR 3 .
v1
v2
1 .)
IR 3 ,
{v1 , v2 , v3 }
v3
IR 3 .
IR 3 ,
v1
{v1 , v2 , v3 }
Note that the resulting vector is a multiple ofthe vector in Example as it must be.
Find an orthogonal basis for the subspace W of given by
v3
Exa m p l e 5 . 3
v2
5.2,
IR 3
Sect
i
o
n
gi
v
es
a
gener
a
l
pr
o
cedur
e
for
pr
o
bl
e
ms
of
t
h
i
s
s
o
rt
.
For
now,
tweofhevectwiorligofiirnsnofdintthhee orforFrtohmmogonalthe equatbasisiobyn ofbruthtee forplacne,e. Thewe havesubspace Wis a plsoaneWtconshroughists
5.3
Solution
IR 3 .
x=y
-
2z,
Section 5. 1 Orthogonality in
IR"
311
It follow' that [ �] and [ - �] "e a b";' foe W, but they Me mthogo­
nalthes. eIt. suffices to find another nonzero vector in W that is orthogonal to either one of
Suppo'e [ �] ;, a vedm ;n W that ;, mthogonal to Then ssyinstceem is in the plane W. Since we also have Solving the linear
we find that and (Check this.) Thus, any nonzero vector of the form
u�
v�
nol
w�
x y + 2z � 0,
u.
w
x + y = 0.
u w = 0,
·
x y + 2z = 0
=0
x+y
�
x = -z
y = z.
w
wil do. To b"pedfie, we wold toke [ - :J- It to ehe<k that ;, an
orthogonal set in Wand, hence, an orthogonal basis for W, since dim W
base.iIsnideed,
s that tthheercoore is adifornatmesulofa
aforvecttAnothesorewihcoorerthadvant
rdeinspatecteasge,tasoofstuhworche folaklibasonwig iwins garthtehaneoreasoryemtthoogonal
comput
establishes.
vectLet or in W. Then betheanuniorqtuehogonal
scalarsbasis for a ssuubschptacehat W of and let be any
are given by
for ...
;""Y
w�
{u, w}
= 2.
Theorem 5 . 2
{v1 , v2 , . . . , vk }
!R n
c 1 , . . . , ck
W ' V;
C; = -V; ' V;
i = 1,
w
,k
Sincesuch that is a basis for W,(frweomknowTheortheatmthere arToe uniesqtaueblisschaltahres
formula for we take the dot product of this linear combination with to obtain
Proof
c 1 , . . . , ck
{v1 , v2 , . . . , vk }
w = c 1 v1 +
C;,
· · ·
+ ckvk
3.29).
w · V; = (c 1 v1 + · · · + ckvk ) • V;
= c 1 (v1 · v;) + · · · + c ; (v; • v;) + · · · + ck (vk • v; )
= C; (V; ' v; )
vj V; = 0
* i.
V; V;,
V; * 0, V; V; * 0.
dessinceired result. for j Since
·
•
V;
Dividing by we obtain the
•
312
Chapter 5 Orthogonality
Exa m p l e 5 . 4
[
FiofnExampl
d the coores dinandates of � ] with respectto the orthogonal basis
Using Theorem we compute
w =
5.1
5.2.
3
B = {v1 , v2 , v3 }
5.2,
Solulion
2+2-3 1
=
V1 • V1 4 + 1 + 1 6
w·V
0+2+3 5
C2 = V " V2 = 0 + 1 + 1 = 2
2 2
w · V3
1-2+3 2
C3 = V • V = 1 + 1 + 1 3
3 3
w · V1
--
c] =
=
-
--
-
--
�
Thus,
(Checkionthasis.) With the notation introduced in Section 3.5, we can also write the above
equat
to findbastheess. e
coorAsCompar
dinnotateesdediattrhetctehelprybegiandocedurnyouninegshiofnoulExampl
dthisstsaertcteitoo5.n,appr4thwieetciothahtterheethprworeovalperkutreyeofquiofortrhteehdogonal
iornthogonal
is thatityeach, weshavetandarthde basfollioswivectngodefir is naiunition.t vector. Combining thissprtaondarpertdybaswitihs
sthetatofisunian torvectthonoroArs.smeAntalofsevectt. ors in is anfor a subspace ioff it is ians aorbasthiogonal
s of
If The fact thatiseachan orthionors a unimalt vectset oofr vectis equiorsv,altheennt to for
j
and
It follows that we can summarize the statement that is orthonormal as
{ o iiff jj
Show that is an orthonormal set in if
[ ] and [ ]
!R n
!R n
Definition
orthonormal set
W
orthonormal basis
S = { q1 , . . . , qd
ll q; ll = 1 .
·
q;
q; . qj = 1
Exa m p l e 5 . 5
qi =
·
S
i*
i=
IR 3
S = { q1 , q2 }
l / v'3
- l / v'3
l / v'3
W
q; qj = 0
q ; q; = 1 .
Remark
i -=fa
!R n
q2 =
l / v'6
2/ v'6
l / v'6
Solution
Section 5. 1 Orthogonality in
We check that
313
lffi "
q1 • q2 = 1 / \/18 - 2/ \/18 + 1 / \/18 = 0
q i . q i = 1 /3 + 1 /3 + 1 /3 = 1
q2 · q2 = 1 /6 + 4/6 + 1 /6 = 1
Exa m p l e 5 . 6
simplIfywenorhavemalanizeoreachthogonal
vectors. et, we can easily obtain an orthonormal set from it: We
Construct an orthonormal basis for from the vectors in Example
malize themSintocegetwe already know that and are an orthogonal basis, we nor­
IR 3
v1, v2 ,
Solution
v3
[
l
[
]
; �
Then is an orthonormal basis for
Since any, byorTheor
thonoremm5.al1s.eItfofwevecthaveorsanis,orinthparonorticmulalar,basortihs,ogonal
,eimt is5.l2inbecomes
early independent
Theor
even simpler.
anyLet vector in Thenbe an orthonormal basis for a subspace of and let be
and this representation is unique.
Apply Theorem 5.2 and use the fact that for
q3 = 11 1r3 =
3
l
-1
1
l / V3
= - 110
l / V3
IR 3 .
{ q 1, q2 , q3 }
Theorem 5 . 3
5. 1 .
{ q 1, q2 , . . . , qk}
W
W.
IR "
w
w = ( w · q1 ) q1 + ( w · qz ) qz + · · · + ( w · qk) qk
q; q; = 1
Proof
·
i = 1, . . . , k.
iclesswhos
younowMatrwiexami
eenie.n eSectcoliuomnsn formSuchanmatortrhiconores havemalsseteveraralisate tfrraectquentive prlyoipern appltiesi,cwhiatiocnsh ,weas
Orthogonal Malrices
5.5.
314
Chapter 5 Orthogonality
Theorem 5 . 4
The columns of an n matrix form an orthonormal set if and only if
We need to show that
{ Q iiff ii jj
Letentry ofdenote tihs ethitehdotcoluproduct
mn of of(athnd,e ithence,
tofhe ithandrowthofe )th colSiunmnce thofe ( i, jit)
h
r
o
w
follows that
by tNow
he defithneitcolionuofmnsmatrforix mulm antiporlicthatonorion. mal set if and only if
{ o iiff ii j
which, by Equation holds if and only if if i j
- { Q if i j
ThisIfcompl
e
t
e
s
t
h
e
pr
o
of
.
the matrix in Theorem is a matrix, it has a special name.
called an An n n matrix whose columns form an orthonormal set is
The most important fact about orthogonal matrices is given by the next theorem.
A square matrix is orthogonal if and only if
only if Byis Theor
inverteibmle and is orthogonalby Theorif andemonly if This is true if and
Show that the following matrices are orthogonal and find their inverses:
and [ cossin -cossin ]
the staandndard basis vectors for which are
clearly orthTheonorcolmalumns. Hence,of ariseorjuthstogonal
[� � � ]
m X
Q TQ = In "
Q
Proof
(Q TQ )ij =
q;
*
1
=
Q
Q TQ
Q T) .
QT
Q,
(Q TQ) ;j = q; · qj
Q
q; . qi =
(2)
*J
=
1
(2),
is an "Ortho­
unfortu­
nate
bi
t
of
terminology.
normalterm,
matributx" would
clstandard.
early be a
better
i
t
is
not
Moreover,
there
is winothtermorthonor­
for a
nonsquare
matrix
mal columns.
Orthogonal matrix
Theorem 5 . 5
( Q TQ )ij
Q
Definition
x
orthogonal matrix.
1
square
Q
Q - 1 = Q T.
Q
Proof
Q
Exa m p l e 5 . 1
5.4
*
=
_
5.4, Q 1
Q- = Q T,
3.13.
B=
Solulion
A
A
A - ' � A' �
{J
{J
Q TQ = I.
{J
{J
IR 3 ,
Section 5. 1 Orthogonality in
For B, we checkcos()diresctinly()thatcos() - sin ()
BrB = [ - sin () cos () ] [ sin () cos() ]
[ - sin ()coscos2 sicosn2 ()() sin () - cos s()insi2n cossin2 ()() cos () ] [ � � ] = I
Therefore, B is orthogonal, by Theorem and
B = B = [ - sin () cos()sin () ]
ational,matanyrix, a matper­rix
obtmutaaintiedonbymatperMatrixmirutsixorintghitnogonal
hExampl
e colu(mnsseee Exerofisanancisidexampl
ente ityMatematofrairxiperxB. Iimns uttgener
tthrarnsoughformthateianglon (ke nown
() in IRas2. Anyan rotation hasin geomet
the prorpery) . tThey thatnextihteistmathaeorriexmofshaowsrotattihoatn
ation matis anricisesometare rchary. Oracttheogonal
prteverheseseeyrprvoreotdotperhogonal
prtieos.ductmats.rIinx facttrans, orfotrhmogonal
rized bymateithrericesonealsofo
Leta. QQisbeoranthogonalmat. rix. The following statements are equivalent:
IR". n !R".
= l xl · yforforever
b.c. lQxQx·l Qy=x
every xiynxandyi
(mn)c)::::}vect(b)::::}ors i(na)IR. To", thdoensox, ·wey =wixlryneed. to make use
of(a)::::}the fact(cWe) tAshwiatsluimefprxoandvethatthyatQarei(sa)::::}or(colthuogonal
r
.
Then
Q
Q
=I
,
and
we
have
(
Qx
·
Qy
=
Q
xfQy
=
xrQrQy
=
xrly
=
xry
=
x
·
y
and yi=n IRl x".l Then,
taking y = x,
.
we((bc)::::})::::}have((baQx)) AsAs· Qxssuumeme= xtthh·atatx, Qxprsool·perOQyxtyl ==(bx)v'· holyQx·fodrs everQxandy=lextVX:-X
Using Exercise in Sectx·yio=n i ( l andx prr lo2per- tyl x(b-), wer l 2)havedenote the ith column of Q.
== ii (( ll QQx(x Qy)rl l22 -- l l QQ(xx -- yQ)rl 2l )2)
=
Qx·Qy
for alNowl x andif yisinthIRe"i.th[Tshitasndarshowsd basthiats vect(b) o::::}r, t(hc)en.] Qe;. Consequently,
= Qe; . Qej = = { o iiff =
Thus, the columns of Q form an orthonormal set, so Q is an orthogonal matrix.
() +
() +
() +
() +
_1
isometry
isos
metron
Theorem 5 . 6
5.5,
COS ()
T
A
Remark
The wordpreserving;'lisince
terallyitmeans
"length
is
deri
v
ed
from
the
Greek
roots
("equal") and ("measure").
315
lffi "
5. 7
nXn
25).
length-preserving
isometry
nXn
Proof
63
q;
1 .2
+
+
+
e;
q; . %
q; =
e; . ej
1
i*j
i j
Chapter 5 Orthogonality
316
matorritchesonorandmalBisents-Exampl
eth5.ei7r, you mayIn notfacti,ceeverthaty
notorthLooki
onlogonaly dongmatattheitrhrixecolorhastuhmnstogonal
for
m
s
o
do
his property, as the next theorem shows.
If Q is an orthogonal matrix, then its rows form an orthonormal set.
From Theorem 5.5(,QweT) know that Q- 1 QQT. Ther( QT) Yefore,
T
sQ-for
o QT ismananororththogonal
mat
r
i
x
.
Thus
,
t
h
e
col
u
mns
of
Q
-which are just the rows of
onor
m
al
s
e
t
.
The
fi
n
al
t
h
eorem
i
n
t
h
i
s
s
e
ct
i
o
n
l
i
s
t
s
s
o
me
ot
h
er
propert
i
e
s
of
or
t
h
ogonal
matrices.
Leta. QQ-be1 isanortorhogonal
thogonal. matrix.
b.c. <lIfet Qis an eigenvalue of Q, then 1.
d. If Q1 and Q2 are orthogonal matrices, then so is Q1 Q2.
ve property (c) and leave the proofs of the remaining properties
as(c)exerLetcWeisebes.wianl preigoenval
and, using Theorem 5.6 (ube), ofweQhavewith corresponding eigenvector Then Q
Since 0, this implies that 1.
o1 - 01 ]
Pr
o
per
t
y
(
c
)
hol
d
s
even
fo
r
compl
e
x
ei
g
enval
u
es
.
The
mat
r
i
x
[
is orthogonal with eigenvalues and - both of which have absolute value 1.
A
Theorem 5 . 1
- 1 = co- 1 ) - 1 =
Proof
Theorem 5 . 8
rows.
=
=
= ::t:: l
A
IAI =
nXn
Proof
A
v.
JJ v JJ = JJ Q v JJ = JJ Av JJ = I A I JJ v JJ
JJ v JJ *
IAI =
Remark
..
I
i
Exercises 5 . 1
In Exercises 1 -6, determine which sets of vectors are
orthogonal.
I.
,_
nHn [ - �l 2. uirn m
[ Jf :H - !l ·· [ :H - H Ul
i,
v = AV,
Section 5. 1 Orthogonality in
In Exercises 7-10, show that the given vectorsform an ortho­
gonal basis for IR 2 or IR 3 • Then use Theorem 5.2 to express
w as a linear combination of these basis vectors. Give the
coordinate vector [w]8 of w with respect to the basis
B = {v1 , vJ of !R 2 or B = v1 , v2 , v3 of!R 3 •
7. v1 =
8 . v1 =
9. v, �
10. v, �
[ � ] [ � J [ �]
[ � ] , [ �J [ � ]
[ +' [ l; [ - :} [: l
[ l, [ -J, [ _n m
_
, v2 =
v2 =
w=
-
w=
�
_
_
�
�
�
w�
0
2/3
- 2/3
1 /3
27.
In Exercises 1 6-21, determine whether the given matrix is
orthogonal. If it is, find its inverse.
(c)
12 '
]
]
[ [ l[ [ l
16.
18.
19.
20.
[o ]
[
u -! :i
cos[ cose 2sien e -sicosn ee - cos- sien2sien e
sin e
cos e l
2
l
[ -l 2 ]
1
I
-2
I
2
I
2
!
(b)
v'3/2
0
0
- v'3/6
0
v6/3
'
'
1 / v6
v'3/6
1 / \/2
- 1 / v6
1 / \/2
- v'3/6
-1
0
1 / \/2 1 / \/2
17.
- 1 / \/2 1 / \/2
0
!
I
2
I
2
I
-2
Qx
x
x
y,
y
2X2
b
-
1 /2
1 /2
15 '
- 1 /2 '
1 /2
2X2
IR 2 .
angle-preserving IR 2 ,
[il [ -iJ
[il [ -n
[!J [ !J [ J l . [ l l
• -i H ! U
13.
Q
Q
28. (a)
11.
]
PrProoveve Theor
e
m
S
(
a
)
.
5.
Theor
e
m
S
(
b
)
.
5.
PrProoveve tTheor
e
m
S
(
d
)
.
5.
h
at
ever
y
per
m
ut
a
t
i
o
n
mat
r
i
x
i
s
or
t
h
ogonal
.
Iobtf aiins edan byortrheogonal
matng trhiex,rprowsoveofthatisanyalsomatrix
ar
r
a
ngi
orLetthogonal
.
beprovectve bethoatrans itnhore tanglhogonal
Ifeebetis twheeenanglmate betandrixwandQyeenilsealtandsoande.
s provesmatthatricteshearlineear transformatioinsn defianfaedctby
ort(hTatthihogonal
iPrs otrveuethinatgeneran oratlh.)ogonal matrix must have
the form a b
[ - ] or [ ]
where [ : ] is a unit vector.
Using matpartri(xa)i,ssofhowthethformat every orthogonal
cos[ sin ee -cossin ee ] or [ cossin ee - cossin ee ]
wher
ethat evere y orthogonal matrix corre­
Show
spondsthtato eianthorerthaogonal
rotation or a rmateflerctixioncor­in
Show
resreflpeondsctiontiona roitfa<ltieotn in if <let and a
22.
23.
24.
25.
26. Q
w�
In Exercises 1 1 - 1 5, determine whether the given orthogo­
nal set of vectors is orthonormal. If it is not, normalize the
vectors to form an orthonormal set.
1 / v6
0
1 / v6
1 / \/2
1 / \/2 - 1 / v6
1 / \/2
0
311
IR"
]
a
a b
b -a
2X2
0 :::::: < 21T.
2X2
2X2
(d)
IR 2
IR 2
Q = - 1.
Q
Q=1
IR 2 .
In Exercises 29-32, use Exercise 28 to determine whether
the given orthogonal matrix represents a rotation or a
reflection. If it is a rotation, give the angle of rotation; if it is
a reflection, give the line of reflection.
[ 1 / \/2 1 / \/2] 30. [ - V3/2 - 1 /2 ]
- 1 /2 V3/2
31. [
1 /2 ]
V3/2
29.
1 / \/2 - 1 / \/2
- 1 /2 V3/2
Chapter 5 Orthogonality
318
33.
Let Prandove thbeat r orthr)ogonal matrices.
Usethenpart (a)itos notproivenverthtatib,lief. <let <let
Let be a unit vector in !Rn. Partition as
A
B
(a)
A(A + B B = A + B.
A+
(b)
34.
x
A+B
B = 0,
x
35.
36.
37.
Let
Q=
[-�1- �- - - C- - I�-xJyy- - - r]
yi I
> m,
I Ax I = ll x ll
n
(a)
B={
•
•
x
mX
A
n
•
x
+ (x
x y = (x
+
+ (x
·
·
·
Parseval's Identity. )
Prquiocvek metthathodisfoorrtfihnogonal
ding an. (Torhiths onorprocedurmal base giivsesfora !Rn
Q
wifretquent
h a prleysuscriebfuledifinrapplst vecticatorions.a)construction that is
triangulmatrairxmat. rix is orthogonal,
tPrPrhenooveveittthhmusatat iitffbean aupperdiagonal
t
h
en
t
h
er
e
i
s
no
mat
r
i
x
n
sLetuch that v1, , vn} beforan alorlthonorin !R m. al basis for !Rn.
Prove that, for any and yin !Rn,
· · v1) (y · v1) · v2) (y · v2) · vn) (y · vn)
(This idoesdentiParsty isecalvalle'sdIdentity imply about the
What
relationship between the dot products · and
x,
X n
n
(b)
-
-
x y
[xla · [y] a?
O rth o g o n a l C o m p l e m ents a n d
O rth o g o n a l Proieclio n s
red in Chapt
eermentThes, andno­
ttIihnoetnhprofisosjaeectctnoriioomn,n alofwevectonegenerovectr atoliozareplonttwaneooconcept
wianotl hbeersextwithatlendedgiweveencount
troiseorttohetogonal
compl
he concept of orthogonal
projection onto a subspace.
Apasnorsesmthalrovectughotrhne ortoiagipln,atnehenisiort isthaogonal
to everW ofy vectasorisinsptan(hatnpl) .aHence,
ne. If thwee plhaveane
s
u
bs
p
ace
vecttwo osur bsofptaceshe otofher. Thiwitshisththeepridoeaperbehity thnatd teverhe foly vectlowionrgofdefionenitiisoorn.thogonal to every
n
n
Let
Wbe
a
s
u
bs
p
ace
of
!R
.
We
say
t
h
at
a
vect
o
r
v
i
n
!R
irss that are
W
i
f
v
i
s
or
t
h
ogonal
t
o
ever
y
vect
o
r
i
n
W.
The
s
e
t
of
al
l
vect
o
orthogonal to Wis calW_j_led th{vie n !Rn : v ·w for all wiW,n denot
e
d
W_j_
.
That
i
s
,
W}
diorIfctWihulogonal
asr atoplWatneo(everit.eh.r,oyparughvectalteholretwioro tighnienW;norinhence,
malandvectee oisrW_Ltthoe W).liMorne, tthheenoverroughever, Wtyhconsevectoriigositrnsvperpen­
on e is
ofFigturhose e vectil uosrtsrwatetshtathiars seitoruatthioogonal
n. to every v on hence, we also have W e_L.
1.
w _j_
is pronounced perp:'
"w
O rth ogonal Complements
IR 3 ,
IR 3
w
w
DefiniliOD
nal to
I
e
and
Figure 5 . 5
e = w_j_
orthogo­
orthogonal complement of
= 0
=
w = e _j_
Exa m p l e 5 . 8
IR 3
=
5.5
€;
precisely
=
Section 5.2 Orthogonal Complements and Orthogonal Projections
319
thogonalementcomplof theementcomplofementa subsofpacea sutbsurpnaceed wasout tthoebeorian­gi­
otnalhersIunbssExampl
upbsace.pace.These Alseoprt, htohepereorcompl
(b) ofATheorofemsets APrandoperttconsiieess ar(ics)etsandtrofuet(hidnei)gener
wir common
l alasloandbeelaruseement
eprfulo.ved(sR. Seeecalas prAppendi
l thoatperthtieexs A.(a)) and
Leta. beis aa ssuubsbsppaceace of!Rof !Rnn..
b.c.
d. If span(W1, . . . , Wk), then Vis in if and only if V W; for all
be a sc(ala)ar.SiThennce w u ·forw allvwi· wn ifos rinall w inLet u and v be in and let
Therefore, (u v) ·w u·w v·w
so uWe valissoinhave
(
)
·w
c
u
·w
fr(bo)mWewhiwichl weprosveeetthhiats prcuoperis itny as CorIt ofoll alorwsy that is a subspace of !Rn.
((dc)) YouYou araree asaskkededtoto prprooveve tthhiiss prprooperperttyyiinn ExerExercciissee
assoWeciatecand winowth anexpres matsomerixfundament
. al relationships involving the subspaces
LetA isAthbee nulanl space ofmatA, randix. Thenthe ortthheogonal
orthogonalcomplcompl
e
ment
of
t
h
e
r
o
w
s
p
ace
of
of the column space ofA
is the null space(ofrowAT(A: ) .L null(A) and (cole(ment
A) .L null(AT)
n
I
f
x
i
s
a
vect
o
r
i
n
!R
everin nuly lr(oAw),ofsoA.weButhavethiess itsabltr,uistehhenedif andtxhiesonlfiinrsyt(rioidfw(entAxAit)y. Toifwhiandprocvehonlitshyequieifsexcondvials eornttidhtentogonal
o xitbeiy, wentog
simply replace A by AT and use the fact that row(AT ) col (A) .
rix hascompl
four suement
bspaces:s inro!Rw(n, Aand), nulthle(Ala)s, tcoltw(Ao )ar, ande ornulthogonal
l (A T ) .
TheThus,first tanwo are ormatthogonal
5.8,
tion
Theorem 5 . 9
5.9.
nB
intersec­
B
W
w.L
( W.L ) .L = W
w n w.L = {o}
W=
W.L
·
i = 1, . . . , k.
Proof
c
0
·
= 0
=
+
+
=
W.L
W.L .
W, 0
= 0
W
+
= 0 + 0 = 0
W .L .
(cu)
=
= c (O) = 0
W.L
W.L .
5.12.
23.
24.
m X n
Theorem 5 . 1 0
m X n
=
=
Proof
.L
= 0,
=
m X n
= 0
380
Chapter 5 Orthogonality
null(AT)
null(A)
.
/
/
�.
row(A)
col(A)
[ffi m
The four fundamental subspaces
Figure 5 . 6
compl
e
ment
s
i
n
The
mat
r
i
x
A
defi
n
es
a
l
i
n
ear
t
r
a
ns
fo
r
m
at
i
o
n
fr
o
m
into Figurwhose
rlaungestraitsescolth(esA)e. iMordeaseoverschemat, thisictalralyns. foThesrmeatfourion sesundsbspnulacesl (Aar)etocalleind
e
i
the
of the matrix A.
Find bases for the four fundamental subspaces of
A=
]
I
[
;
and verify Theorem
column space,In Exampl
and nulelsspace ofA. Weandfound twehat comput
row(A) e=d sbaspan(esufor1, u2t,hu3e )r,owherw speace,
, U3
=
u2 =
Also, null (A) = span(x1, x2 ), where
= , =
Toeachshxowj, whithatch(riosw(anAeas) 1y- =exernulcils(eA. )(,Withyis enough
is this sutoffischiowentt?h) at every U; is orthogonal to
�m.
�m
�m.
mX
n
5.6
mX
fundamental subspaces
Exa m p l e 5 . 9
0
�n
n
3
-1 0
2 1 -2
1 6 1
5.10.
3.45, 3.47,
Solulion
ll 1
[1 0
0 -1],
X1
�
3.48,
[O
2 0 3]
-1
-2
1
0
0
1
-3
0
-4
Xz
= [0
0 0
4]
Section 5.2 Orthogonal Complements and Orthogonal Projections
381
The column space ofA is col (A) span(a1, a2, a3), where
=
We stil need to compute the null space ofAr. Row reduction produces
-3
63
[Ar 3 6
6 3
So,It foliflowsisthinat the null space of Ar, then y1 -y4, y2 - 6y4, and y3 - 3y4.
2
4 0
1 -1 2 1 0
0
0
1
1 -2 1 0
-1
0
i O]
y
�
1
0
0
0
0
0 0 1 0
0
0
0
0
0 0 0 0
0 0 0 0
=
=
null(A') { [ =:�]} sp� ( [ =m
and it is easy to check that this vector is orthogonal to a1, a2, and a3.
The method of Example is easily adapted to other situations.
Let be the subspace of spanned by
-3 ,
,
3
Find a basis for
space of The subspace spanned by w1, w2, and w3 is the same as the column
A = -3
3
�
�
5.9
Exa m p l e 5 . 1 0
IR 5
W
1
5
0
5
W1 =
Wz =
-1
1
2
-2
W3 =
W_j_ .
Solution
W
1 -1 0
1 -1
5 2 4
0 -2 - 1
5
5
0
-1
4
-1
5
382
Chapter 5 Orthogonality
T
Ther
e
for
e
,
by
Theor
e
m
Wl
_
=
(
c
ol
(
A
)
J
_
=
nul
l
(
A
), and we may proceed as in
the previous example. We compute
IA' o J [ - :
J [
J
= and = It
folHence,lows thiats in W1-if and only if =
=
= span
and these two vectors form a basis for WJ_ .
Recall that, in the projection of a vector (vu•vonto)a nonzero vector u is given by
proj0(v) = u·u u
Furcantdecompos
hermore, tehve asvector perp0(v) = v - proj0(v) is orthogonal to proj0(v), and we
as shIfownwe leint WFig=ursepan(u), then w = proju(v) is in Wand w1- = perpu(v) is in W1- . We
a way ofto"dW-namel
ecomposiyn,g"v v=inwto thwle _s.uWem ofnowtwogenervectoarlisz, eonethifrs iodmeaWand
tthhere oteforhere haveorthogonal
to
basdefiins edforasW. ForLetanyWvectbe aorsuvbsinpace ofthe and let {u1, ... , uk} be anvorthogonalWis
proj w(v) (UU11••U1 )u, ( Uk"Uk·Uk)uk
The
v
W is the vector
perpw (v) = v - projw (v)
tsoensr Each
(eo)r., Therequisummand
veforaleent, lwiyi,ntthhthetheone-edefinotdnaiitmtiiooensnn ofofiotprnalheojprswu(ebsvcedi)piaces nalgssodefipaannedprniotjioectn,byioweint-ontcaninowroura siitneprgleevivec­ous
5.10,
l
-3 5 0 5 O
2 -2 3 0
-1 4 -1 5 0
�
y
- 3y4 - 4y5 , y2
y1
wl_
�
1 0 0 3 4 O
0 1 0 1 3 0
0 0 1 0 2 0
-y4 - 3y5 , y3 - 2y5 .
-4
-3
-2
0
-3
-1
0
1
0
- 3y4 - 4y5
-y4 - 3y5
- 2y5
Y4
Ys
O rth ogonal Proieclions
v
IR 2 ,
--
u
v proj0(v) + perp0(v)
Figure 5 . 1
=
5.7.
!R n .
+
Definition
!R n
!R n ,
=
orthogonal projection of
V
--
component of orthogonal to
+ ···+
V
--
onto
Section 5.2 Orthogonal Complements and Orthogonal Projections
383
w
Figure 5.8
P = P1 + Pz
ofurSineitces prthiolejuvectescttriaootnsress tonthiarsoseione-toruatthdiogonal
ionmwiensthi,otWnalhe orsutsbshpogonal
panaces(u1, tuprh2at)o,jparecteimutoprn ofouj alwvl(yontv)or, opt1hWiogonalsprthoej.0sFi1u(vmg)­,
andAsp2 a spprecioja0l,(casv) .e of the definition ofproj w(v), we now also have a nice geometric
erminolbasogy,is
ti{nvhtat1e, rvtp2h,reoretate,imovd,nstofathteTheor
senthateimf is inInthteersmubss ofpaceourWprofesent whinotcahtiohasn andorthtogonal
U;
5.8
=
=
=
=
5.2.
!R n ,
w
•
.
•
Thus,
oforthogonal projections onto mutually orthogo­
nal one-The ddefiisimdecompos
ensnitioionnalaboveesdubsinspteoacesemsa suoftmo W.depend
on
t
h
e
choi
c
e
of
or
t
h
ogonal
bas
i
s
;
t
h
at
i
s
,
differentFor" prnow,oj w(lve)t'sandbe
perconta dipffew(ntervewint) . tForbash antiusexampl
nat{u;e,ly..., eth,. iu£s }isfornotWthwoule casde,appear
as we wito lgisvoeona "prove.
w
Exa m p l e 5 . 1 1
Let W be tbe plone in with equation
and let v � [ : ] . Find tbe
orthogonal projection ofv onto Wand the component of v orthogonal to W.
In Example we found an orthogonal basis for W. Taking
II'
Solution
we have
x
-
y + 2z � 0,
5.3,
ll1
" V
= 2
U1 • U1 = 2
Uz • V = - 2
Uz • Uz =
3
-
384
Chapter 5 Orthogonality
Therefore,
proj w (v) = ( ) + ( )
� i[ � l - f : J Ul
and perpw(v) = v - proj w (v) = [ - �3 ] [ -i� ] [ -�; ]
nce it sattoisW,fiessitnhcee equat
is equalIt isleasy easy tyotsoeseetehatthatprperojwp(vw() vis)iins orW,thsiogonal
it is aioscnalofartmulhe pltiapne.le ofIt
thrnornal vootm [ - : l to W. igmo
u1 · v
-- u 1
U 1 . U1
( S" F
u2 · v
- u2
U U
z. z
5.9.)
perpw(v)
v projw(v) + perpw(v)
Figure 5 . 9
=
Theorem 5 . 1 1
respTheect tnexto a suthbseorpaceemandshowsits torhatthweogonalcan alcomplwaysefimentnd a.decomposition of a vector with
ubspacew_j_ofin W_j_andsuchletthvatbe a vector in Then there are unique
vectLet oWrsbewian sWand
v = w + w_j_
We need to show two things: that such a decomposition and that it is
for Let
orthogonal basis
w =Toprowsjhwow+(vw_j_)exiands=telprence,t ow_j_j wwe(=v)perchoos
p+ perw(epvanw) .(Then
v) = proj w(v) + (v - proj w(v) = v
The Orthogonal Decomposition Theorem
IW
Proof
unique.
!R n .
exists
{u1, . . . , uk}
W
Section 5.2 Orthogonal Complements and Orthogonal Projections
385
Clu1,ear. .ly. ,, wuk. Toprshojoww (vth) atis wlin. W,is insinWlce. ,ititiiss aenough
linear combi
nattihoatn wlof.thiseorbasthiogonal
s vectotros
t
o
s
h
ow
each of the basis vectors U;, by Theorem 5.9(d) . We compute
U; wl. u;· per(v -pwpr(vo)j w(v)
u;· (v - (:1.";Ju1 - - (�k.";Juk)
Uu;·; " u;V ) (u' · u)'
UU11·"Vu1 ) (u' · u ) - (-u-·' v - (-u-·v - - (-uui' ·-· uvi ) (u-· u.)
that wl. is in Wl. and completes the existence
parsincetToofu;sthhujowe prtohof.e founir quenesThis ofs tprhiosvesdecompos
ppose wew_Lhave anotW1 +hwt,er de­so
composition v W1 +wt, where w1 is in Wandwtitioin,s inletwl's s.uThenw+
But sinthceatwth-is common
w1 is in Wand
these earme 5.su9bs(cp) ]aces. Thus,), we
know
vectorwtis in-Wwl. isWlin. Wl. (b[ecaus
usingeTheor
=
•
=
=
U;
•
=
···
=
=
t
0-···
z
-···
-···
z
z
-· · ·- 0
= u; · v - u; · v = 0
·
=0
j -=F i.
=
=
n
= {O}
the Exampl
subspacee 5.of1IR1 3ilgiuvsentrabytedththeeplOranethwiogonalth equatDecompos
ion - iti+on Theoretm.he orWhenthogonalW is
dernmpo,;tion of v [ -! ] wdh mped to W ;, w + w", whece
w pmj w(v) Ul Md w" pecpw(v) H l
s of thve)ordothogonal
decomposon thiteiochoin guarce aofnteoresththogonal
at the defibasinsi.tiTheons
ofOrtprhTheogonal
ojwuni(v)qDecompos
anduenesperpw(
not
depend
i
t
i
o
n
Theor
e
m
al
s
o
al
l
o
ws
us
t
o
pr
o
ve
pr
o
per
t
y
(
b
)
of
Theo­
rem 5.e9m.. We state that property here as a corol ary to the Orthogonal Decomposition
Theor
If Wis a subspace of then
x y
�
C o r o l l a rv 5 . 1 2
. �
�
�
!R n ,
2z = 0,
�
�
386
Chapter 5 Orthogonality
f w is inWWand(W_Lxis) _Lin. NowW_L ,lteht env bewinx(W_L ) _LBut. BythTheor
is nowemimplieswethcanat wwris iitne
(W_Lv w) _L . IHence,
w_Lv w_Lfor (u(wnique)w_Lvect) wo_Lrs wiwn Wand
w_L
i
n
W_L
.
But
now
_L w_L w_L w_L w_L w_L w_L
w
and,so w_Lsince tTherhe reeverforsee, vincluwsionw_Lis alsow,trandue, wethusconclvisuindeW.thThiat (sWsh_Lows) _L thW,at (asWr_Le)qui_L red.W
There eims al5.so1 3.a nice relationship between the dimensions of Wand W_L , expres ed in
Theor
If W is a subspace of then dim W dim W _L
anworthogonal
onalWe clbasaimLetistforhatw_LBi...s. Thenan, ukor} betdihmogonal
kbasandis fodibasrmiw_Ls for WandLet lBet {v1, ......, v1,} Ukbe,vanl, ...orth, v1og-}.
We first note thatU,; 'siVnj ce eachforiis in Wand.. .,kandj
each vj is in W_L ,
Thus,, iBf viiss ana vectortohrogonal
shete and,Orthhence,
iDecompos
s linearly iintidependent
, bytelTheor
emat v
Next
i
n
t
ogonal
o
n
Theor
e
m
s
us
t
h
wtion ofw_Lthfoe rvectsomeorswiU; andn Wandw_L canw_LbeinwrW_Lit .enSinascea wlincanear combi
be wrint atenioasn ofa litnheare vectcombiors vnj,a­v
andcan beso wris aitbaseniass fora linearItcombi
follodiwsnmatthiwoatnkofdithme vectw_Ldimors in orB. Therefore, B spans also
As
a
l
o
vel
y
bonus
,
when
we
appl
y
t
h
i
s
r
e
s
u
l
t
t
o
t
h
e
fundament
a
l
s
u
bs
p
aces
of
a
matCororilxa, rwey get a quick proof of the Rank Theorem (Theorem restated here as
C
Proof
= +
0= ·
= 0.
Theorem 5 . 1 3
=
·
+
·
= +
=
=
+
·
= 0.
5. 1 1,
=0+
·
=
·
C
!R n ,
=
+
Proof
{u 1 ,
=
!R n .
u;
=0
n
= l.
= {u 1 ,
= 1,
= 1, . . . , l
5. 1 .
=
!R n ,
+
!R n .
+l=
+
!R n
!R n ,
=
n
3.26),
5. 14.
C o r o l l a rv 5 . 1 4
=
·
The Rank Theorem
IfA is an matrix, thenrank(A) nullity(A)
em
so diNotm eWInthTheor
rat awenk(emgetA) anda counttdiakemeW_LrWpart riodnulw(entlAiitty)y.(Then
Aby) . tTheakiW_LnrgesuWlnult folll(coloAws.)(,Aby) Theor
[and therefore
W_L null (AT )J :
rank(A) nullity(A
mX
n
+
Proof
=
5.13,
=
=
n
=
=
5.10,
=
=
+
r) = m
Section 5.2 Orthogonal Complements and Orthogonal Projections
381
have, weilhaveustranotted essometablisofhedthtehatadvanteveryagessubsofpaceworkinganwiorth­
orthogonal
thSectogonaliobasnsbasis, enorsand. However
ven a mete hodThesfor conse is turuesctarineg tshuechsuabjbasectisof(etxcept
parsecttiioculn. ar exampleshave, suchweasgiExampl
he nextin
5.1
5.2
has
5.3).
I
Exercises 5 . 2
In Exercises 1 -6, find the orthogonal complement W_j_ of W
and give a basis for W_j_ .
1. W =
2. W =
3. w �
4. W �
5. W �
6. W �
{ [;] }
}
{ [;]
{ [�l
)
{ [:J
)
{ [�l
)
{ [�] � +. )
:
:
2x - y = 0
3x + 4y = 0
x+y-z� 0
2x - y + 3z � 0
x � I, y � - I, z � 3t
x � t, y �
z � 21
In Exercises 7 and 8, find bases for the row space and null
space of A. Verify that every vector in row(A) is orthogonal
to every vector in null(A).
7. A =
[ � : _! ]
[ -� � -� � !l
-1 -1
8. A =
In Exercises 1 1 - 1 4, let W be the subspace spanned by the
given vectors. Find a basis for W_j_ .
1
2 2 -2 0 1
-3 - 1 3 4 5
In Exercises 9 and 1 0, find bases for the column space of
A and the null space of A T for the given exercise. Verify
that every vector in col(A) is orthogonal to every vector
in null(A T).
10.
8
9.
7
Exercise
Exercise
4
2
6
2
2
=
=
W
,
W
=
,
W
2
0 3
-1 z
14. 1
1
-1
2
-1
-3
In Exercises 1 5- 1 8, find the orthogonal projection of v onto
the subspace W spanned by the vectors u;. (You may assume
that the vectors U; are orthogonal.)
Chapter 5 Orthogonality
388
t
h
at
v
=
I
s
i
t
neces
s
a
r
i
l
y
t
r
u
e
t
h
at
i
s
i
n
W
J
.
?
EiLetth{erv1pr, ...ove, vthn}atbeit anis torrutehorogonalfind abascountis foerre!Rxampl
eand l.et
v
n
v = [ - � l W = span( [�] )
Wpan(= vspk+an(l' ...v1, ,...vn),?vEik)t·hIers itprnecesove tsharatilyit tirsuteruthe atorWlfin.d=a
scount
e
r
e
xampl
e
.
J
W
([
J
)
[
:
!Rn, x
n!R . W
Pr
o
ve
t
h
at
xi
s
i
n
W
i
f
and
onl
y
i
f
pr
o
j
w
(
x
)
=
x.
(
)
H
to W if and only if
[ � ]. [ - : ]
[prPrProoojvevew(ttxhh)atat=prxiosjorw(tphrogonal
o
j
w
(
x
)
=
pr
o
j
w
(
x
)
.
Letlet xSbe= a{vvect1, ...or ,invk!R} nbe. an orthonormal set i n !Rn, and
[ -; l W ( [J [ -m
Prol vex l 2th2:at lx ·v1 l2 lx ·v l 2 lx ·vkl2
23. PrProoveve Theor
e
m
9
(
c
)
.
5.
2
(
T
hi
s
i
n
equal
i
t
y
i
s
cal
l
e
d
Theor
e
m
9
(
d
)
.
5.
LetthatW andbe a subsarepaceorthofogonal
!Rn andvectv aovectrs wiotrhin !Rinn. Wand
Suppose onlProyveiftxihats iBesn sspean(l's ISn).equality is an equality if and
w + w' .
In Exercises 1 9-22, find the orthogonal decomposition of
with respect to W
26.
19.
� 'P"•
20. v �
21.
22.
w'
In Exercises 27-29, let
a vector in
27.
28.
0.
29.
30.
w � 'P""
F
� 'P""
F
be a subspace of
(a)
+
24.
25.
and let be
+···+
Bessel's Inequality. )
(b)
w
w'
w
T h e G r a m - S c h m i d t Process
and t h e OR Facto rizati o n
tmosIhnonorthtiuss mseefulalct)iobasofn,aliwesl formatpranyerisxentfactsubsaospriimacezatpliofeons.met!Rn.hThiod sformetconshodtrwiuctl itnhgenanleoradthusogonal
to one(oofr torhe­
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vectExamplor ate 5.a 3ti.me. We wil il ustrate the basic construction with the subspace W from
Let W = span(x1, x2 ), where
The Gram-Schmidl Process
Exa m p l e 5 . 1 2
Construct an orthogonal basis for W.
1, we getto ax1se(Fcondigurevect5. 1o0)r.that is orthogonal to it by taking
the componentStartiofngxwi2 orththxogonal
Solulion
Section 5.3 The Gram-Schmidt Process and the Factorization
QR
389
w
Constructing orthogonal to
Figure 5.10
v2
x1
Algebraically, we set v1 = x1 , so
v2 = perpxJ xz ) = x2 - proj x, ( xz )
( )
= x2 - � xI
X 1 • xi
Then {v1 , v2 } is an orthogonal set of vectors in W. Hence, {v1 , v2 } is a linearly independent set and therefore a basis for W, since dim W = 2.
Remark
Theorem 5 . 1 5
[il
Observe that this method depends on the order of the original basis
mtocs. Jn Exrunpl' 5 . 1 2, if we had taken x , �
�
[ -fl
4
and x, �
we would have
obtained a different orthogonal basis for W. (Verify this.)
The generalization of this method to more than two vectors begins as in
Example 5 . 1 2 . Then the process is to iteratively construct the components of subse­
quent vectors orthogonal to all of the vectors that have already been constructed. The
method is known as the Gram-Schmidt Process.
The Gram-Schmidt Process
Let {x 1 , . . . , xd be a basis for a subspace W of !R n and define the following:
_ ( : ) _ ...
xk
vz
Vz Vz
v2
Then for each i = 1 , . . . , k, {v1 , . . . , v; } is an orthogonal basis for W;. In particular,
{v1 , , vk } is an orthogonal basis for W.
.
•
.
390
Chapter 5 Orthogonality
Jorgen
PedersenwasGram
(1850-1916)
a Danishwhoactuary
(insurance
stati
s
tician)
was
isurement.
nterested inHethefirstscience
of
mea­
published
the
process
that
bears
his
name
in
an 1883Schmidt
paper on(1876-1959)
least squares.was
Erhard
astudiGerman
mathematician
whod
e
d
under
the
great
Davi
Hilbert
and is considered
one of
ofmathematics
the founders
of
the
branch
known as functional
analysis.
His
contribution
to thein a
Gram-Schmidt
Process
came
integral
inof1907thewhipaper
cmethod
h heonwrotemore
outexpltheequations,
details
i
c
itly than
Gram had done.
Stated succinctly, Theorem 5 . 1 5 says that every subspace of ll�r has an orthogonal
basis, and it gives an algorithm for constructing such a basis.
Proof
We will prove by induction that, for each i = 1, . . . , k, {v1 , . . . , v;} is an or­
thogonal basis for W;.
Since v1 = x1 , clearly {v i } is an (orthogonal) basis for W1 = span(x 1 ) . Now assume
that, for some i < k, {v1 , . . . , v;} is an orthogonal basis for W;. Then
V;+ J = X;+ J
_(
) _(
VI • X;+ 1
V1
V1 . V1
) _ ... _ (
V2 • X;+ 1
Vz
Vz . Vz
)
V; • X;+ 1
V;
V; . V;
By the induction hypothesis, {v1 , . . . , v;} is an orthogonal basis for span(x1 , . . . , x;) =
W;. Hence,
V;+ 1 = X;+ 1 - proj w, C x;+ 1 ) = perp w, C X;+ 1 )
So, by the Orthogonal Decomposition Theorem, V;+ 1 is orthogonal to W;. By
definition, v1 , . . . , v; are linear combinations of x1 , . . . , X; and, hence, are in W;.
Therefore, {v1 , . . . , V;+ d is an orthogonal set of vectors in W;+ i ·
Moreover, V;+ i * 0, since otherwise X;+ 1 = proj w, C X; + 1 ) , which in turn implies
that X;+ 1 is in W;. But this is impossible, since W; = span(x 1 , . . . , X;) and {x 1 , . . . , X;+ i }
is linearly independent. (Why?) We conclude that {v1 , . . . , V;+ d is a set of i + ! linearly independent vectors in W;+ i · Consequently, {v1 , . . . , V;+ d is a basis for W;+ 1 ,
since dim W;+ 1 = i + 1 . This completes the proof.
If we require an orthonormal basis for W, we simply need to normalize the
orthogonal vectors produced by the Gram-Schmidt Process. That is, for each i, we
replace V; by the unit vector q; = ( l / ll v; ll )v; .
Exa m p l e 5 . 1 3
Apply the Gram-Schmidt Process to construct an orthonormal basis for the subspace
W = span(x1 , x2 , x3 ) of IR 4 , where
First we note that {x1 , x2 , x3 } is a linearly independent set, so it forms a basis
for W. We begin by setting v1 = x1 . Next, we compute the component of x2 orthogo­
nal to W1 = span(v1 ) :
Solulion
Section 5.3 The Gram-Schmidt Process and the Factorization
QR
391
For hand calculations, it is a good idea to "scale" v2 at this point to eliminate fractions.
When we are finished, we can rescale the orthogonal set we are constructing to obtain
an orthonormal set; thus, we can replace each V; by any convenient scalar multiple
without affecting the final result. Accordingly, we replace v2 by
We now find the component of x3 orthogonal to
W2 span ( x1 , xi ) span ( v1 , v2 )
=
=
=
span ( v1 , v� )
using the orthogonal basis {v1 , v�}:
Agllin, we mrnle and "'";
�
[
� 2 � !l
v,
We now have an orthogonal basis {v1 , v�, v� } for W. (Check to make sure that
these vectors are orthogonal.) To obtain an orthonormal basis, we normalize each
vector:
q3
=
-1� 1 [ -1/V6] [ -V6/6 ]
[
C : ) ( �) 2 21 /�V6V6 �/V6/63
v
1 � 11 �
=
=
Then {q 1 , q2 , q3 } is an orthonormal basis for W.
392
Chapter 5 Orthogonality
One of the important uses of the Gram-Schmidt Process is to construct an orthogo­
nal basis that contains a specified vector. The next example illustrates this application.
Exa m p l e 5 . 1 4
Find an orthogonal basis for IR 3 that contains the vector
Solulion
�
We first find any basis for IR 3 containing v1 • If we take
then {v1 , x2 , x3 } is clearly a basis for IR 3 . (Why?) We now apply the Gram-Schmidt
Process to this basis to obtain
and finally
Then {v1 , v�, v� } is an orthogonal basis for IR 3 that contains v1 .
Similarly, given a unit vector, we can find an orthonormal basis that contains it by
using the preceding method and then normalizing the resulting orthogonal vectors.
R e m a r k When the Gram-Schmidt Process is implemented on a computer, there
is almost always some roundoff error, leading to a loss of orthogonality in the vec­
tors q;. To avoid this loss of orthogonality, some modifications are usually made. The
vectors V; are normalized as soon as they are computed, rather than at the end, to
give the vectors q;, and as each q; is computed, the remaining vectors xj are modified
to be orthogonal to q;. This procedure is known as the Modified Gram-Schmidt
Process. In practice, however, a version of the QR factorization is used to compute
orthonormal bases.
The OB Facto rization
2
If A is an m X n matrix with linearly independent columns (requiring that m n),
then applying the Gram-Schmidt Process to these columns yields a very useful fac­
torization of A into the product of a matrix Q with orthonormal columns and an
Section 5.3 The Gram-Schmidt Process and the Factorization
QR
393
upper triangular matrix R. This is the QR factorization, and it has applications to the
numerical approximation of eigenvalues, which we explore at the end of this section,
and to the problem of least squares approximation, which we discuss in Chapter 7.
To see how the QR factorization arises, let a 1 , . . . , an be the (linearly independent)
columns of A and let q 1 , . . . , qn be the orthonormal vectors obtained by applying the
Gram-Schmidt Process to A with normalizations. From Theorem 5.15, we know that,
for each i = 1, . . . , n,
W; = span ( a 1 , . . . , a; ) = span ( q 1 , . . . , q; )
Therefore, there are scalars r 1 ;, r2 ;, • . • , r;; such that
That is,
a, = r l l q,
az = r 1 2 q 1 + Yzz qz
which can be written in matrix form as
r 1n
r� n
0
rnn
l
= QR
Clearly, the matrix Q has orthonormal columns. It is also the case that the diago­
nal entries of R are all nonzero. To see this, observe that if r;; = 0, then a; is a linear
combination of q 1 , . . . , q; _ 1 and, hence, is in W; _ 1 . But then a; would be a linear com­
bination of a 1 , . . . , a; _ 1 , which is impossible, since a 1 , . . . , a; are linearly independent.
We conclude that r;; -=fa 0 for i = 1, . . . , n. Since R is upper triangular, it follows that it
must be invertible. (See Exercise 23. )
We have proved the following theorem.
Theorem 5 . 1 6
The QR Factorization
Let A be an m X n matrix with linearly independent columns. Then A can be fac­
tored as A = QR, where Q is an m X n matrix with orthonormal columns and R is
an invertible upper triangular matrix.
R e m a rks
We can also arrange for the diagonal entries of R to be positive. If any r;; < 0,
simply replace q; by - q; and r;; by - r;;.
•
The requirement that A have linearly independent columns is a necessary one.
To prove this, suppose that A is an m X n matrix that has a QR factorization, as in The­
orem 5.16. Then, since R is invertible, we have Q = AR - 1 • Hence, rank(Q) = rank(A),
by Exercise 61 in Section 3.5. But rank( Q) = n , since its columns are orthonormal and,
therefore, linearly independent. So rank(A) = n too, and consequently the columns of
A are linearly independent, by the Fundamental Theorem.
•
Chapter 5 Orthogonality
394
=
=
•
The QR factorization can be extended to arbitrary matrices in a slightly
modified form. If A is m X n, it is possible to find a sequence of orthogonal matrices
Q1, , Om - I such that Om - I · · · Q2 Q1A1 is an upper triangular m X n matrix R. Then
A QR, where Q ( O m - I · · · Q2 Q1) - is an orthogonal matrix. We will examine this
approach in Exploration: The Modified QR Factorization.
.
Exa m p l e 5 . 1 5
•
.
=A [ - � � �1
Find a QR factorization of
-1 0 1
1
2
The columns of A are just the vectors from Example 5.13. The orthonormal
basis for col(A) produced by the Gram-Schmidt Process was
Solulion
= [ = :;: 1 , = [ ��:: 1 , = [ - ��:1
= =[
v: 1
=
=
= =IR =
[
l
= =
=:
l
[
�
l
q1
qz
Q
V6/3
Vs/ 10
1/2
so
q3
[ q i qz q 3 ]
1/2 3 Vs/10 - /6
- 1/2 3 Vs/10
- 1/2 Vs/10 V6/6
1/2 Vs/10 V6/3
From Theorem 5.16, A QR for some upper triangular matrix R. To find R, we
use the fact that Q has orthonormal columns and, hence, Q TQ I. Therefore,
Q TA
We compute
R
..
I
Exercises 5 . 3
Q'A
R
1/2
- 1/2 - 1 /2
1 /2
3 Vs/ 10 3 Vs/10 Vs/10 \/5/ 10
V6/6 V6/3
- V6/6
0
1
1 /2
Vs 3 Vs/2
0 V6/2
� [:
In Exercises 1 -4, the given vectors form a basis for IR 2 or IR 3 •
Apply the Gram-Schmidt Process to obtain an orthogonal
basis. Then normalize this basis to obtain an orthonormal
basis.
Q TQR
= [ _�l [�]
[
[
3 � x� x�
J, n, m
2. X1
.
..
Xz =
1 2
1
0
1
Section 5.3 The Gram-Schmidt Process and the Factorization
QR
Exercises 5 and 6, the given vectors form a basis for a
subspace W of IR 3 or IR 4. Apply the Gram-Schmidt Process
to obtain an orthogonal basis for W
**
*]
In Exercises 13 and 1 4, fill in the missing entries of Q
to make Q an orthogonal matrix.
1/ \/2 1 / \/3
13. Q
0
1 / \/3
- 1/ \/2 1 / \/3
1/2 2/ Vi4
1/2 l/ Vi4
14. Q
0
1/2
1/2 - 3/ Vi4
=
In
395
=
[
[
**
** **: ]
In Exercises 1 5 and 1 6, find a QR factorization of the
matrix in the given exercise.
16. Exercise 10
15. Exercise 9
In Exercises 7 and 8, find the orthogonal decomposition of v
with respect to the subspace W
7. v �
8.
F
[ -:}
[ i}
w "' i• Emd" ,
] ['
i -! ]
=
=
W as in Exmise 6
Use the Gram-Schmidt Process to find an orthogonal basis
for the column spaces of the matrices in Exercises 9 and 1 0.
9.
In Exercises 1 7 and 1 8, the columns of Q were obtained by
apply ing the Gram-Schmidt Process to the columns ofA.
Find the upper triangular matrix R such that A QR.
l
8 2
3
�
17. A
7 -1 ' Q
3
2
1
-2
--3 3
18. A
=
u
]
=
[ -j - � [ �
3
4
,Q
=
:3
1 /v'6 1/
2/ V6
- 1 V6 1 / \/3
1 / \/3 _
[: � �]
19. If A is an orthogonal matrix, find a QR factorization
of A.
20. Prove that A is invertible if and only if A QR, where
Q is orthogonal and R is upper triangular with nonzero
[:J
In Exercises 21 and 22, use the method suggested by
Exercise 20 to compute A i for the matrix A in the given
exercise.
22. Exercise 15
21. Exercise 9
23. Let A b e an m X n matrix with linearly independent
1 1 . Find an orthogonal basis for IR 3 that contains the
veoto'
12. Find an orthogonal basis for IR 4 that contains the
vectors
=
entries on its diagonal.
-
columns. Give an alternative proof that the upper
triangular matrix R in a QR factorization of A must
be invertible, using property (c) of the Fundamental
Theorem.
24. Let A be an m X n matrix with linearly independent
columns and let A = QR be a QR factorization of A.
Show that A and Q have the same column space.
Exp lorations 0
T h e M o difi e d QR Factorizat i o n
When the matrix A does not have linearly independent columns, the Gram-Schmidt
Process as we have stated it does not work and so cannot be used to develop a gen­
eralized QR factorization of A. There is a modification of the Gram-Schmidt Process
that can be used, but instead we will explore a method that converts A into upper
triangular form one column at a time, using a sequence of orthogonal matrices. The
method is analogous to that of LU factorization, in which the matrix L is formed
using a sequence of elementary matrices.
The first thing we need is the "orthogonal analogue" of an elementary matrix;
that is, we need to know how to construct an orthogonal matrix Q that will trans­
form a given column of A-call it x-into the corresponding column of R-call it y.
I Qx ll
By Theorem 5.6, it will be necessary that ll x ll
Figure 5. 1 1 suggests a
way to proceed: We can reflect x in a line perpendicular to x y. If
= = l r l-.
y
figure 5 . 1 1
- uj_ = [ �: ]
u
[ l--2d,2dd212 --2d21dd22 ] = I - 2uu .
is the unit vector in the direction of x
y, then
-
is orthogonal to
u,
and
we can use Exercise 26 in Section 3.6 to find the standard matrix Q of the reflection
in the line through the origin in the direction of J_ .
1 . Show that Q =
2.
Compute Q for
T
1
( a)
u = [fl = [ : J , r = [ � ]
u
= I - 2uur
(b) x
We can generalize the definition of Q as follows. If is any unit vector in !R n , we
define an n X n matrix Q as
Q
396
Such a matrix is called a Householder matrix (or an elementary reflector) .
3. Prove that every Householder matrix Q satisfies the following properties:
(a) Q is symmetric.
(b) Q is orthogonal.
(c) Q 2 = I
4. Prove that if Q is a Householder matrix corresponding to the unit vector
u, then
-v if v is in span ( u )
Qv =
v if v · u = 0
AlstononeHouseholder
(1904-1993)
was
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the firstoftoalgpresent
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ving linear systems.
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The Theory
of Matrices in Numerical Analysis
5.
Compute Q foe u �
{
[ �]
-
<md wdfy Prnblem' 3 ond 4.
6. Let x * y with ll x ll = llrll and set u = ( 1/ ll x - rll ) ( x - y). Prove that the
corresponding Householder matrix Q satisfies Q x = y. [Hint: Apply Exercise 57 in
Section 1 .2 to the result in Problem 4.]
7. Find Q and verify Problem 6 for
We are now ready to perform the triangularization of an m X n matrix A, column
by column.
8. Let x be the first column of A and let
Show that if Q 1 is the Householder matrix given by Problem 6, then Q 1 A is a matrix
with the block form
where A 1 is (m - l) X ( n - 1 ).
If we repeat Problem 8 on the matrix A 1 , we use a Householder matrix P2 such that
where A 2 is ( m - 2 ) X ( n - 2 ).
9. Set Q =
2
[ � :J.
Show that Q 2 is an orthogonal matrix and that
391
10. Show that we can continue in this fashion to find a sequence of orthogonal
matrices Qi, . . . , Q m - i such that Q m - i Q2 QiA = R is an upper triangular m X n
matrix (i.e., r;1 = 0 if i > j).
1 1 . Deduce that A = QR with Q = Qi Q2 Q m - i orthogonal.
1 2. Use the method of this exploration to find a QR factorization of
· · ·
· · ·
3
3
-4
-1
-5
Ap proxi m ating Eigenval u e s
with t h e QR Algo rith m
SeeLoan,G. H. Golub and C. F. Van
(Baltimore:Press,
Johns1983).
Hopkins
University
Matrix Computations
One of the best (and most widely used) methods for numerically approximating the
eigenvalues of a matrix makes use of the QR factorization. The purpose of this ex­
ploration is to introduce this method, the QR algorithm, and to show it at work in a
few examples. For a more complete treatment of this topic, consult any good text on
numerical linear algebra. (You will find it helpful to use a CAS to perform the calcula­
tions in the problems below.)
Given a square matrix A, the first step is to factor it as A = QR (using whichever
method is appropriate). Then we define Ai = RQ.
1. First prove that Ai is similar to A. Then prove that Ai has the same eigen­
values as A.
2.
If A =
[ � �]
, find A 1 and verify that it has the same eigenvalues as A.
Continuing the algorithm, we factor A i as Ai = QiR 1 and set A 2 = RiQi. Then we
factor A 2 = Q2 R 2 and set A 3 = R 2 Q2 , and so on. That is, for k 2: 1 , we compute Ak =
QkRk and then set Ak i = RkQk.
+
3. Prove that Ak is similar to A for all k 2: 1.
4. Continuing Problem 2, compute A 2 , A 3 , A 4 , and A5, using two-decimal-place
accuracy. What do you notice?
It can be shown that if the eigenvalues of A are all real and have distinct absolute
values, then the matrices Ak approach an upper triangular matrix U.
5. What will be true of the diagonal entries of this matrix U?
6. Approximate the eigenvalues of the following matrices by applying the QR
algorithm. Use two-decimal-place accuracy and perform at least five iterations.
( a)
( c)
[� � ]
[� �]
�[ � - � ] [ � � ]
[ ]
(b)
-4
0
1
-
( d)
-2
2
4
2
2
7. Apply the QR algorithm to the matrix A =
-1
Why?
398
3
-2
. What happens?
8. Shift the eigenvalues of the matrix in Problem 7 by replacing A with
B = A + 0.9I. Apply the QR algorithm to B and then shift back by subtracting 0.9
from the (approximate) eigenvalues of B. Verify that this method approximates the
eigenvalues of A.
9. Let Q0 = Q and R 0 = R. First show that
QO Q I . . . Qk - lA k = AQ0Q 1 . . . Qk - 1
for all k 2: 1 . Then show that
[Hint: Repeatedly use the same approach used for the first equation, working from
the "inside out:'] Finally, deduce that ( Q0Q 1 Qk) (Rk R 1 R0) is the QR factorization
of A k+ i .
· · ·
· · ·
399
400
Chapter 5 Orthogonality
Orthogonal Diagonalizalion ot svmmetric Matrices
[o ]
We saw in Chapter 4 that a square matrix with real entries will not necessarily have real
-1
eigenvalues. Indeed, the matrix
has complex eigenvalues i and - i. We also
1
0
discovered that not all square matrices are diagonalizable. The situation changes
dramatically if we restrict our attention to real symmetric matrices. As we will show
in this section, all of the eigenvalues of a real symmetric matrix are real, and such a
matrix is always diagonalizable.
Recall that a symmetric matrix is one that equals its own transpose. Let's begin by
studying the diagonalization process for a symmetric 2 X 2 matrix.
Exa m p l e 5 . 1 6
If possible, diagonalize the matrix A =
[
]
1
2
.
2 -2
The characteristic polynomial of A is A 2 + A - 6 = ( A + 3 )(A - 2), from
which we see that A has eigenvalues A1 = - 3 and A 2 = 2. Solving for the correspond­
ing eigenvectors, we find
Solulion
v1 =
[ -� �]
[ _�]
and
v2 =
[�]
respectively. So A is diagonalizable, and if we set P = [ v1
p - 1AP =
v2 ] , then we know that
= D.
However, we can do better. Observe that v1 and v2 are orthogonal. So, if we nor­
malize them to get the unit eigenvectors
[
l / Vs
U1 = -2/ Vs
and then take
]
and
[
u2 =
[ 2/l/ VsVs]
]
l / Vs 2/ Vs
-2/ Vs l / Vs
we have Q - 1AQ = D also. But now Q is an orthogonal matrix, since { u1, u2 } is an
1
orthonormal set of vectors. Therefore, Q - = Q T, and we have Q TAQ = D. (Note that
Q = [ u 1 Uz ] =
checking is easy, since computing Q - 1 only involves taking a transpose!)
The situation in Example 5.16 is the one that interests us. It is important enough
to warrant a new definition.
D efi n iii 0 n
A square matrix A is orthogonally diagonalizable if there exists an
orthogonal matrix Q and a diagonal matrix D such that Q TAQ = D.
We are interested in finding conditions under which a matrix is orthogonally
diagonalizable. Theorem 5 . 1 7 shows us where to look.
Section 5.4 Orthogonal Diagonalization of Symmetric Matrices
Theorem 5 . 1 1
401
If A is orthogonally diagonalizable, then A is symmetric.
Proof
If A is orthogonally diagonalizable, then there exists an orthogonal ma­
trix Q and a diagonal matrix D such that Q TAQ = D. Since Q - 1 = Q T, we have Q TQ =
I = QQ T, so
But then
since every diagonal matrix is symmetric. Hence, A is symmetric.
Theorem 5 . 1 7 shows that the orthogonally diagonalizable matrices are
all to be found among the symmetric matrices. It does not say that every symmetric
matrix must be orthogonally diagonalizable. However, it is a remarkable fact that this
indeed is true! Finding a proof for this amazing result will occupy us for much of the
rest of this section.
Remark
We next prove that we don't need to worry about complex eigenvalues when work­
ing with symmetric matrices with real entries.
Theorem 5 . 1 8
If A is a real symmetric matrix, then the eigenvalues of A are real.
Recall that the complex conjugate of a complex number z = a + bi is the number
z = a - bi (see Appendix C). To show that z is real, we need to show that b = 0. One
way to do this is to show that z = z, for then bi = - bi (or 2bi = O), from which it
follows that b = 0.
We can also extend the notion of complex conjugate to vectors and matrices by,
for example, defining A to be the matrix whose entries are the complex conjugates of
the entries of A; that is, if A = [a;) , then A = [ au ] . The rules for complex conjugation
extend easily to matrices; in particular, we have AB = AB for compatible matrices
A and B.
Proof Suppose that A is an eigenvalue of A with corresponding eigenvector v. Then
- Av = Av, and, taking complex conjugates, we have Av = Av. But then
Av = Av = Av = Av = Av
since A is real. Taking transposes and using the fact that A is symmetric, we have
vTA = vTA T = (Avf = ( Avf = Avr
Therefore,
A ( vrv) = vr( Av) = vr(Av) = ( vrA ) v = ( Avr ) v = A( vrv)
[ � l __ [ � l
or ( A - A) ( vrv) = o.
Now if v =
a,
.
b, i
an + bn i
, then v -
a,
.
b,i
an - bni
, so
402
Chapter 5 Orthogonality
since v * 0 (because it is an eigenvector). We conclude that A - A = 0, or A = A.
Hence, A is real.
Theorem 4.20 showed that, for any square matrix, eigenvectors corresponding
to distinct eigenvalues are linearly independent. For symmetric matrices, something
stronger is true: Such eigenvectors are orthogonal.
Theorem 5 . 1 9
If A is a symmetric matrix, then any two eigenvectors corresponding to distinct
eigenvalues of A are orthogonal.
Let v1 and v2 be eigenvectors corresponding to the distinct eigenvalues
A 1 * A 2 so that Av1 = A 1 v1 and Av2 = A 2v2 . Using A T = A and the fact that x y = xTy
for any two vectors x and y in !R n , we have
Proof
·
(vfA T) Vz
( vfA ) v2 = vf (Av2 )
= vi ( A 2v2 )
A 2 ( vfv2 ) = A 2 ( v1 • v2 )
Hence, (A 1 - A 2 ) (v 1 v2 ) = 0. But A 1 - A 2 * 0, so v1 v2 = 0, as we wished to show.
•
Exa m p l e 5 . 11
•
Verify the result of Theorem 5.19 for
The characteristic polynomial of A is - A 3 + 6A 2 - 9A + 4 = - (A - 4)
2
(A - 1 ) , from which it follows that the eigenvalues of A are A 1 = 4 and A 2 = 1 . The
corresponding eigenspaces are
Solulion
�
·
(Check this.) We easily verify that
from which it follows that every vector in £4 is orthogonal to every vector in £ 1 .
(Why?)
Roman
Note thot
[ - �] [ - i ]
•
� !. Thos, 'igenvedo;s conesponding to the
same eigenvalue need not be orthogonal.
Section 5.4 Orthogonal Diagonalization of Symmetric Matrices
403
We can now prove the main result of this section. It is called the Spectral Theo­
rem, since the set of eigenvalues of a matrix is sometimes called the spectrum of the
matrix. (Technically, we should call Theorem 5.20 the Real Spectral Theorem, since
there is a corresponding result for matrices with complex entries.)
Theorem 5 . 2 0
The Spectral Theorem
Let A be an n X n real matrix. Then A is symmetric if and only if it is orthogonally
diagonalizable.
We have already proved the "if" part as Theorem 5 . 1 7. To prove the "only if"
implication, we proceed by induction on n. For n = 1 , there is nothing to do, since a
1 X 1 matrix is already in diagonal form. Now assume that every k X k real symmet­
ric matrix with real eigenvalues is orthogonally diagonalizable. Let n = k + 1 and let
A be an n X n real symmetric matrix with real eigenvalues.
Let A 1 be one of the eigenvalues of A and let v 1 be a corresponding eigenvector.
Then v1 is a real vector (why?) and we can assume that v1 is a unit vector, since
otherwise we can normalize it and we will still have an eigenvector corresponding
to A 1 . Using the Gram-Schmidt Process, we can extend v1 to an orthonormal basis
{v1 , v2 , . . . , vn } of !R n . Now we form the matrix
Q l = [v1 Vz vn J
Proof
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404
Chapter 5 Orthogonality
since vf (A 1 v1 ) = A 1 ( vfv1 ) = A 1 (v1 · v1 ) = A 1 and vf (A 1 v1 ) = A 1 ( vfv1 ) = A 1 (v; · v1 ) = 0
for i of- 1, because {v1 , v2 , , vn } is an orthonormal set.
But
•
.
.
so B is symmetric. Therefore, B has the block form
�
..-...
and A 1 is symmetric. Furthermore, B is similar to A (why?), so the characteristic poly­
nomial of B is equal to the characteristic polynomial of A, by Theorem 4.22. By
Exercise 39 in Section 4.3, the characteristic polynomial of A 1 divides the character­
istic polynomial of A. It follows that the eigenvalues of A 1 are also eigenvalues of A
and, hence, are real. We also see that A 1 has real entries. (Why?) Thus, A 1 is a k X k
real symmetric matrix with real eigenvalues, so the induction hypothesis applies to it.
Hence, there is an orthogonal matrix P2 such that PiA 1 P2 is a diagonal matrix-say,
D 1 . Now let
Then Q2 is an orthogonal (k + l)X(k + 1) matrix, and therefore so is Q = Q 1 Q2 .
Consequently,
Q rAQ = ( Q 1 Q 2 fA( Q 1 Q z ) = ( Q IQ f ) A ( Q 1 Q z ) = Q I( Q fAQ 1 )Q z = Q IB Q 2
2
which is a diagonal matrix. This completes the induction step, and we conclude that,
for all n 1, an n X n real symmetric matrix with real eigenvalues is orthogonally
diagonalizable.
Exa m p l e 5 . 1 8
Orthogonally diagonalize the matrix
This is the matrix from Example 5.17. We have already found that the
eigenspaces of A are
Solution
Section 5.4 Orthogonal Diagonalization of Symmetric Matrices
[ -�] r -il
n l [! ]
405
We need three orthonormal eigenvectors. First, we apply the Gram-Schmidt Process to
and
to obtain
and
The new wctm, which h., been rnmtructed to be orthogoml to
,._...
(why?) and '° ;, o,thogonal to
[
[:l
[ -�l
;, ''ill ;n E ,
Thu,, we haw th'" mutually mthogonal
]
vectors, and all we need to do is normalize them and construct a matrix Q with these
vectors as its columns. We find that
Q
=
1/ v3 - 1/ v2 - 1 / v6
l/ v3
0
2/ v6
v2
v3
l/
l/
- 1 / v6
and it is straightforward to verify that
Q 'AQ �
[� � �]
The Spectral Theorem allows us to write a real symmetric matrix A in the form
A = QDQ T, where Q is orthogonal and D is diagonal. The diagonal entries of D
are just the eigenvalues of A, and if the columns of Q are the orthonormal vectors
q1, . . . , qn , then, using the column-row representation of the product, we have
This is called the spectral decomposition of A. Each of the terms A;q; qT is a rank 1
matrix, by Exercise 62 in Section 3.5, and q;qT is actually the matrix of the projec­
tion onto the subspace spanned by q;. (See Exercise 2 5.) For this reason, the spectral
decomposition
is sometimes referred to as the projection form of the Spectral Theorem.
406
Chapter 5 Orthogonality
Exa m p l e 5 . 1 9
Find the spectral decomposition of the matrix A from Example 5 . 1 8.
Solulion
[ ]
From Example 5 . 1 8, we have:
qi =
Therefore,
l / VJ
l / VJ ,
l / VJ
q3 =
l / VJ] =
[
[ ]
- 1 / \/6
2/ \/6
- 1 / \/6
]
� �]
1 /3
1 /3
1 /3
1 /3 1 /3
1 /3 1 /3
1 /3 1 /3
1 /2
-1 2
[�
[
-1 2 0
so
=4
[ttt ttt tttl [
+
1 /2
1 /6 - 1 /3
2/3
- 1 /3
1 /6 - 1 /3
1 /6
- 1 /3
1 /6
]
! 0
0 0
-! 0
which can be easily verified.
In this example, ,\ 2 = ,\ 3 , so we could combine the last two terms A 2 q2 qi + ,\ 3 q3 qr
to get
The rank 2 matrix q2 qi + q3 qr is the matrix of a projection onto the two-dimensional
subspace (i.e., the plane) spanned by qi and q3 . (See Exercise 26.)
4
Observe that the spectral decomposition expresses a symmetric matrix A explic­
itly in terms of its eigenvalues and eigenvectors. This gives us a way of constructing a
matrix with given eigenvalues and (orthonormal) eigenvectors.
Exa m p l e 5 . 2 0
Find a 2 X 2 matrixwith eigenvalues ,\ 1 = 3 and ,\ 2 = - 2 and corresponding eigenvectors
Section 5.4 Orthogonal Diagonalization of Symmetric Matrices
401
We begin by normalizing the vectors to obtain an orthonormal basis
{q 1 , q2 }, with
Solution
Now, we compute the matrix A whose spectral decomposition is
A = A 1 q 1 q f + A 2 qz qi
[i] [ i]
!1
[ ] [ -�]
[-J �]
-2
[t � ]
=3
-
16
!s
= 3 1 2 2516 - 2 251 2
25
=
�
I
3. A =
5. A =
7. A =
9. A =
25
25
It is easy to check that A has the desired properties. (Do this.)
Exercises 5 . 4
Orthogonally diagonalize the matrices in Exercises 1 - 1 0
by finding a n orthogonal matrix Q and a diagonal
matrix D such that Q TAQ = D.
I. A =
25
l
[ -� t
[� :J
[ V21 V2]
[� : :]
u : -�]
O
2. A =
4. A =
6. A =
[-1 -1]
[9 ]
[ � ! �]
[� � ;]
3
-2
I 0. A =
[ ab ab ] .
[� � n
3
12. If b oF 0, orthogonally diogonolire A =
-2
13. Let A and B be orthogonally diagonalizable n X n
matrices and let c be a scalar. Use the Spectral
Theorem to prove that the following matrices are
orthogonally diagonalizable:
(c) A 2
(a) A + B
(b) cA
14. If A is an invertible matrix that is orthogonally diago­
nalizable, show that A- 1 is orthogonally diagonalizable.
15. If A and B are orthogonally diagonalizable and AB =
BA, show that AB is orthogonally diagonalizable.
16. If A is a symmetric matrix, show that every eigenvalue
of A is nonnegative if and only if A = B 2 for some
symmetric matrix B.
6
[ � � � ! ] [ � � � �:
8. A =
1 1 . If b * 0 , orthogonally diagonalize A =
In Exercises 1 7-20, find a spectral decomposition of the
matrix in the given exercise.
408
Chapter 5 Orthogonality
17. Exercise 1
19. Exercise 5
18. Exercise 2
20. Exercise 8
In Exercises 21 and 22, find a symmetric 2 X 2 matrix with
eigenvalues A 1 and A 2 and corresponding orthogonal
eigenvectors v1 and v2 •
2 1 . A 1 = - l , A 2 = 2 , v1 =
22. A 1 = 3, A 2 = - 3, v1 =
[�] [_�]
[�] [ -�]
, v2 =
·
, v2 =
In Exercises 23 and 24, find a symmetric 3 X 3 matrix with
eigenvalues A 1 , A 2 , and ,\ 3 and corresponding orthogonal
eigenvectors v1 , v2 , and v3 .
23 A, � 1 , A, � 2, A, � 3, v, �
25. Let q be a unit vector in !R n and let W be the subspace
spanned by q. Show that the orthogonal projection of a
vector v onto W (as defined in Sections 1 .2 and 5.2) is
given by
proj w ( v) = ( qq T) v
and that the matrix of this projection is thus qq T.
[Hint: Remember that, for x and y in !R n , x y = xTy.]
26. Let {q 1 , . . . , qd be an orthonormal set of vectors in !R n
and let W be the subspace spanned by this set.
(a) Show that the matrix of the orthogonal projection
onto W is given by
p = q 1 qf + . . . + qkql
[}, [ - : ] .
�
(b) Show that the projection matrix P in part (a) is
symmetric and satisfies P 2 = P.
(c) Let Q = [ q 1 · · · qk ] be the n X k matrix whose
columns are the orthonormal basis vectors of W.
Show that P = QQ T and deduce that rank(P) = k.
27. Let A be an n X n real matrix, all of whose eigenvalues
are real. Prove that there exist an orthogonal matrix Q
and an upper triangular matrix T such that Q TAQ = T.
This very useful result is known as Schur's Triangular­
ization Theorem. [Hint: Adapt the proof of the Spec­
tral Theorem.]
28. Let A be a nilpotent matrix (see Exercise 56 in Sec­
tion 4.2). Prove that there is an orthogonal matrix Q
such that Q T AQ is upper triangular with zeros on its
diagonal. [Hint: Use Exercise 27.]
A p p l icati o n s
Quad ratic Forms
An expression of the form
ax 2 + by 2 + cxy
is called a quadratic form in x and y. Similarly,
ax 2 + by 2 + cz 2 + dxy + exz + fyz
is a quadratic form in x, y, and z. In words, a quadratic form is a sum of terms, each of
which has total degree two in the variables. Therefore, 5 x 2 - 3y 2 + 2xy is a quadratic
form, but x 2 + y 2 + x is not.
We can represent quadratic forms using matrices as follows:
ax 2 + by 2 + cxy = [x y]
[ ; � J [;J
c 2
c 2
Section 5.5 Applications
and
ax 2 + by 2 + cz 2 + dxy + exz + fyz
�
=
[�
d/2 e/2
[x y z] d 2 b f/2
e/2 f/2 c
409
][ ]
x
y
z
(Verify these.) Each has the form xTAx, where the matrix A is symmetric. This obser­
vation leads us to the following general definition.
D e fi n it i o n
form
A
quadratic form in n variables is a function f : !R n ---+ IR of the
where A is a symmetric n X n matrix and x is in !R n . We refer to A as the matrix
associated with f
Exa m p l e 5 . 2 1
What is the quadratic form with associated matrix A
Solution
If x
=
[:J
=
[
]
2 -3
?
-3 5
then
Observe that the off-diagonal entries a1 2 a 2 1 -3 of A are combined to give
the coefficient - 6 of x1x2 • This is true generally. We can expand a quadratic form in
n variables xTAx as follows:
=
xTAx
=
=
a 11 x� + a 22x� + · · · + a nnx� + 2: 2a ;jxi xj
i<j
Thus, if i * j, the coefficient of X; Xj is 2a ij .
Exa m p l e 5 . 2 2
Find the matrix associated with the quadratic form
f (x 1 , x 2 , x3 )
=
2x � - xi + 5xf + 6x1 x 2 - 3x 1 x3
The coefficients of the squared terms x;2 go on the diagonal as a;;, and the
coefficients of the cross-product terms x;xj are split between a ij and aji· This gives
Solution
3
-1
0
-�]
410
Chapter 5 Orthogonality
so
f(x ,. x,, x, )
�
[x, x, x, J
as you can easily check.
[ _� - ! -�J[ ::l
In the case of a quadratic form f(x, y) in two variables, the graph of z = j(x, y) is
a surface in IR 3 . Some examples are shown in Figure 5 . 1 2 .
Observe that the effect o f holding x o r y constant i s t o take a cross section of
the graph parallel to the yz or xz planes, respectively. For the graphs in Figure 5 . 1 2,
all of these cross sections are easy to identify. For example, in Figure 5 . 1 2 (a), the
cross sections we get by holding x or y constant are all parabolas opening upward,
so f(x, y) 0 for all values of x and y. In Figure 5 . 1 2 (c), holding x constant gives
parabolas opening downward and holding y constant gives parabolas opening upward,
producing a saddle point.
2
z
z
x
x
(a) z = 2x2 + 3y2
y
(b)
z = - 2x2 - 3y2
z
z
x
y
y
(c) z = 2x2 - 3y2
Graphs of quadratic forms f
Figure 5 . 1 2
(x, y)
(d) z = 2x2
Section 5. 5 Applications
411
What makes this type of analysis quite easy is the fact that these quadratic forms
have no cross-product terms. The matrix associated with such a quadratic form is a
diagonal matrix. For example,
2x2 - 3y2 = [x y ]
[ � - � J [;J
In general, the matrix of a quadratic form is a symmetric matrix, and we saw in Sec­
tion 5.4 that such matrices can always be diagonalized. We will now use this fact to
show that, for every quadratic form, we can eliminate the cross-product terms by
means of a suitable change of variable.
Let f(x) = xTAx be a quadratic form in n variables, with A a symmetric n X n
matrix. By the Spectral Theorem, there is an orthogonal matrix Q that diagonalizes A;
that is, Q TAQ = D, where D is a diagonal matrix displaying the eigenvalues of A. We
now set
X = Qy Or, equivalently, y = Q - 1 X = Q TX
Substitution into the quadratic form yields
xTAx = (Q yfA(Q y)
= yTQ TAQy
= yTDy
which is a quadratic form without cross-product terms, since D is diagonal. Further­
more, if the eigenvalues of A are A1, . . . , A n , then Q can be chosen so that
If y = [y l
becomes
Yn ] T, then, with respect to these new variables, the quadratic form
yTDy = A 1Y12 + · · . + A ny;
This process is called diagonalizing a quadratic form. We have just proved the fol­
lowing theorem, known as the Principal Axes Theorem. (The reason for this name
will become clear in the next subsection.)
Theorem 5 . 2 1
The Principal Axes Theorem
Every quadratic form can be diagonalized. Specifically, if A is the n X n symmet­
ric matrix associated with the quadratic form xTAx and if Q is an orthogonal
matrix such that Q TAQ = D is a diagonal matrix, then the change of variable
x = Qy transforms the quadratic form xTAx into the quadratic form yTDy,
which has no cross-product terms. If the eigenvalues of A are A1,
, A n and
y = [y 1
Yn f , then
•
xTAx = yTDy = A 1y � + · · · + A ny;
.
.
412
Chapter 5 Orthogonality
Exa m p l e 5 . 2 3
5
Find a change o f variable that transforms the quadratic form
f (x1 , x2 ) = x f + 4x1 x2 + 2x�
into one with no cross-product terms.
Solulion
The matrix off is
A=
6 2/Vs1.
l /Vs]
[ - 2/Vs
- [ l /Vs]
6
/Vs
o
l
]
[ 2/Vs
[
2/Vs
1]
/Vs
l [:: ] [;:]
[ � �] [;:] 6
with eigenvalues A1 = and A 2 =
(Check this.) If we set
Corresponding unit eigenvectors are
and
qI
�
[� �]
q2 =
and D =
Q=
then Q TAQ = D. The change of variable x = Qy, where
x=
converts f into
0
and y =
f (y) = f(y 1 , yi } = [ Y1 Y2 l
= y f + Yi
The original quadratic form xTAx and the new one yTDy (referred to in the Princi­
pal Axes Theorem) are equal in the following sense. In Example 5.23, suppose we want
[ - � ].
- 1 5 - 1 - 1 2 11
1
/Vs
/Vs
l
l
]
]
]
[]
[
[
[ 2/Vs
2/Vs
/Vs
/Vs
l 6 6( 1/Vs)2 /Vs)2 55/5 11
to evaluate f( x) = xTAx at x =
j(
, 3) = (
We have
)2 + 4(
) ( 3) + ( 3)2 =
In terms of the new variables,
so
Y 1 = y = Q TX =
y2
f(yl , yi } = y f + y� =
3
+ (-7
=
=
-7
=
exactly as before.
The Principal Axes Theorem has some interesting and important consequences.
We will consider two of these. The first relates to the possible values that a quadratic
form can take on.
2.1.
5.
D e fi n i t i o n
A quadratic form f(x) = xTAx is classified as one of the following:
positive de.finite iff( x) > 0 for all x -=fa 0
positive semidefinite iff( x) ::=:: 0 for all x
3. negative de.finite ifj(x) < 0 for all x -=fa 0
4. negative semidefinite iff( x) :s 0 for all x
indefinite ifj(x) takes on both positive and negative values
Section 5. 5 Applications
413
A symmetric matrix A is called positive definite, positive semidefinite, nega­
tive definite, negative semidefinite, or indefinite if the associated quadratic form
f( x) = xTAx has the corresponding property.
The quadratic forms in parts (a), (b ), (c), and (d) of Figure 5 . 1 2 are positive definite,
negative definite, indefinite, and positive semidefinite, respectively. The Principal Axes
Theorem makes it easy to tell if a quadratic form has one of these properties.
Theorem 5 . 2 2
Let A be an n X n symmetric matrix. The quadratic form f( x) = xTAx is
a. positive definite if and only if all of the eigenvalues of A are positive.
b. positive semidefinite if and only if all of the eigenvalues of A are nonnegative.
c. negative definite if and only if all of the eigenvalues of A are negative.
d. negative semidefinite if and only if all of the eigenvalues of A are non positive.
e. indefinite if and only if A has both positive and negative eigenvalues.
You are asked to prove Theorem 5.22 in Exercise 27.
Exa m p l e 5 . 2 4
Classify f(x, y, z) = 3.x2 + 3y2 + 3z2 - 2xy - 2xz - 2yz as positive definite, negative
definite, indefinite, or none of these.
Solution
-�
-[ � = �
The matrix associated with f is
]
3
-1 -1
which has eigenvalues 1 , 4, and 4. (Verify this.) Since all of these eigenvalues are posi­
tive,f is a positive definite quadratic form.
If a quadratic form f( x) = xTAx is positive definite, then, since f(O) = 0, the
minimum value of f(x) is 0 and it occurs at the origin. Similarly, a negative definite
quadratic form has a maximum at the origin. Thus, Theorem 5.22 allows us to solve
certain types of maxima/minima problems easily, without resorting to calculus. A type
of problem that falls into this category is the constrained optimization problem.
It is often important to know the maximum or minimum values of a quadratic
form subject to certain constraints. (Such problems arise not only in mathematics
but also in statistics, physics, engineering, and economics.) We will be interested in
finding the extreme values of f( x) = xTAx subject to the constraint that I x ii = 1 .
In the case of a quadratic form in two variables, we can visualize what the problem
means. The graph of z = f(x, y) is a surface in IR 3 , and the constraint ll x ll = 1 restricts
the point (x, y) to the unit circle in the xy-plane. Thus, we are considering those
points that lie simultaneously on the surface and on the unit cylinder perpendicular
to the xy plane. These points form a curve lying on the surface, and we want the high­
est and lowest points on this curve. Figure 5 . 1 3 shows this situation for the quadratic
form and corresponding surface in Figure 5 . 1 2 (c).
414
Chapter 5 Orthogonality
z
y
The
intersection of 1
cylinder
Figure 5 . 1 3
x2 + y2 =
z =
2x2 - 3y2
with the
In this case, the maximum and minimum values off(x, y) = 2x 2 - 3y 2 (the high­
est and lowest points on the curve of intersection) are 2 and - 3, respectively, which
are just the eigenvalues of the associated matrix. Theorem 5.23 shows that this is
always the case.
Theorem 5 . 2 3
Let f(x) = xTAx be a quadratic form with associated n X n symmetric matrix A.
Let the eigenvalues of A be A 1 :::::: A 2 :::::: :::::: A w Then the following are true, subject
to the constraint II xii = 1 :
· · ·
a . A i 2: f(x) 2: A n
b. The maximum value of f(x) is A 1 , and it occurs when x is a unit eigenvector
corresponding to A 1 .
c. The minimum value of f(x) is A n , and it occurs when x is a unit eigenvector
corresponding to Aw
As usual, we begin by orthogonally diagonalizing A. Accordingly, let Q be an
orthogonal matrix such that Q TAQ is the diagonal matrix
Proof
Then, by the Principal Axes Theorem, the change of variable x = Qy gives xTAx =
yTDy. Now note that y = Q Tx implies that
since Q T = Q - 1 . Hence, using x x = xTx, we see that llrll = Wr = � =
ll x ll = 1 . Thus, if x is a unit vector, so is the corresponding y, and the values of xTAx
and yTDy are the same.
·
Section 5. 5 Applications
415
(a) To prove property (a), we observe that if y = [y 1 · · · Yn ] T, then
f ( x) = xTAx = yTD y
= A1Yl + A zyi + · · · + A ny;
:::; A 1y f + A 1Yi + · · · + A 1y;
= A 1 (y f + Yi + · · · + y; )
= A 1 llrll 2
= A1
Thus, f(x) s A 1 for all x such that ll x ll = The proof that f(x) 2: A n is similar.
(See Exercise 37.)
(b) If q 1 is a unit eigenvector corresponding to A 1 , then Aq 1 = A 1 q 1 and
1.
f ( q 1 ) = q fAq 1 = q fA 1 Q 1 = A 1 ( q fq 1 ) = A 1
This shows that the quadratic form actually takes on the value A 1 , and so, by prop­
erty (a), it is the maximum value off(x) and it occurs when x = q 1 .
(c) You are asked to prove this property in Exercise 38.
Exa m p l e 5 . 2 5
2xi
1,
Find the maximum and minimum values of the quadratic form f(x 1 , x 2 ) = 5x i +
4x 1 x2 +
subject to the constraint xi + xi = and determine values of x 1 and x 2
for which each of these occurs.
1,
In Example 5.23, we found thatf has the associated eigenvalues A 1 = 6 and
with corresponding unit eigenvectors
Solution
A2 =
-_ [ 2/Vs
1/Vs]
1 1/Vs
Qi
1/Vs]
-_ [ - 2/Vs
2/Vs 1/Vs.
- 2/Vs.
and Qz
Therefore, the maximum value off is 6 when x 1 =
and x2 =
The mini­
mum value off is when x 1 =
and x2 =
(Observe that these extreme
values occur twice-in opposite directions-since - q 1 and - q2 are also unit eigen­
vectors for A 1 and A 2 , respectively.)
Graphing Quad ratic Equations
The general form of a quadratic equation in two variables x and y is
ax 2 + by 2 + cxy + dx + ey + f = 0
where at least one of a, b, and c is nonzero. The graphs of such quadratic equations are
called conic sections (or conics), since they can be obtained by taking cross sections
of a (double) cone (i.e., slicing it with a plane) . The most important of the conic sec­
tions are the ellipses (with circles as a special case), hyperbolas, and parabolas. These
are called the nondegenerate conics. Figure 5 . 1 4 shows how they arise.
It is also possible for a cross section of a cone to result in a single point, a straight
line, or a pair of lines. These are called degenerate conics. (See Exercises 59-64.)
The graph of a nondegenerate conic is said to be in standard position relative to
the coordinate axes if its equation can be expressed in one of the forms in Figure 5 . 1 5.
416
Chapter 5 Orthogonality
Ellipse
Circle
Parabola
Hyperbola
The nondegenerate conics
Figure 5 . 1 4
x
2
y2
Ellipse or Circle: 2 + 2 = l; a , b > 0
a
b
y
y
y
b
a
-b
-a
a<b
a = b
b
-b
a>b
Hyperbola
y
y
b
-b
x2
a2
_
.i_ - 1 , a, b > 0
b2
_
r2
b
_
x2
- 1 , a, b > 0
a2
Parabola
y
y
y
y
2
y = ax , a > 0
2
y = ax , a < 0
x = ay 2 , a > 0
x = ay 2 , a < 0
Nondegenerate conics in standard position
Figure 5 . 1 5
Section 5. 5 Applications
Exa m p l e 5 . 2 6
411
If possible, write each of the following quadratic equations in the form of a conic in
standard position and identify the resulting graph.
(a) 4x 2 + 9y 2 = 36
(c) 4x 2 - 9y = 0
(b) 4x 2 - 9y 2 + 1 = 0
Solulion
(a) The equation 4x 2 + 9y 2 = 36 can be written in the form
x2 y z
+
= 1
9
4
-
-
so its graph is an ellipse intersecting the x-axis at ( ± 3, 0) and the y-axis at (O, ± 2) .
(b) Th e equation 4x2 - 9y2 + 1 = 0 can b e written i n the form
y2 x2
1- 1 = 1
9
4
so its graph is a hyperbola, opening up and down, intersecting the y-axis at (O, ± t) .
(c) Th e equation 4x2 - 9y = 0 can b e written i n the form
4
y = -x 2
9
so its graph is a parabola opening upward.
If a quadratic equation contains too many terms to be written in one of the forms
in Figure 5 . 1 5, then its graph is not in standard position. When there are additional
terms but no xy term, the graph of the conic has been translated out of standard
position.
Exa m p l e 5 . 2 1
Identify and graph the conic whose equation is
x 2 + 2y 2 - 6x + Sy + 9
Solulion
=
0
We begin by grouping the x and y terms separately to get
( x 2 - 6x) + ( 2y 2 + Sy) - 9
=
or
( x 2 - 6x) + 2 (y 2 + 4y)
=
-9
Next, we complete the squares on the two expressions in parentheses to obtain
( x 2 - 6x + 9 ) + 2 (y 2 + 4y + 4 ) = - 9 + 9 + S
or
( x - 3 ) 2 + 2 (y + 2 ) 2 = S
We now make the substitutions x' = x - 3 and y' = y + 2, turning the above equa­
tion into
( x ' ) 2 (y ' ) 2
( x' ) 2 + 2 (y' ) 2 = S or
+
= I
s
4
--
--
418
Chapter 5 Orthogonality
This is the equation of an ellipse in standard position in the x 'y ' coordinate system,
intersecting the x' -axis at ( ± 2 \/2, O) and the y'-axis at (O, ± 2) . The origin in the x'y'
coordinate system is at x = 3, y = - 2, so the ellipse has been translated out of stan­
dard position 3 units to the right and 2 units down. Its graph is shown in Figure 5 . 1 6.
y'
y
2
x
-2
-2
-
x
'
4
translated ellipse
Figure 5 . 1 6
A
If a quadratic equation contains a cross-product term, then it represents a conic
that has been rotated.
Exa m p l e 5 . 2 8
Identify and graph the conic whose equation is
5x 2 + 4xy + 2y 2 = 6
The left-hand side of the equation is a quadratic form, so we can write it in
matrix form as xTAx = 6, where
Solulion
A=
[� �]
In Example 5.23, we found that the eigenvalues of A are 6 and 1 , and a matrix Q that
orthogonally diagonalizes A is
Q=
.......
[
l / Vs
2/ Vs
l / Vs - 2 / Vs
]
Observe that <let Q = - 1 . In this example, we will interchange the columns o f this
matrix to make the determinant equal to + 1 . Then Q will be the matrix of a rotation,
by Exercise 28 in Section 5. 1 . It is always possible to rearrange the columns of an
orthogonal matrix Q to make its determinant equal to + 1 . (Why?) We set
Q=
instead, so that
[
l / Vs 2/ Vs
- 2/ Vs l / Vs
]
Section 5. 5 Applications
419
The change ofvariable x = Qx' converts the given equation into the form (x' ) TDx' = 6
by means of a rotation. If x' =
[; : ]
x'
, then this equation is just
(x') 2
(x') 2 + 6(y') 2 = 6 or
+ (y') 2 = 1
6
which represents an ellipse in the y' coordinate system.
y
3
To graph this ellipse, we need to know which vectors play the roles of e; =
[x' �]
and e� =
of the
in the new coordinate system. (These two vectors locate the positions
and y' axes.) But, from x = Qx' , we have
Qe; =
and
1
Qez =
x'
A rotated ellipse
Figure 5 . 1 1
[�]
/Vs
l
l /Vs 2/Vs
l
]
]
[ - 2/Vs
[
[
l /Vs - 2/Vs]
2/Vs
l /Vs 2/Vs
Q
]
]
[
[
[ - 2/Vs
l /Vs l /Vs]
0
1
=
These are just the columns q 1 and q2 o f Q , which are the eigenvectors o f A ! The fact
that these are orthonormal vectors agrees perfectly with the fact that the change of
variable is just a rotation. The graph is shown in Figure 5 . 1 7.
4
You can now see why the Principal Axes Theorem is so named. If a real symmet­
ric matrix A arises as the coefficient matrix of a quadratic equation, the eigenvectors
of A give the directions of the principal axes of the corresponding graph.
It is possible for the graph of a conic to be both rotated and translated out of stan­
dard position, as illustrated in Example 5.29.
Exa m p l e 5 . 2 9
5x 4xy 2 - -x
Vs28 - Vs4 4
4
[ � � ] [ - � -�]
5.28
/Vs 2/Vs
l- 2/Vs
]
[
/Vs
l
5.28,
x'
l /Vs 2/Vs
]
[ - Vs28 - Vs4 ] [ - 2/Vs
[
l /Vs ] - 4x' - 12y'
Identify and graph the conic whose equation is
2+
+ y2
-
y+ =0
Solution
The strategy is to eliminate the cross-product term first. In matrix form,
the equation is xTAx + Bx + = 0, where
and B =
A=
The cross-product term comes from the quadratic form xTAx, which we diagonalize
as in Example
by setting x = Qx' , where
Q=
Then, as in Example
xTAx = ( x' fDx' = (x') 2 + 6(y') 2
But now we also have
B x = BQx =
I
y'
=
420
Chapter 5 Orthogonality
y
\
2
-2
y'
-1
y"
x
-1
-2
�
-4
x
'
x
"
Figure 5 . 1 8
Thus, in terms of x' and y', the given equation becomes
(x') 2 + 6(y ') 2 - 4x' - 12y ' + 4 = 0
To bring the conic represented by this equation into standard position, we need
to translate the x'y' axes. We do so by completing the squares, as in Example 5.27.
We have
or
((x') 2 - 4x' + 4) + 6((y ') 2 - 2y ' + 1 ) = -4 + 4 + 6 = 6
(x' - 2) 2 + 6(y ' - 1) 2 = 6
This gives us the translation equations
x " = x' - 2 and y " = y ' In the x "y " coordinate system, the equation is simply
(x " ) 2 + 6(y ") 2 = 6
4
which is the equation of an ellipse (as in Example 5.28). We can sketch this ellipse by
first rotating and then translating. The resulting graph is shown in Figure 5.18.
The general form of a quadratic equation in three variables x, y, and z is
ax 2 + by 2 + cz 2 + dxy + exz + fyz + gx + hy + iz + j = 0
where at least one of a, b, . . . , f is nonzero. The graph of such a quadratic equation
is called a quadric surface (or quadric) . Once again, to recognize a quadric we need
Section 5. 5 Applications
x--,,2 y2 z2
c
z
x2 y2 z2
c
z
Ellipsoid: :::2 + b2 + :::2 = 1
a
Hyperboloid of one sheet:
a "'
+ b2
-
421
:::2 = I
y
x
x2 -+--ty22 -z22
c
Hyperboloid of two sheets : -;;,
z
y
x
Elliptic paraboloid:
z
x
Quadric surfaces
Figure 5 . 1 9
=
z x-;;,2 yb22
=
-
Elliptic cone:
I
z
z2 x-;;,2 y2
y
x
Hyperbolic paraboloid:
+
z
y
+ b2
=
z x2 y2
=
-;;, - p
422
Chapter 5 Orthogonality
t o put it into standard position. Some quadrics i n standard position are shown in
Figure 5.19; others are obtained by permuting the variables.
Exa m p l e 5 . 3 0
Identify the quadric surface whose equation is
5x 2 + l ly 2 + 2z 2 + 16xy + 20xz - 4yz
Solulion
]
=
36
The equation can be written in matrix form as xTAx = 36, where
8 10
1 1 -2
-2 2
We find the eigenvalues of A to be 18, 9, and - 9, with corresponding orthogonal
eigenvectors
respectively. We normalize them to obtain
and form the orthogonal matrix
=
[ ! ! =l l
=
-
Note that in order for Q to be the matrix of a rotation, we require <let Q 1 , which
is true in this case. (Otherwise, <let Q - 1 , and swapping two columns changes the
sign of the determinant.) Therefore,
and, with the change of variable x = Qx', we get xTAx = (x' )Dx' = 36, so
18(x') 2 + 9(y') 2 - 9(z') 2
=
(x') 2 (y ') 2
+
36 or
2
4
--
--
-
(z') 2
4
--
=
1
From Figure 5.19, we recognize this equation as the equation of a hyperboloid of one
sheet. The x', y', and z' axes are in the directions of the eigenvectors q 1, q2 , and q3 ,
respectively. The graph is shown in Figure 5.20.
Section 5. 5 Applications
423
z
z'
Anonstandard
hyperboloidposition
of one sheet in
Figure 5.20
I
We can also identify and graph quadrics that have been translated out of standard
position using the "complete-the-squares method" of Examples 5.27 and 5.29. You
will be asked to do so in the exercises.
Exercises 5 . 5
Q u a d ratic F o r m s
In Exercises 1 -6, evaluate the quadratic form f(x) = xTAx
for the given A and x.
1. A =
2. A =
3. A =
4. A =
5. A =
6. A =
[ � ! l [;]
[ � � l [::]
[ _ � ! l- [ ! ]
u -} � [�]
}
�
:]
[
u
[ : :Jx � m
x=
x=
-
0
2
0
2
2
0
x=
In Exercises 7-12, find the symmetric matrix A associated
with the given quadratic form.
7. xl + 2xf + 6x 1 x2
8. X1 X2
9. 3x 2 - 3xy - y2
10. x � - x � + 8X 1 X2 - 6XzX3
1 1 . 5xl - xi + 2x� + 2x 1 x2 - 4x 1 x3 + 4x2x3
12. 2x 2 3y 2 + z 2 - 4xz
-
Diagonalize the quadratic forms in Exercises 13-18 by
finding an orthogonal matrix Q such that the change of
variable x = Qy transforms the given form into one with no
cross-product terms. Give Q and the new quadratic form.
13. 2x l + 5x� 4x 1 x2
14. x 2 + 8xy + y 2
15. 7xl + x i + xj + 8X 1 X2 + 8X 1 X3 - 16XzX3
16. x l + x i + 3xj - 4x 1 x2
17. x 2 + z 2 - 2xy + 2yz
18. 2xy + 2xz + 2yz
-
424
Chapter 5 Orthogonality
Classify each of the quadratic forms in Exercises 1 9-26 as
positive definite, positive semidefinite, negative definite,
negative semidefinite, or indefinite.
20. x f + xi - 2x, x2
19. x f + 2x}
2
2
21. - 2x - 2y + 2xy 22. x 2 + y 2 + 4xy
23. 2x f + 2x i + 2x� + 2x1 x2 + 2x1 x3 + 2x2x3
24. x f + x i + x� + 2x1 x3 25. x i + x � - x � + 4x 1 x2
26. -x 2 - y 2 - z 2 - 2xy - 2xz - 2yz
27. Prove Theorem 5.22.
28. Let A =
[ � �]
be a symmetric 2 X 2 matrix. Prove
that A is positive definite if and only if a > 0 and
<let A > 0. [Hint: ax 2 + 2bxy + dy 2 =
( � y (d - �2 )y 2.]
a x+ y
+
29. Let B be an invertible matrix. Show that A = B TB is
positive definite.
30. Let A be a positive definite symmetric matrix. Show
that there exists an invertible matrix B such that A =
B TB . [Hint: Use the Spectral Theorem to write A =
QDQ T. Then show that D can be factored as c Tc for
some invertible matrix C.]
31. Let A and B be positive definite symmetric n X n
matrices and let c be a positive scalar. Show that the
following matrices are positive definite.
(a) cA
(b) A 2
(c) A + B
_
,
(d) A (First show that A is necessarily invertible.)
32. Let A be a positive definite symmetric matrix. Show
that there is a positive definite symmetric matrix B
such that A = B 2 • (Such a matrix B is called a square
root of A.)
In Exercises 33-36, find the maximum and minimum val­
ues of the quadratic form f(x) in the given exercise, subject
to the constraint ll x ll = 1, and determine the values of x for
which these occur.
33. Exercise 20
34. Exercise 22
35. Exercise 23
36. Exercise 24
37. Finish the proof of Theorem 5.23 (a) .
38. Prove Theorem 5.23 ( c).
G r a p h i n g Q u a d ratic E q u a t i o n s
In Exercises 39-44, identify the graph of the given equation.
39. x 2 + 5y 2 = 25
40. x 2 - y 2 - 4 = 0
41. x 2 - y - 1 = 0
42. 2x 2 + y 2 - 8 = 0
44. x = - 2y 2
43. 3x 2 = y 2 - 1
In Exercises 45-50, use a translation of axes to put the conic
in standard position. Identify the graph, give its equation in
the translated coordinate system, and sketch the curve.
45. x 2 + y 2 - 4x - 4y + 4 = 0
46. 4x 2 + 2y 2 - Sx + l2y + 6 = 0
47. 9x 2 - 4y 2 - 4y = 37 48. x 2 + lOx - 3y = - 13
49. 2y 2 + 4x + Sy = 0
50. 2y 2 - 3x 2 - 1 8x - 20y + 1 1 = 0
In Exercises 51 -54, use a rotation of axes to put the conic in
standard position. Identify the graph, give its equation in the
rotated coordinate system, and sketch the curve.
51. x 2 + xy + y 2 = 6 52. 4x 2 + lOxy + 4y 2 = 9
53. 4x 2 + 6xy - 4y 2 = 5 54. 3x 2 - 2xy + 3y 2 = 8
In Exercises 55-58, identify the conic with the given equa­
tion and give its equation in standard form.
55. 3x 2 - 4xy + 3y 2 - 28 v'2x + 22 Vly + 84 = 0
56. 6x 2 - 4xy + 9y 2 - 20x - lOy - 5 = 0
57. 2xy + 2 \/2x - 1 = 0
58. x 2 - 2xy + y 2 + 4 V2x - 4 = 0
Sometimes the graph of a quadratic equation is a straight
line, a pair of straight lines, or a single point. We refer to
such a graph as a degenerate conic. It is also possible that
the equation is not satisfied for any values of the variables,
in which case there is no graph at all and we refer to the
conic as an imaginary conic. In Exercises 59-64, identify
the conic with the given equation as either degenerate or
imaginary and, where possible, sketch the graph.
59. x 2 - y 2 = 0
60. x 2 + 2y 2 + 2 = 0
62. x 2 + 2xy + y 2 = 0
61. 3x 2 + y 2 = 0
63. x 2 - 2xy + y 2 + 2 Vlx - 2 Vly = 0
64. 2x 2 + 2xy + 2y 2 + 2 \/2x - 2 Vly + 6 = 0
65. Let A be a symmetric 2 X 2 matrix and let k be a
scalar. Prove that the graph of the quadratic equation
xTA x = k is
(a) a hyperbola if k * 0 and <let A < 0
(b) an ellipse, circle, or imaginary conic if k * 0 and
det A > 0
(c) a pair of straight lines or an imaginary conic if
k * 0 and <let A = 0
(d) a pair of straight lines or a single point if k = 0
and det A * 0
(e) a straight line if k = 0 and <let A = 0
[Hint: Use the Principal Axes Theorem.]
Chapter Review
In Exercises 66-73, identify the quadric with the given
equation and give its equation in standard form.
66. 4x 2 + 4y 2 + 4z 2 + 4xy + 4xz + 4yz = 8
67. x 2 + y 2 + z 2 - 4yz = 1
68. -x 2 - y 2 - z 2 + 4xy + 4xz + 4yz = 12
69. 2xy + z = 0
70. 16x 2 + 100y 2 + 9 z 2 - 24xz - 60x - 80z = 0
71. x 2 + y 2 - 2z 2 + 4xy - 2xz + 2yz - x + y + z = 0
72. 10x 2 + 25y 2 + 10z 2 - 40xz + 20 \/2x + soy +
20 \/2z = 1 5
73. l lx 2 + l ly 2 + 14z 2 + 2xy + 8xz - 8yz - 12x +
12y + 12z = 6
425
74. Let A be a real 2 X 2 matrix with complex eigenvalues
A = a :±: bi such that b =F 0 and I A I = 1 . Prove that
every trajectory of the dynamical system xk + i = Axk
lies on an ellipse. [Hint: Theorem 4.43 shows that if v
is an eigenvector corresponding to A = a - bi, then
the matrix P = [ Re v Im v] is invertible and
- -1
A=P
P . Set B = (PPT) - 1 . Show that the
[: �J
quadratic xTBx = k defines an ellipse for all k > 0,
and prove that if x lies on this ellipse, so does Ax.]
Chapter Review
Kev Defi nitions and Concepts
fundamental subspaces
of a matrix, 380
Gram-Schmidt Process, 389
orthogonal basis, 370
orthogonal complement
of a subspace, 378
orthogonal matrix, 374
orthogonal projection, 382
orthogonal set of vectors, 369
Orthogonal Decomposition
Theorem, 384
orthogonally diagonalizable
matrix, 400
orthonormal basis, 372
Review Questions
1 . Mark each of the following statements true or false:
(a) Every orthonormal set of vectors is linearly
independent.
(b) Every nonzero subspace of u;g n has an orthogonal
basis.
(c) If A is a square matrix with orthonormal rows,
then A is an orthogonal matrix.
(d) Every orthogonal matrix is invertible.
(e) If A is a matrix with det A = 1 , then A is an
orthogonal matrix.
(f) If A is an m X n matrix such that (row(A))_j_ = u;g n ,
then A must be the zero matrix.
(g) If W is a subspace of u;g n and v is a vector in u;g n such
that projw(v) = 0, then v must be the zero vector.
(h) If A is a symmetric, orthogonal matrix, then A 2 = I.
(i) Every orthogonally diagonalizable matrix is invertible.
orthonormal set of vectors, 372
properties of orthogonal
matrices, 374-376
QR factorization, 393
Rank Theorem, 386
spectral decomposition, 405
Spectral Theorem, 403
(j) Given any n real numbers A 1 , , A n , there exists
a symmetric n X n matrix with A 1 , , A n as its
eigenvalues.
2. Find all values of a and b such that
.
•
.
.
\ [H [ J [fl )
.
i< an mthogonal <et of vedm.
3. Find the coordinate vector [ v] of v =
respect to the orthogonal basis
B
•
8
[ -�]
� ml [ J [ - � l) or n'
2
with
Chapter 5 Orthogonality
426
4. The coordinate vector of a vector v with respect to an
-3
orthonormal basis B = {v1 , v2 } of lR 2 is [v] 8 =
.
1/2
3/5
If v1 =
, find all possible vectors v.
4/ 5
[ ]
[ ]
[ - �j�
[ � :]
5. Show that
�
]
1 2
- [ ! J · � - [ iJ· � - [ rJ
A- [ ! � ]
m [J
- ml· - }
A [ � �l
A. A.
15. ( a) Apply the Gram-Schmidt Process to
2
is an
4/7 Vs - 1 5/7 Vs 2/7 Vs
orthogonal matrix.
6. If
��
2 7
with respect to
�
is an orthogonal matrix, find all possible
values of a, b, and c.
7. If Q is an orthogonal n X n matrix and {v1 , , vk } is
an orthonormal set in !R n , prove that { Q v1 , , Q vk }
is an orthonormal set.
8. If Q is an n X n matrix such that the angles
L ( Q x, Q y) and L ( x, y) are equal for all vectors x and
y in IR " , prove that Q is an orthogonal matrix.
•
•
•
.
•
.
In Questions 9-12, find a basis for W _j_ .
9. W is the line in IR 2 with general equation
2x - Sy = 0
10. W is the line in IR 3 with parametric equations
x=t
y = 2t
z = -t
to find an orthogonal basis for W = span{x 1 , x2 , xJ.
(b) Use the result of part (a) to find a QR factorization
of
16. Find an orthogonal basis for IR 4 that contains the
vecto"
'" d
17. Find an orthogonal basis for the subspace
w
18. Let
+ x, + x, + x.
=
2
-1 1
0 om'
-
·
2
( a) Orthogonally diagonalize
(b) Give the spectral decomposition of
A [ � � -� � �]
-
-
13. Find bases for each of the four fundamental subspaces of
=
1
2
3 -5
4
8
6 -1
14. Find the orthogonal decomposition of
v
- [ -�]
9
7
19. Find a symmetric matrix with eigenvalues ,\ 1 = ,\ 2 = 1,
A 3 = - 2 and eigenspaces
20. If {v1 , v2 ,
•
prove that
ues c 1 , c2 ,
•
A
.
•
.
•
, v" } is an orthonormal basis for !R n and
is a symmetric matrix with eigenval­
, e n and corresponding eigenvectors
Vector Spaces
6.0
Algebra is generous; she often gives
more than is asked of her.
-Jean le Rond d'Alembert
In Carl B. Boyer
Wiley,
( 1 7 1 7- 1 783)
A History of Mathematics
1 968, p. 48 1
I n t ro d u ctio n : Fib o n acci i n (Vecto r> S p a c e
The Fibonacci sequence was introduced in Section 4.6. It i s the sequence
0, 1, 1, 2, 3, 5, 8, 13, . . .
of nonnegative integers with the property that after the first two terms, each term
is the sum of the two terms preceding it. Thus 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3,
2 + 3 = 5, and so on.
If we denote the terms of the Fibonacci sequence by f0,f1,f2 ,
, then the entire
sequence is completely determined by specifying that
Jo = O,f1 = 1 and fn = fn - l + fn - l for n 2
By analogy with vector notation, let's write a sequence x0, x1, x2 , x3 , . . . as
x = [xo , X1, X2 , X3 , . . . )
The Fibonacci sequence then becomes
f = [f0, J1, J2 ,f3 ,
) = [O, 1, 1, 2, . . . )
We now generalize this notion.
.
2
•
.
•
•
.
2
A Fibonacci-type sequence is any sequence x [x0, x1, x 2 , x 3 ,
such that x0 and x 1 are real numbers and Xn = xn - l + Xn - z for n 2.
D e fi n it i o n
=
•
.
•
)
For example, [1, \/2, 1 + \/2, 1 + 2 \/2, 2 + 3 \/2, . . . ) is a Fibonacci-type sequence.
Write down the first five terms of three more Fibonacci-type sequences.
By analogy with vectors again, let's define the sum of two sequences x = [x0, x1,
x2 , ) and y = [y0, y 1, y2 , ) to be the sequence
Problem 1
•
.
.
•
.
•
If c is a scalar, we can likewise define the scalar multiple of a sequence by
ex = [ cx0 , cx 1 , cx2 ,
•
•
•
)
421
428
Chapter Vector Spaces
6
(a) Using your examples from Problem 1 or other examples, compute
the sums of various pairs of Fibonacci-type sequences. Do the re­
sulting sequences appear to be Fibonacci-type?
(b) Compute various scalar multiples of your Fibonacci-type se­
quences from Problem 1. Do the resulting sequences appear to be
Fibonacci-type?
Problem 3 (a) Prove that if x and y are Fibonacci-type sequences, then so is x + y.
(b) Prove that if x is a Fibonacci-type sequence and c is a scalar, then
ex is also a Fibonacci-type sequence.
Let's denote the set of all Fibonacci-type sequences by Fib. Problem 3 shows that,
like 11;r, Fib is closed under addition and scalar multiplication. The next exercises
show that Fib has much more in common with 11;r.
Problem 4 Review the algebraic properties of vectors in Theorem 1 . 1 . Does Fib
satisfy all of these properties? What Fibonacci-type sequence plays the role of O? For a
Fibonacci-type sequence x, what is - x? Is - x also a Fibonacci-type sequence?
n
Problem 5 In !R , we have the standard basis vectors e1, e 2 ,
, e n - The Fibonacci
sequence f = [O, 1, 1, 2, . . . ) can be thought of as the analogue of e2 because its first
two terms are 0 and 1 . What sequence e in Fib plays the role of e1?
What about e3 , e4, ? Do these vectors have analogues in Fib?
Problem 6 Let x = [ x0 , x1, x2 ,
) be a Fibonacci-type sequence. Show that x is a
linear combination of e and f.
Problem 1 Show that e and f are linearly independent. (That is, show that if
ce + df = 0, then c = d = 0.)
Problem 8 Given your answers to Problems 6 and 7, what would be a sensible
value to assign to the "dimension" of Fib ? Why?
Problem 9 Are there any geometric sequences in Fib? That is, if
2
[ 1 , r, r , r 3 , . . . )
is a Fibonacci-type sequence, what are the possible values of r?
Problem 10 Find a "basis" for Fib consisting of geometric Fibonacci-type sequences.
Problem 11 Using your answer to Problem 10, give an alternative derivation of
Binet's formula [formula ( 5) in Section 4.6] :
Problem 2
•
.
•
.
•
•
.
•
.
)
(
(
)
1_ 1 Vs n
1 1 + Vs n
Vs
2
Vs
2
for the terms of the Fibonacci sequence f = [f0 , f1 , f2 , ) . [Hint: Express f in terms of
j,
n=
The
LucasLucas
sequence
is named after
Edouard
(see page
336).
_...
__
_
_
-
.
•
.
the basis from Problem 10.]
The Lucas sequence is the Fibonacci-type sequence
l = [ 10 , 11 , 12 , 13 , ) = [2, 1, 3, 4, . . . )
Problem 12 Use the basis from Problem 1 0 to find an analogue of Binet's formula
for the nth term Zn of the Lucas sequence.
Problem 13 Prove that the Fibonacci and Lucas sequences are related by the identity
fn - 1 + fn + I = Zn for n 2: 1
[Hint: The Fibonacci-type sequences C = [ 1 , 1, 2, 3, . . . ) and f + = [ 1 , 0, 1, 1, . . . )
form a basis for Fib. (Why?) ]
I n this Introduction, we have seen that the collection Fib o f all Fibonacci-type
sequences behaves in many respects like IR 2 , even though the "vectors" are actually
infinite sequences. This useful analogy leads to the general notion of a vector space
that is the subject of this chapter.
.
•
.
Section Vector Spaces and Subspaces
6. 1
429
Vecto r S p a ces a n d S u b s p aces
I n Chapters 1 and 3, we saw that the algebra o f vectors and the algebra o f matrices
are similar in many respects. In particular, we can add both vectors and matrices,
and we can multiply both by scalars. The properties that result from these two opera­
tions (Theorem 1 . 1 and Theorem 3.2) are identical in both settings. In this section,
we use these properties to define generalized "vectors" that arise in a wide variety
of examples. By proving general theorems about these "vectors:' we will therefore
simultaneously be proving results about all of these examples. This is the real power
of algebra: its ability to take properties from a concrete setting, like !R n , and abstract
them into a general setting.
Let V be a set on which two operations, called addition and scalar
multiplication, have been defined. If u and v are in V, the sum of u and v is denoted
by u + v, and if c is a scalar, the scalar multiple of u by c is denoted by cu. If the
following axioms hold for all u, v, and w in V and for all scalars c and d, then V is
D e fi n it i o n
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called a vector space and its elements are called vectors.
Closure under addition
Commutativityvity
Associati
Closure under scalar multiplication
Distributi
Distributivviittyy
1 . u + v is in V.
2. u + v = v + u
3. (u + v) + w = u + (v + w)
4. There exists an element 0 in V, called a zero vector, such that u + 0 = u.
5. For each u in V, there is an element - u in V such that u + ( - u) = 0.
6. cu is in V.
7. C ( U + V) = CU + CV
8. (c + d)u = cu + du
9. c (du) = (cd) u
10. lu = u
( 1 858- 1 932) .
Calcolo Geometrico,
U, n,
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1 9 1 8,
( 1 88 5 - 1 955)
Space, Time, Matter,
R e m a rks
By "scalars" we will usually mean the real numbers. Accordingly, we should
refer to V as a real vector space (or a vector space over the real numbers). It is also pos­
sible for scalars to be complex numbers or to belong to "ll.P , where p is prime. In these
cases, V is called a complex vector space or a vector space over "ll.P , respectively. Most of
our examples will be real vector spaces, so we will usually omit the adjective "real:' If
something is referred to as a "vector space:' assume that we are working over the real
number system.
In fact, the scalars can be chosen from any number system in which, roughly
speaking, we can add, subtract, multiply, and divide according to the usual laws of
arithmetic. In abstract algebra, such a number system is called a field.
•
The definition of a vector space does not specify what the set V consists
of. Neither does it specify what the operations called "addition" and "scalar multi­
plication'' look like. Often, they will be familiar, but they need not be. See Example 6.6
and Exercises 5-7.
•
We will now look at several examples of vector spaces. In each case, we need to
specify the set V and the operations of addition and scalar multiplication and to verify
axioms 1 through 10. We need to pay particular attention to axioms 1 and 6 (closure),
430
Chapter Vector Spaces
6
axiom 4 (the existence of a zero vector in V), and axiom 5 (each vector in V must have
a negative in V).
Exa m p l e 6 . 1
For any n 2: 1 , !R n is a vector space with the usual operations of addition and scalar
multiplication. Axioms 1 and 6 follow from the definitions of these operations, and
the remaining axioms follow from Theorem 1 . 1 .
Exa m p l e 6 . 2
The set of all 2 X 3 matrices is a vector space with the usual operations of matrix
addition and matrix scalar multiplication. Here the "vectors" are actually matrices.
We know that the sum of two 2 X 3 matrices is also a 2 X 3 matrix and that multiply­
ing a 2 X 3 matrix by a scalar gives another 2 X 3 matrix; hence, we have closure.
The remaining axioms follow from Theorem 3.2. In particular, the zero vector 0 is the
2 X 3 zero matrix, and the negative of a 2 X 3 matrix A is just the 2 X 3 matrix -A.
There is nothing special about 2 X 3 matrices. For any positive integers m and n ,
the set of all m X n matrices forms a vector space with the usual operations of matrix
addition and matrix scalar multiplication. This vector space is denoted Mm n ·
Exa m p l e 6 . 3
Let <;IP 2 denote the set of all polynomials of degree 2 or less with real coefficients.
Define addition and scalar multiplication in the usual way. (See Appendix D.) If
p(x)
=
a 0 + a 1 x + a 2x 2 and q(x)
=
b0 + b 1 x + b2x 2
are in <!P 2 , then
has degree at most 2 and so is in <!P 2 • If c is a scalar, then
cp(x)
=
ca0 + ca 1 x + ca 2x 2
is also in <!P 2 . This verifies axioms 1 and 6.
The zero vector 0 is the zero polynomial-that is, the polynomial all of whose
coefficients are zero. The negative of a polynomial p(x) = a0 + a1x + a 2x 2 is the poly­
nomial -p ( x) = - a0 - a1x - a 2x 2 • It is now easy to verify the remaining axioms. We
will check axiom 2 and leave the others for Exercise 12. With p(x) and q(x) as above,
we have
p (x) + q(x)
=
=
=
=
=
(a0 + a 1 x + a 2x 2 ) + (b0 + b 1 x + b2x 2 )
(a0 + b0) + (a 1 + b 1 )x + (a 2 + b2 )x2
( b0 + a0) + (b 1 + a 1 )x + (b2 + a 2 )x2
( b0 + b 1 x + b2x 2 ) + (a0 + a 1 x + a 2x 2 )
q(x) + p(x)
where the third equality follows from the fact that addition of real numbers is
commutative.
Section Vector Spaces and Subspaces
6. 1
431
In general, for any fixed n 2: 0, the set <!J' n of all polynomials of degree less than or
equal to n is a vector space, as is the set <!J' of all polynomials.
Exa m p l e 6 . 4
Let ?Jf denote the set of all real-valued functions defined on the real line. Iff and g are
two such functions and c is a scalar, then f + g and cf are defined by
(f + g) (x)
=
f (x) + g(x) and (cj) (x)
=
cf (x)
In other words, the value off + g at x is obtained by adding together the values off
and g at x [Figure 6.1 (a) J . Similarly, the value of cf at x is just the value off at x mul­
tiplied by the scalar c [Figure 6. 1 (b) J . The zero vector in ?Jf is the constant function
f0 that is identically zero; that is, f0 (x) 0 for all x. The negative of a functionf is the
function -f defined by ( -j) (x) -f(x) [Figure 6.l (c) ] .
Axioms 1 and 6 are obviously true. Verification o f the remaining axioms is left as
Exercise 1 3 . Thus, ?Jf is a vector space.
=
=
y
y
(x, 2/(x))
\
I
(x, f(x) + g (x))
I
2f
(x, - 3/(x))
(b)
(a)
y
/-f(x))
-f
(x,
(c)
The graphs of (a) f, and f (b) f, 2f, and -3f, and (c) f and -f
Figure 6 . 1
g,
+ g,
432
Chapter Vector Spaces
6
In Example 6.4, we could also have considered only those functions defined on
some closed interval [a, b] of the real line. This approach also produces a vector space,
denoted by ?f [a, b] .
Exa m p l e 6 . 5
Exa m p l e 6 . 6
The set "1l_ of integers with the usual operations is not a vector space. To demon­
strate this, it is enough to find that one of the ten axioms fails and to give a specific
instance in which it fails (a counterexample). In this case, we find that we do not have
closure under scalar multiplication. For example, the multiple of the integer 2 by the
scalar t is (t)(2) = t , which is not an integer. Thus, it is not true that ex is in "1l_ for
every x in "1l_ and every scalar c (i.e., axiom 6 fails).
Let V = IR 2 with the usual definition of addition but the following definition of scalar
multiplication:
Then, for example,
so axiom 10 fails. [In fact, the other nine axioms are all true (check this), but we do
not need to look into them, because V has already failed to be a vector space. This
example shows the value of looking ahead, rather than working through the list of
axioms in the order in which they have been given.]
Exa m p l e 6 . 1
Let C 2 denote the set of all ordered pairs of complex numbers. Define addition and
scalar multiplication as in IR 2 , except here the scalars are complex numbers. For
example,
] [ ] [ - 2-+31� i ]
[ 2l -+ 3ii ] = [ ((ll -- i)i)((2l -+ 3i)i) ] = [ - 1 2- Si ]
( 1 - i)
[
and
- 3 + 2i
1+i
+
.
4
2 - 31
6
Using properties of the complex numbers, it is straightforward to check that all ten
axioms hold. Therefore, C 2 is a complex vector space.
In general, e n is a complex vector space for all n
Exa m p l e 6 . 8
2 1.
4
If p is prime, the set z; (with the usual definitions of addition and multiplication by
scalars from "ll_p ) is a vector space over ZP for all n 2: 1 .
4
Section Vector Spaces and Subspaces
6. 1
433
Before we consider further examples, we state a theorem that contains some use­
ful properties of vector spaces. It is important to note that, by proving this theorem
for vector spaces in general, we are actually proving it for every specific vector space.
Theorem 6 . 1
Let V be a vector space, u a vector in V, and c a scalar.
a. Ou = 0
b. co = 0
c. ( - l)u = -u
d . I f cu = 0, then c = 0 o r u = 0.
Proof We prove properties (b) and ( d) and leave the proofs of the remaining proper­
ties as exercises.
(b) We have
cO = c(O + 0) = cO + cO
by vector space axioms 4 and 7. Adding the negative of cO to both sides produces
co + ( -co) = (co + co) + ( -co)
which implies
by axioms and 3
by axiom
by axiom
5
o = co + (co + ( -co))
5
= co + 0
4
= co
(d) Suppose cu = 0. To show that either c = 0 or u = 0, let's assume that c * 0. (If
c = 0, there is nothing to prove.) Then, since c * 0, its reciprocal l /c is defined, and
u = lu
1
-( cu)
c
1
-0
c
0
by axiom
10
by axiom
9
by property (b)
We will write u - v for u + ( -v), thereby defining subtraction of vectors. We will
also exploit the associativity property of addition to unambiguously write u + v + w
for the sum of three vectors and, more generally,
for a linear combination of vectors.
Subspaces
We have seen that, in !R n , it is possible for one vector space to sit inside another one,
giving rise to the notion of a subspace. For example, a plane through the origin is a
subspace of IR 3 . We now extend this concept to general vector spaces.
434
Chapter Vector Spaces
6
A subset W of a vector space V is called a subspace of V if W is
itself a vector space with the same scalars, addition, and scalar multiplication as V.
D e fi n i t i o n
As in !R n , checking to see whether a subset W of a vector space V is a subspace of
V involves testing only two of the ten vector space axioms. We prove this observation
as a theorem.
Theorem 6 . 2
Let V be a vector space and let W be a nonempty subset of V. Then W is a sub­
space of V if and only if the following conditions hold:
a. If u and v are in W, then u + v is in W.
b. If u is in W and c is a scalar, then cu is in W.
Assume that W is a subspace of V. Then W satisfies vector space axioms 1 to
10. In particular, axiom 1 is condition (a) and axiom 6 is condition (b) .
Conversely, assume that W is a subset of a vector space V, satisfying condi­
tions (a) and (b) . By hypothesis, axioms 1 and 6 hold. Axioms 2, 3, 7, 8, 9, and 10 hold
in W because they are true for all vectors in V and thus are true in particular for those
vectors in W. (We say that W inherits these properties from V.) This leaves axioms 4
and 5 to be checked.
Since W is nonempty, it contains at least one vector u. Then condition (b) and
Theorem 6. l (a) imply that Ou = 0 is also in W. This is axiom 4.
If u is in V, then, by taking c = - 1 in condition (b ), we have that -u = ( - l)u is
also in W, using Theorem 6.l (c) .
Proof
Remark
Since Theorem 6.2 generalizes the notion of a subspace from the con­
text of !R n to general vector spaces, all of the subspaces of !R n that we encountered in
Chapter 3 are subspaces of !R n in the current context. In particular, lines and planes
through the origin are subspaces of IR 3 .
Exa m p l e 6 . 9
Exa m p l e 6 . 1 0
We have already shown that the set <!/' n of all polynomials with degree at most n is a
vector space. Hence, <!/' n is a subspace of the vector space <!/' of all polynomials.
4
Let W be the set of symmetric n X n matrices. Show that W is a subspace of Mnn -
Clearly, W is nonempty, so we need only check conditions (a) and (b) in
Theorem 6.2. Let A and B be in W and let c be a scalar. Then A T = A and B T = B,
from which it follows that
Solulion
Therefore, A + B is symmetric and, hence, is in W. Similarly,
(cAf = cA T = cA
so cA is symmetric and, thus, is in W. We have shown that W is closed under addition
and scalar multiplication. Therefore, it is a subspace of Mnn ' by Theorem 6.2.
4
Section Vector Spaces and Subspaces
6. 1
Exa m p l e 6 . 1 1
435
Let C(;S be the set of all continuous real-valued functions defined on IR and let 0J be
the set of all differentiable real-valued functions defined on IR. Show that C(;S and 0J are
subspaces of �, the vector space of all real-valued functions defined on IR.
From calculus, we know that iff and g are continuous functions and c is a
scalar, then f + g and cf are also continuous. Hence, C(;S is closed under addition and
scalar multiplication and so is a subspace of �. Iff and g are differentiable, then so are
f + g and cf Indeed,
Solution
(j + g) '
=
f ' + g ' and ( cf) '
=
c (j')
So 0J is also closed under addition and scalar multiplication, making it a subspace
of �.
...+
It is a theorem of calculus that every differentiable function is continuous. Conse­
quently, 0J is contained in C(;S (denoted by 0J C C(;S), making 0J a subspace ofC(;S . It is also
the case that every polynomial function is differentiable, so rt/' C 0J , and thus rt/' is a
subspace of 0J . We therefore have a hierarchy of subspaces of �, one inside the other:
This hierarchy is depicted in Figure 6.2.
The hierarchy of subspaces of�
Figure 6 . 2
There are other subspaces of � that can be placed into this hierarchy. Some of
these are explored in the exercises.
In the preceding discussion, we could have restricted our attention to functions
defined on a closed interval [a, b] . Then the corresponding subspaces of � [a, b]
would be C(;S [a, b] , 0J [a, b] , and rt/' [a, b] .
Exa m p l e 6 . 1 2
Let S be the set of all functions that satisfy the differential equation
f" + f 0
=
That is, S is the solution set of Equation ( 1 ). Show that S is a subspace of �-
(1)
436
Chapter Vector Spaces
6
Solulion
S is nonempty, since the zero function clearly satisfies Equation ( 1 ). Let
f and g be in S and let c be a scalar. Then
(f + g) " + (f + g) = (j" + g " ) + (f + g)
= (f" + f) + (g " + g)
=O+O
=O
which shows that f + g is in S. Similarly,
(cf)" + cf = cf" + cf
= c (f " + f)
= co
=O
so cf is also in S.
Therefore, S is closed under addition and scalar multiplication and is a subspace
of ?F.
The differential Equation ( 1) is an example of a homogeneous linear differential
equation. The solution sets of such equations are always subspaces of ?F. Note that in
Example 6.12 we did not actually solve Equation ( 1) (i.e., we did not find any specific
solutions, other than the zero function) . We will discuss techniques for finding solu­
tions to this type of equation in Section 6.7.
As you gain experience working with vector spaces and subspaces, you will notice
that certain examples tend to resemble one another. For example, consider the vector
spaces IR 4 , <!/' 3 , and M22 • Typical elements of these vector spaces are, respectively,
u�
Indejathevuwords
all overof Yogi
again:'Berra, "It's
Exa m p l e 6 . 1 3
m
p (x) � a + bx + ex' + dx', and A �
[: !]
Any calculations involving the vector space operations of addition and scalar multi­
plication are essentially the same in all three settings. To highlight the similarities, in
the next example we will perform the necessary steps in the three vector spaces side
by side.
(a) Show that the set W of all vectors of the form
is a subspace of IR 4 •
(b) Show that the set W of all polynomials of the form a + bx - bx 2 + ax 3 is a
subspace of <!!' 3 .
( c) Show that the set W of all matrices of the form
[ _ : �] is a subspace of M22 .
Section Vector Spaces and Subspaces
6. 1
a=b= b=
uv
Solution
(a) W is nonempty because it con­
tains the zero vector 0. (Take
0.) Let and be in W-say,
a= b=
AA = B � �] a
[
B = [ � �]
A+B= [ -a(b++ ba ++ d]
(b) W is nonempty because it con­
tains the zero polynomial. (Take
O.) Letp (x) and q(x) be in W-say,
p ( x)
ax 3
and
q ( x)
[
ab ++ l
u+v= -ab+
[ ba ++ l
- a(b++
u+v k
Then
Then
p ( x)
c
d
-d
c
= a + bx -bx2 +
=+ +
+ = +(a(+b +
-+ ((ba ++
dx - dx2
c
q ( x)
(c) W is nonempty because it con­
tains the zero matrix 0. (Take
0. ) Let and be in W-say,
cx 3
and
c)
_
c
c ) x3
so
is also in W (because it has
the right form).
Similarly, if is a scalar, then
_
Then
d) x
d ) x2
c
d
d)
c
431
d)
+ k
A+B k
k = ka + kbx - kbx2 + kax3
kA = [ -kakb kakb]
kA
so p(x) q(x) is also in W (because it
has the right form).
Similarly, if is a scalar, then
c
so
is also in W (because it has
the right form).
Similarly, if is a scalar, then
p ( x)
so ku is in W.
Thus, W is a nonempty subset of
IR 4 that is closed under addition and
scalar multiplication. Therefore, W is
a subspace of iR 4 , by Theorem 6.2.
so
is in W.
Thus, W is a nonempty subset of
M22 that is closed under addition and
scalar multiplication. Therefore, W is
a subspace of M22 , by Theorem 6.
so kp (x) is in W.
Thus, W is a nonempty subset of
<;IP 3 that is closed under addition and
scalar multiplication. Therefore, W is
a subspace of <;if 3 by Theorem 6.2.
4
Example 6. 1 3 shows that it is often possible to relate examples that, on the surface,
appear to have nothing in common. Consequently, we can apply our knowledge of
!R n to polynomials, matrices, and other examples. We will encounter this idea several
times in this chapter and will make it precise in Section 6.5.
Exa m p l e 6 . 1 4
+=
=
If V is a vector space, then V is clearly a subspace of itself. The set {O}, consisting of
only the zero vector, is also a subspace of V, called the zero subspace. To show this, we
simply note that the two closure conditions of Theorem 6.2 are satisfied:
0
0
0 and c 0
0 for any scalar c
The subspaces {O} and V are called the trivial subspaces of V.
438
Chapter Vector Spaces
6
An examination of the proof of Theorem 6.2 reveals the following useful fact:
If W is a subspace of a vector space V, then W contains the zero vector 0 of V.
This fact is consistent with, and analogous to, the fact that lines and planes are sub­
spaces of !R 3 if and only if they contain the origin. The requirement that every subspace
must contain 0 is sometimes useful in showing that a set is not a subspace.
Exa m p l e 6 . 1 5
Let W be the set of all 2 X 2 matrices of the form
Is W a subspace of M22 ?
Solulion
Each matrix in W has the property that its ( 1 , 2) entry is one more than its
( 1 , 1) entry. Since the zero matrix
0
=
[� �]
does not have this property, it i s not in W. Hence, W is not a subspace o f M22 •
Exa m p l e 6 . 1 6
Let W be the set of all 2 X 2 matrices with determinant equal to 0. Is W a subspace
of M22 ? (Since <let 0 0, the zero matrix is in W, so the method of Example 6. 15 is
of no use to us.)
=
Solulion
Let
A
Then <let A
=
<let B
=
=
[ � �] and B [ � � ]
=
0, so A and B are in W. But
A +B
=
[� � ]
so det (A + B) = 1 =fa 0, and therefore A + B is not in W. Thus, W is not closed under
addition and so is not a subspace of M22 .
Spanning sets
The notion of a spanning set of vectors carries over easily from !R n to general vector
spaces.
D e fi n it i o n If S {v1 , v2 , . . • , vd is a set of vectors in a vector space V, then
the set of all linear combinations of v1 , v2 , . . . , vk is called the span of v1 , v2 , . . . , vk
and is denoted by span (v1 , v2 , . . . , vk) or span ( S ) . If V = span (S ), then S is called
a spanning set for V and V is said to be spanned by S.
=
Section Vector Spaces and Subspaces
6. 1
Exa m p l e 6 . 1 1
439
Show that the polynomials 1, x, and x 2 span l!J' 2 •
By its very definition, a polynomial p (x) = a + bx + cx2 is a linear combi­
nation of 1, x, and x2 • Therefore, l!J' 2 = span ( l , x, x 2 ) .
Solution
Example 6. 1 7 can clearly b e generalized t o show that l!J' n = span ( l , x, x2 , . . . ,
n
x ) . However, no finite set of polynomials can possibly span l!J', the vector space of
all polynomials. (See Exercise 44 in Section 6.2.) But, if we allow a spanning set
to be infinite, then clearly the set of all nonnegative powers of x will do. That is,
l!J' = span ( l , x, x2 , • • • ) .
Exa m p l e 6 . 1 8
Show that M23 = span (E 1 1 , E 1 2 , E 1 3 , E2 1 , E22 , E23 ), where
El l =
E2 1 =
[� �]
[� �]
0
0
E1 2 =
0
0
E22 =
[� �]
[� �]
0
0
E1 3 =
E23 =
[� �]
[ � �]
0
0
0
0
(That is, E;j is the matrix with a 1 in row i, column j and zeros elsewhere.)
Solution
We need only observe that
Extending this example, we see that, in general, Mm n is spanned by the mn matri­
ces Eij , where i = 1 , . . . , m and j = 1, . . . , n.
Exa m p l e 6 . 1 9
In l!J' 2 , determine whether r (x) = 1 - 4x + 6x 2 is in span(p(x), q(x)), where
p ( x) = 1 - x + x 2 and q ( x) = 2 + x - 3x 2
We are looking for scalars c and d such that cp(x) + dq(x) = r (x) . This
means that
c ( l - x + x 2 ) + d ( 2 + x - 3x 2 ) = 1 - 4x + 6x 2
Solution
Regrouping according powers of x, we have
( c + 2 d) + ( - c + d) x + ( c - 3d) x 2 = 1 - 4x + 6x 2
Equating the coefficients of like powers of x gives
c + 2d = 1
- c + d = -4
c - 3d = 6
440
Chapter Vector Spaces
6
which is easily solved to give c = 3 and d =
r(x) is in span(p(x), q(x)). (Check this.)
Exa m p l e 6 . 2 0
-
1 Therefore, r(x) = 3p(x) - q(x), so
.
In ?F, determine whether sin 2x is in span(sin x, cos x) .
We set c sin x + d cos x = sin 2x and try to determine c and d so that this
equation is true. Since these are functions, the equation must be true for all values of
x. Setting x = 0, we have
S o lution
c sin O + d cos O = sin O or c ( O ) + d ( l ) = O
from which we see that d = 0. Setting x = 1T /2, we get
c sin ( 7r/2 ) + d cos ( 'TT / 2 ) = sin ( 7T ) or c ( l ) + d ( O ) = 0
giving c = 0. But this implies that sin 2x = O (sin x) + O(cos x) = 0 for all x, which
is absurd, since sin 2x is not the zero function. We conclude that sin 2x is not in
span (sin x, cos x).
Remark
It is true that sin 2x can be written in terms of sin x and cos x. For
example, we have the double angle formula sin 2x = 2 sin x cos x. However, this is not
a linear combination.
Exa m p l e 6 . 2 1
In M22 , describe the span of A =
Solution
[ � � l B = [ � � l and C = [ � �] .
Every linear combination of A, B, and C is of the form
[� �] + d [ � � ] + e[ � �]
c+d c+e
]
= [
c+ e
d
cA + dB + eC = c
This matrix is symmetric, so span (A, B, C) is contained within the subspace of sym­
metric 2 X 2 matrices. In fact, we have equality; that is, every symmetric 2 X 2 matrix is
[; �] be a symmetric 2 2 matrix. Setting
[yx y ] = [ cc ++ de c +d e ]
in span(A, B, C). To show this, we let
X
z
and solving for c and d, we find that c = x - z, d = z, and e = - x + y + z. Therefore,
�
(Check this.) It follows that span(A, B, C) is the subspace of symmetric 2 X 2 matrices .
.+
Section Vector Spaces and Subspaces
6. 1
441
As was the case in !R n , the span of a set of vectors is always a subspace of the vector
space that contains them. The next theorem makes this result precise. It generalizes
Theorem 3.19.
Theorem 6 . 3
Let V i, v2 ,
•
.
.
, vk b e vectors in a vector space V.
a. span (vi, v2 ,
b. span (vi, v2 ,
•
.
•
.
•
.
, vk) is a subspace of V.
, vk) is the smallest subspace of V that contains Vi, v2 ,
•
.
.
, vk.
(a) The proof of property (a) is identical to the proof of Theorem 3 . 1 9, with
Proof
!R n replaced by V.
(b) To establish property (b ), we need to show that any subspace of V that contains
Vi, v2 , , vk also contains span( vi, v2 , , vk) · Accordingly, let W be a subspace of V
that contains Vi, v2 , . . . , vk. Then, since W is closed under addition and scalar multi­
plication, it contains every linear combination Ci Vi + c 2v2 + + ckvk of Vi, v2 ,
,
vk. Therefore, span (vi, v2 , , vk) is contained in W.
.
•
.
•
•
•
· · ·
.
1
•
•
.
.
.
Exercises 6 . 1
In Exercises 1 - 1 1, determine whether the given set, together
with the specified operations of addition and scalar multi­
plication, is a vector space. If it is not, list all of the axioms
that fail to hold.
1 . The set of all vectors in IR 2 of the form
[: l with the
usual vector addition and scalar multiplication
2. The set of all vectors
[;] in IR2 with x 2: 0, y 2: 0 (i.e.,
the first quadrant), with the usual vector addition and
scalar multiplication
3. The set of all vectors
[�] in IR2 with xy 2: 0 (i.e., the
union of the first and third quadrants), with the usual
vector addition and scalar multiplication
4 . The set of all vectors
[; ]
in IR 2 with x 2: y, with the
usual vector addition and scalar multiplication
5. IR 2 , with the usual addition but scalar multiplication
defined by
6. IR 2 , with the usual scalar multiplication but addition
defined by
[] [] [
x,
x
+ 2
Y1
Yz
=
l]
x , + x2 +
Yi + Yz + 1
7. The set of all positive real numbers, with addition EB
defined by x EB y = xy and scalar multiplication 0
defined by c 0 x = xc
8. The set of all rational numbers, with the usual addition
and multiplication
9. The set of all upper triangular 2 X 2 matrices, with the
usual matrix addition and scalar multiplication
[: �l
10. The set of all 2 X 2 matrices of the form
where ad = 0, with the usual matrix addition and
scalar multiplication
1 1 . The set of all skew-symmetric n X n matrices, with the
usual matrix addition and scalar multiplication
(see page 162).
12. Finish verifying that <!f 2 is a vector space
(see Example 6.3).
13. Finish verifying that ge is a vector space
(see Example 6.4) .
442
Chapter Vector Spaces
6
E:v In Exercises 1 4- 1 7, determine whether the given set, together
with the specified operations of addition and scalar multi­
plication, is a complex vector space. If it is not, list all of
the axioms that fail to hold.
14. The set of all vectors in C 2 of the form
[ �] , with the
usual vector addition and scalar multiplication
15. The set Mm n ( C) of all m X n complex matrices, with
the usual matrix addition and scalar multiplication
16. The set C 2 , with the usual vector addition but scalar
30. V = Mnn ' W = {A in Mnn : <let A = 1 }
31. V = Mnn ' W is the set of diagonal n X n matrices
32. V = Mnn > W is the set of idempotent n X n matrices
33. V = Mnn ' W = {A in Mnn : AB = BA}, where B is a
given (fixed) matrix
34. V = C!P 2 , W = {bx + cx 2 }
35. V = C!P 2 , W = {a + bx + cx 2 : a + b + c = O}
36. V = C!P 2 , W = {a + bx + cx 2 : abc = O}
z
z
multiplication defined by c 1 = � 1
37. V = C!P, W is the set of all polynomials of degree 3
CZ2
Z2
17. !R n , with the usual vector addition and scalar multiplication 38. V = ':ffe , W = {f in '2F : f ( -x) = f(x)}
39. V = ':ffe , W = {f in '2F :f( -x) = -f(x)}
In Exercises 1 8-21, determine whether the given set, together 40. V = ':ffe , w = {f in '2F :f(O) = l }
with the specified operations of addition and scalar. multipli. ';'Y .· f( ) -_ }
. is. a ector space �ver th e in. d"zca ted lL · If "t is no t, l"is t ·tt;. 41. v _- '!:Y, w _- {f m
catzon,
p
�
.
.
Jlh 42. V _- '!:Y, W is. the set of all mtegrable
funct10ns
all of the axzoms that fail to hold.
CZl! :f'(x) 2:
for all x}
18. The set of all vectors in "11._ � with an even number of Jlh 43. V = CZJJ , W = {f in
l
(2
l s, over "ll 2 with the usual vector addition and scalar � 44. V = ':ffe , W = ct; , the set of all functions with
[] [ ]
77
a;-;
u.::
1
a;-;
0 0
0
continuous second derivatives
multiplication
Jlh 45. V = ':ffe , W = {f in '2F : = lim f (x) = 00}
19. The set of all vectors in "11._ � with an odd number of
X->0
l s, over "ll 2 with the usual vector addition and scalar
46. Let V be a vector space with subspaces U and W. Prove
multiplication
that U n W is a subspace of V.
20. The set Mm n ( "llp ) of all m X n matrices with entries
47. Let V be a vector space with subspaces U and W. Give
from "11._P ' over "11._P with the usual matrix addition and
an example with v = [R 2 to show that U U W need not
scalar multiplication
be a subspace of V.
21. "ll 6 , over "ll 3 with the usual addition and multiplication
48. Let V be a vector space with subspaces U and W.
(Think this one through carefully!)
Define the sum of U and W to be
22. Prove Theorem 6.l (a) .
U + W = { u + w : u is in U, w is in W}
23. Prove Theorem 6.l (c) .
(a) If V = IR 3 , U is the x-axis, and W is the y-axis,
what is U + W ?
In Exercises 24-45, use Theorem 6.2 to determine whether
(b) If U and W are subspaces o f a vector space V,
W is a subspace of V
prove that U + W is a subspace of V.
49. If U and V are vector spaces, define the Cartesian
product of U and V to be
24. v � �' . w �
25. v � �' . w �
U X V = { (u, v) : u is in U and v is in V}
Prove that U X V is a vector space.
50. Let W be a subspace of a vector space V. Prove that
26. v = IR 3 , w =
Li = { (w, w) : w is in W } is a subspace of V X V.
a+b+l
mii
fl [ � l)
27. v � �' . w �
lUJ l
trn
1 1
In Exercises 51 and 52, let A =
and
1
1
l -1
B=
. Determine whether C is in span(A, B).
1
l 2
3 -5
51. c =
52. c =
5 -1
3 4
[
0]
[ ]
[
]
[
]
Section Linear Independence, Basis, and Dimension
6.2
i ] [o ]
In Exercises 53 and 54, let p(x) = 1 - 2x, q(x) = x - x 2,
and r(x) = - 2 + 3x + x 2 • Determine whether s(x) is in
span(p(x), q(x), r(x) ).
53. s(x) = 3 - 5x - x 2 54. s ( x) = 1 + x + x 2
In Exercises 55-58, let f(x) = sin 2x and g(x) = cos 2x.
Determine whether h(x) is in span(j(x), g(x) ).
55. h (x) = 1
56. h (x) = cos 2x
57. h (x) = sin 2x
58. h ( x) = sin x
59. Is M22 spanned by
[� � l [� �l [ � �l [� �]
-
Writi n g Project
443
1 ' 1
-1
?
0
61. Is <!!' 2 spanned by 1 + x, x + x 2 , 1 + x 2 ?
62. Is <!!' 2 spanned by 1 + x + 2 x 2 , 2 + x + 2 x 2 ,
- 1 + x + 2x 2 ?
63. Prove that every vector space has a unique zero
?
vector.
64. Prove that for every vector v in a vector space V,
there is a unique v ' in V such that v + v ' = 0.
The Rise of Vector Spaces
As noted in the sidebar on page 429, in the late 1 9th century, the mathematicians
Hermann Grassmann and Giuseppe Peano were instrumental in introducing the
idea of a vector space and the vector space axioms that we use today. Grassmann's
work had its origins in barycentric coordinates, a technique invented in 1 827 by
August Ferdinand Mobius (of Mobius strip fame). However, widespread acceptance
of the vector space concept did not come until the early 20th century.
Write a report on the history of vector spaces. Discuss the origins of the notion of
a vector space and the contributions of Grassmann and Peano. Why was the math­
ematical community slow to adopt these ideas, and how did acceptance come about?
1 . Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (Third Edition)
(Hoboken, NJ: Wiley, 20 1 1 ) .
2. Jean-Luc, Dorier ( 1 995), A General Outline o f the Genesis o f Vector Space
Theory, Historia Mathematica 22 ( 1 995), pp. 227-26 1 .
3. Victor J. Katz, A History of Mathematics: A n Introduction (Third Edition) (Read­
ing, MA: Addison Wesley Longman, 2008).
f
l i n e a r I n d e p e n d e n ce . Basis , a n d D i m e n s i o n
In this section, we extend the notions of linear independence, basis, and dimension
to general vector spaces, generalizing the results of Sections 2.3 and 3.5. In most cases,
the proofs of the theorems carry over; we simply replace !R n by the vector space V.
linear Independence
D e fi n it i o n A set of vectors {v1, v2 , . . . , vd in a vector space V is linearly de­
pendent if there are scalars c1, c2 , . . . , c k, at least one of which is not zero, such that
A set of vectors that is not linearly dependent is said to be linearly independent.
444
Chapter Vector Spaces
6
As in IJ�r , {v1 , v2 ,
.
•
, vd is linearly independent in a vector space V if and only if
•
c 1 v1 + c 2v2 + · · · + ckvk = 0 implies c 1 = 0, c2 = 0, . . . , c k = 0
We also have the following useful alternative formulation of linear dependence.
Theorem 6 . 4
A set of vectors {v1 , v2 , , vk } in a vector space V is linearly dependent if and only
if at least one of the vectors can be expressed as a linear combination of the others.
.
Proof
•
.
The proof is identical to that of Theorem 2.5.
As a special case of Theorem 6.4, note that a set of two vectors is linearly depen­
dent if and only if one is a scalar multiple of the other.
Exa m p l e 6 . 2 2
In l!f 2 , the set { l + x + x 2 , 1 - x + 3x 2 , 1 + 3x - x 2 } is linearly dependent, since
2 ( 1 + x + x 2 ) - ( 1 - x + 3x 2 ) = 1 + 3x - x 2
Exa m p l e 6 . 2 3
In M22 , let
A=
[� � l B = [� - 01 ] , C = [ 21 O1 J
Then A + B = C, so the set {A, B, C} is linearly dependent.
Exa m p l e 6 . 2 4
Exa m p l e 6 . 2 5
In ':IF, the set {sin 2 x, cos 2 x, cos 2x} is linearly dependent, since
cos 2x = cos 2x - sin2x
Show that the set { l , x, x2 , . . . , x n } is linearly independent in <!P w
, e n are scalars such that
Co · 1 + C 1 X + Cz X 2 + · · · + C n X n = 0
Then the polynomial p (x) = c0 + c 1 x + c2 x2 + + c n x n is zero for all values of x. But
a polynomial of degree at most n cannot have more than n zeros (see Appendix D).
So p (x) must be the zero polynomial, meaning that c0 = c 1 = c2 = = en = 0.
Therefore, {l, x, x2 , . . . , x n } is linearly independent.
Solution 1
Suppose that c0 , c 1 ,
.
•
.
· · ·
· · ·
l�
We begin, as in the first solution, by assuming that
p ( x ) = Co + C 1 X + C z X 2 + · · · + C n X n = Q
Since this is true for all x, we can substitute x = 0 to obtain c0 = 0. This leaves
C 1 X + CzX 2 + · · · + C n X n = 0
Solution 2
Section Linear Independence, Basis, and Dimension
6.2
445
Taking derivatives, we obtain
+ ncnxn - I = 0
and setting x = 0, we see that c 1 = 0. Differentiating 2c 2 x + 3c 3 x 2 + + ncn x n - I = 0
and setting x = 0, we find that 2c 2 = 0, so c 2 = 0. Continuing in this fashion, we
find that k!ck = 0 for k = 0, . . . , n. Therefore, c0 = c 1 = c 2 = = e n = 0, and { 1 , x,
C 1 + 2CzX + 3C3X2 +
·
·
·
· · ·
· · ·
x2 , . . . ' x n } is linearly independent.
Exa m p l e 6 . 2 6
4
In <;!/' 2 , determine whether the set { 1 + x, x + x 2 , 1 + x 2 } is linearly independent.
Solution
Let c 1 , c 2 , and c 3 be scalars such that
Then
This implies that
C1 +
c3 = 0
=0
C1 + Cz
C 2 + C3 = 0
the solution to which is c 1 = c2 = c 3 = 0. It follows that { 1 + x, x + x 2 , 1 + x 2 } is
linearly independent.
Remark
Compare Example 6.26 with Example 2.23(b). The system of equations
that arises is exactly the same. This is because of the correspondence between <;!/' 2 and
IR 3 that relates
I+
r n
[iJ
x + x' -
[:J
1 + x' -
m
and produces the columns of the coefficient matrix of the linear system that we have
to solve. Thus, showing that { 1 + x, x + x 2 , 1 + x 2 } is linearly independent is equiva­
lent to showing that
is linearly independent. This can be done simply by establishing that the matrix
[i : �]
has rank 3, by the Fundamental Theorem of Invertible Matrices.
446
Chapter Vector Spaces
6
Exa m p l e 6 . 2 1
In ?JP , determine whether the set {sin x, cos x} is linearly independent.
The functions f(x) = sin x and g(x) = cos x are linearly dependent if and
only if one of them is a scalar multiple of the other. But it is clear from their graphs
that this is not the case, since, for example, any nonzero multiple ofj(x) = sin x has
the same zeros, none of which are zeros ofg(x) = cos x.
This approach may not always be appropriate to use, so we offer the following
direct, more computational method. Suppose c and d are scalars such that
S o lulion
c sin x + d cos x = 0
Setting x = 0, we obtain d = 0, and setting x = n/2, we obtain c = 0. Therefore, the
set {sin x, cos x} is linearly independent.
Although the definitions of linear dependence and independence are phrased
in terms of finite sets of vectors, we can extend the concepts to infinite sets as
follows:
A set S of vectors in a vector space V is linearly dependent if it contains finitely
many linearly dependent vectors. A set of vectors that is not linearly dependent is
said to be linearly independent.
Note that for finite sets of vectors, this is just the original definition. Following is an
example of an infinite set of linearly independent vectors.
Exa m p l e 6 . 2 8
In <lP , show that S = { l , x, x2 , . . . } is linearly independent.
Suppose there is a finite subset T of S that is linearly dependent. Let x m be
n
the highest power of x in T and let x be the lowest power of x in T. Then there are
scalars en, Cn+ l ' . . . , c m , not all zero, such that
Solulion
But, by an argument similar to that used in Example 6.25, this implies that en =
Cn + i =
= cm = 0, which is a contradiction. Hence, S cannot contain finitely many
linearly dependent vectors, so it is linearly independent.
· · ·
Bases
The important concept of a basis now can be extended easily to arbitrary vector
spaces.
D e fi n it i o n
A subset l3 of a vector space V is a basis for V if
1 . l3 spans V and
2. l3 is linearly independent.
Section Linear Independence, Basis, and Dimension
6.2
Exa m p l e 6 . 2 9
Exa m p l e 6 . 3 0
441
If e; is the ith column of the n X n identity matrix, then { e1, e2 , . . . , en } is a basis for !R n ,
called the standard basis for !R n .
{ l , x, x 2 , • . • , x n } is a basis for '!P n ' called the standard basis for '!P n -
Exa m p l e 6 . 3 1
The set [ = {E 11 , . • . , E 1n , E 21 , • • • , E 2n , Em1 , • • • , Emn } is a basis for Mm n ' where the
matrices Eij are as defined in Example 6. 1 8. [ is called the standard basis for Mm n ­
We have already seen that [ spans Mm n - It is easy to show that [ is linearly inde­
pendent. (Verify this! ) Hence, [ is a basis for Mm n -
Exa m p l e 6 . 3 2
Show that B = { l + x, x + x 2 , 1 + x 2 } is a basis for '!1' 2 •
We have already shown that B is linearly independent, in Example 6.26. To
show that B spans '!1' 2 , let a + bx + cx 2 be an arbitrary polynomial in '!1' 2 • We must
show that there are scalars c1, c2 , and c 3 such that
Solution
or, equivalently,
(c 1 + c3 ) + (c 1 + c2 )x + (c2 + c 3 )x 2 = a + bx + cx 2
Equating coefficients of like powers of x, we obtain the linear system
whi<h has a solution, since the weffideot matcix
[i : �]
has rnok 3 aod, hence,
is invertible. (We do not need to know what the solution is; we only need to know that
it exists.) Therefore, B is a basis for '!P 2 •
Ro11ark
Obmve that the matrix
O
[i �]
is the key to Example 6.32. We rnn
immediately obtain it using the correspondence between '!1' 2 and IR 3 , as indicated in
the Remark following Example 6.26.
448
Chapter Vector Spaces
6
Exa m p l e 6 . 3 3
Show that B = { l , x, x 2 , • • • } is a basis for <!P .
In Example 6.28, we saw that B is linearly independent. It also spans <!P ,
since clearly every polynomial is a linear combination of (finitely many) powers of x.
S o lulion
Exa m p l e 6 . 3 4
.+
Find bases for the three vector spaces in Example 6. 13:
Once again, we will work the three examples side by side to highlight the
similarities among them. In a strong sense, they are all the same example, but it will
take us until Section 6.5 to make this idea perfectly precise.
Solulion
(b) Since
(a) Since
we have W1 = span (u, v), where
we have W2 = span (u(x), v(x)),
where
u ( x) = 1 + x 3
and
Since {u, v} is clearly linearly in­
dependent, it is also a basis for W1 .
[ J [0 J [ 0 0J
(c) Since
a + bx - bx 2 + ax 3
= a ( l + x 3 ) + b (x - x 2 )
v ( x) = x - x 2
Since {u (x), v (x) } is clearly lin­
early independent, it is also a basis
for W2 .
1 O
1
a b
=a
+b
-b a
1
-1
[0 0J
we have W3 = span( U, V), where
U=
1
1
and V =
[ 0J
O 1
-1
Since { U, V} is clearly linearly in­
dependent, it is also a basis for W3 •
.+
Coordin a1es
Section 3.5 introduced the idea of the coordinates of a vector with respect to a basis
for subspaces of !R n . We now extend this concept to arbitrary vector spaces.
Theorem 6 . 5
Let V be a vector space and let B be a basis for V. For every vector v in V, there is
exactly one way to write v as a linear combination of the basis vectors in B.
Proof
The proofis the same as the proof ofTheorem 3.29. It works even ifthe basis B is
infinite, since linear combinations are, by definition, finite.
Section Linear Independence, Basis, and Dimension
6.2
449
The converse of Theorem 6.5 is also true. That is, if B is a set of vectors in a vector
space V with the property that every vector in V can be written uniquely as a linear
combination of the vectors in B, then B is a basis for V (see Exercise 30). In this sense,
the unique representation property characterizes a basis.
Since representation of a vector with respect to a basis is unique, the next definition
makes sense.
D e fi n it i o n Let B = {v1 , v2 ,
, vn } be a basis for a vector space V. Let v be a
vector in V, and write v = c 1 v1 + c2v2 + + cnvn - Then c 1 , c2 , , e n are called the
coordinates of v with respect to B, and the column vector
.
•
.
· · ·
•
.
•
is called the coordinate vector of v with respect to B.
Observe that if the basis B of V has n vectors, then [ v l B is a (column) vector in !R n .
Exa m p l e 6 . 3 5
Find the coordinate vector [p ( x) ] B of p (x) = 2 - 3x + 5x 2 with respect to the stan­
dard basis B = { l , x, x 2 } of (!J> 2 .
Solution
The polynomial p(x) is already a linear combination of 1, x, and x2 , so
This is the correspondence between (!J> 2 and IR 3 that we remarked on after
Example 6.26, and it can easily be generalized to show that the coordinate vector of a
polynomial
with respect to the standard basis B = { l , x, x 2 , . . . , x " } is just the vector
Remark
The order in which the basis vectors appear in B affects the order of
the entries in a coordinate vector. For example, in Example 6.35, assume that the
450
Chapter Vector Spaces
6
standard basis vectors are ordered as !3 ' = {x 2 , x, l}. Then the coordinate vector of
p (x) = 2 - 3x + 5x2 with respect to !3 ' is
[p(x) ] a �
Exa m p l e 6 . 3 6
Find the coordinate vector [A ] B of A =
l3 = {E 11 , E 1 2 , E2 1 > E22 } of M22 •
Solulion
[
Hl
]
2 -1
with respect to the standard basis
4
3
Since
we have
This is the correspondence between M22 and IR 4 that we noted before the intro­
duction to Example 6. 1 3 . It too can easily be generalized to give a correspondence
between Mmn and !R mn .
Exa m p l e 6 . 3 1
Find the coordinate vector [p(x) ] 8 of p (x) = 1 + 2x - x 2 with respect to the basis
C = {l + x, x + x 2 , 1 + x 2 } of \)}> 2 .
Solulion
We need to find c1, c2 , and c3 such that
c 1 ( 1 + x) + c2 (x + x 2 ) + c 3 ( 1 + x 2 ) = 1 + 2x - x 2
or, equivalently,
(c 1 + c 3 ) + (c 1 + c2 ) x + ( c2 + c3 ) x 2 = 1 + 2x - x 2
As in Example 6.32, this means we need to solve the system
c1 +
1
2
c2 + c3 = - 1
C1 + c2
c3 =
whose solution is found to be c1 = 2, c2 = 0, c3 = - 1 . Therefore,
Section Linear Independence, Basis, and Dimension
6.2
[ Since this result says that p (x) = 2 ( 1 + x)
correct.]
451
( 1 + x2 ), it is easy to check that it is
4
-
The next theorem shows that the process of forming coordinate vectors is com­
patible with the vector space operations of addition and scalar multiplication.
Theorem 6 . 6
Let B = {v1 , Vz , . . . , vn } be a basis for a vector space V. Let u and v be vectors in
V and let e be a scalar. Then
a. [ u + v l s = [ u l s + [v l s
b. [ c u ] s = e [ u ] s
Proof
We begin by writing u and v in terms of the basis vectors-say, as
Then, using vector space properties, we have
and
so
l ] [l l]
[ ] [l
d1
e1
dz
ez
.. + ..
.
dn
en
e i + d1
e z + dz
.
.
e n + dn
and
[ cu ] 8 =
ee l
eez
:
een
= e
el
ez
:
= e [u]8
en
An easy corollary to Theorem 6.6 states that coordinate vectors preserve linear
combinations:
( 1)
You are asked to prove this corollary in Exercise 3 1 .
The most useful aspect of coordinate vectors is that they allow us to transfer
information from a general vector space to !R n , where we have the tools of Chapters 1
to 3 at our disposal. We will explore this idea in some detail in Sections 6.3 and 6.6.
For now, we have the following useful theorem.
452
Chapter Vector Spaces
6
Theorem 6 . 1
Let B = {v1 , v2 ,
, v"} b e a basis for a vector space V and let u1, . . . , uk
be vectors in V. Then {u1, . . . , ud is linearly independent in V if and only if
{ [ u 1 ] 13, . . . , [ uk J 13} is linearly independent in IR " .
•
Proof
•
•
Assume that {u1, . . . , ud is linearly independent in V and let
in IR " . But then we have
c d u 1 ] 13 + · · · + cd ud 13 = 0
[ c 1 u 1 + · · · + ckuk ] 13 = 0
using Equation ( 1 ), so the coordinates of the vector c1u1 + + ckuk with respect to
B are all zero. That is,
c 1 u 1 + · · · + ckuk = Ov 1 + Ov2 + · · · + Ov" = 0
· · ·
The linear independence of {u1 , . . . , uk} now forces c1 = c2 =
{ [ u 1 ] 13, . . . , [ uk J 13} is linearly independent.
· · ·
= ck = 0, so
The converse implication, which uses similar ideas, is left as Exercise 32.
Observe that, in the special case where U; = V;, we have
V; = 0 · V I + · · · + 1 · V; + · · · + 0 • Vn
Dimension
The definition of dimension is the same for a vector space as for a subspace of !R n -the
number of vectors in a basis for the space. Since a vector space can have more than one
basis, we need to show that this definition makes sense; that is, we need to establish
that different bases for the same vector space contain the same number of vectors.
Part (a) of the next theorem generalizes Theorem 2.8.
Theorem 6 . 8
Let B = {v 1 , v2 , . . . , vn } be a basis for a vector space V.
a. Any set of more than n vectors in V must be linearly dependent.
b. Any set of fewer than n vectors in V cannot span V.
(a) Let {u1, . . . , um} be a set of vectors in V, with m > n. Then { [ u 1 ] 13, . . . ,
[ um J 13} is a set of more than n vectors in !R n and, hence, is linearly dependent,
by Theorem 2.8. This means that {u1, . . . , um} is linearly dependent as well, by
Proof
Theorem 6.7.
(b) Let {u1, . . . , um} be a set of vectors in V, with m < n. Then S = { [ u 1 J 13, . . . , [ um J 13}
is a set of fewer than n vectors in !R n . Now span(u 1 , . . . , um) = V if and only if
span(S) = !R n (see Exercise 33). But span ( S ) is just the column space of the n X m
matrix
A = [ [ u 1 ] 13 · · · [ um J 13 ]
so dim(span(S )) = dim(col(A)) :s m < n. Hence, S cannot span !R n , so {u1, . . . , um}
does not span V.
Now we extend Theorem 3.23.
Section Linear Independence, Basis, and Dimension
6.2
Theorem 6 . 9
453
The Basis Theorem
If a vector space V has a basis with n vectors, then every basis for V has exactly n
vectors.
The proof of Theorem 3.23 also works here, virtually word for word. However, it
is easier to make use of Theorem 6.8.
Let B be a basis for V with n vectors and let B' be another basis for V with m
vectors. By Theorem 6.8, m ::::: n; otherwise, B' would be linearly dependent.
Now use Theorem 6.8 with the roles of B and B' interchanged. Since B' is a
basis of V with m vectors, Theorem 6.8 implies that any set of more than m vectors
in V is linearly dependent. Hence, n ::::: m, since B is a basis and is, therefore, linearly
independent.
Since n ::::: m and m ::::: n, we must have n = m, as required.
Proof
The following definition now makes sense, since the number of vectors in a
(finite) basis does not depend on the choice of basis.
D e fi n it i o n A vector space V is called finite-dimensional if it has a basis con­
sisting of finitely many vectors. The dimension of V, denoted by dim V, is the num ber of vectors in a basis for V. The dimension of the zero vector space { 0} is defined
to be zero. A vector space that has no finite basis is called infinite-dimensional.
Exa m p l e 6 . 3 8
Since the standard basis for !R n has n vectors, dim !R n = n. In the case of IR 3 , a one­
dimensional subspace is just the span of a single nonzero vector and thus is a line
through the origin. A two-dimensional subspace is spanned by its basis of two
linearly independent (i.e., nonparallel) vectors and therefore is a plane through the
origin. Any three linearly independent vectors must span IR 3 , by the Fundamental
Theorem. The subspaces of IR 3 are now completely classified according to dimension,
as shown in Table 6. 1 .
Ta b l e 6 . 1
dim V
3
2
0
Exa m p l e 6 . 3 9
v
IR 3
Plane through the origin
Line through the origin
{O}
The standard basis for <!J> n contains n + 1 vectors (see Example 6.30), so dim <!J> n =
n + 1.
454
Chapter Vector Spaces
6
Exa m p l e 6 . 4 0
Exa m p l e 6 . 4 1
The standard basis for Mm n contains mn vectors (see Example 6.3 1), so
dim Mm n = mn.
Both <!P and 9F are infinite-dimensional, since they each contain the infinite linearly
independent set { l , x, x2 , } (see Exercise 44).
•
Exa m p l e 6 . 4 2
•
•
Find the dimension of the vector space W of symmetric 2 X 2 matrices (see
Example 6. 10).
Solulion
A symmetric 2 X 2 matrix is of the form
[� �] [� �] [� �] [� �]
{ [ � � l [ � �l [ � � ] }
=a
+b
+c
so W is spanned by the set
s
=
If S is linearly independent, then it will be a basis for W. Setting
we obtain
from which it immediately follows that a = b = c = 0. Hence, S is linearly indepen­
dent and is, therefore, a basis for W. We conclude that dim W = 3.
The dimension of a vector space is its "magic number:' Knowing the dimension
of a vector space V provides us with much information about V and can greatly sim­
plify the work needed in certain types of calculations, as the next few theorems and
examples illustrate.
Theorem 6 . 1 0
Let V be a vector space with dim V = n. Then:
a. Any linearly independent set in V contains at most n vectors.
b. Any spanning set for V contains at least n vectors.
c. Any linearly independent set of exactly n vectors in V is a basis for V.
d. Any spanning set for V consisting of exactly n vectors is a basis for V.
e. Any linearly independent set in V can be extended to a basis for V.
f. Any spanning set for V can be reduced to a basis for V.
Section Linear Independence, Basis, and Dimension
6.2
455
The proofs of properties (a) and (b) follow from parts (a) and (b) of Theo­
rem 6.8, respectively.
(c) Let S be a linearly independent set of exactly n vectors in V. If S does not span V,
then there is some vector v in V that is not a linear combination of the vectors in S.
Inserting v into S produces a set S' with n + 1 vectors that is still linearly independent
(see Exercise 54). But this is impossible, by Theorem 6.S(a). We conclude that S must
span V and therefore be a basis for V.
(d) Let S be a spanning set for V consisting of exactly n vectors. If S is linearly
dependent, then some vector v in S is a linear combination of the others. Throwing v
away leaves a set S' with n 1 vectors that still spans V (see Exercise 55). But this is
impossible, by Theorem 6.S(b ) . We conclude that S must be linearly independent and
therefore be a basis for V.
(e) Let S be a linearly independent set of vectors in V. If S spans V, it is a basis for
V and so consists of exactly n vectors, by the Basis Theorem. If S does not span V,
then, as in the proof of property (c), there is some vector v in V that is not a linear
combination of the vectors in S. Inserting v into S produces a set S' that is still linearly
independent. If S' still does not span V, we can repeat the process and expand it into
a larger, linearly independent set. Eventually, this process must stop, since no linearly
independent set in V can contain more than n vectors, by Theorem 6.S(a). When the
process stops, we have a linearly independent set S* that contains S and also spans V.
Therefore, S* is a basis for V that extends S.
(f) You are asked to prove this property in Exercise 56.
Proof
-
You should view Theorem 6. 1 0 as, in part, a labor-saving device. In many
instances, it can dramatically decrease the amount of work needed to check that a set
of vectors is linearly independent, a spanning set, or a basis.
Exa m p l e 6 . 4 3
In each case, determine whether S is a basis for V.
(a) V = rzf 2 , S = { l + x, 2 - x + x 2, 3x - 2x2,
(b) V = M22 , S =
{ [ � �], [ � - �] , [ � �] }
-
1 + 3x + x2}
_
(c) V = rzf 2 , S = { l + x, x + x2, 1 + x2}
(a) Since dim ( \]f 2 ) = 3 and S contains four vectors, S is linearly depen­
dent, by Theorem 6. l O(a). Hence, S is not a basis for \)f 2 .
(b) Since dim(M22 ) = 4 and S contains three vectors, S cannot span M22 , by Theo­
rem 6. l O(b). Hence, S is not a basis for M22 .
( c) Since dim ( \]f 2 ) = 3 and S contains three vectors, S will be a basis for rzf 2 if it is lin early independent or if it spans \]f 2 , by Theorem 6.10( c) or (d) . It is easier to show that
S is linearly independent; we did this in Example 6.26. Therefore, S is a basis for \]f 2 •
(This is the same problem as in Example 6.32-but see how much easier it becomes
using Theorem 6. 1 0 ! )
Solution
Exa m p l e 6 . 4 4
Extend { l + x , 1
-
x } t o a basis fo r \)f 2 .
First note that { l + x, 1 x} is linearly independent. (Why?) Since dim
((lf 2 ) = 3, we need a third vector-one that is not linearly dependent on the first two.
Solution
-
Chapter Vector Spaces
6
456
We could proceed, as in the proof of Theorem 6.I O(e), to find such a vector using trial
and error. However, it is easier in practice to proceed in a different way.
We enlarge the given set of vectors by throwing in the entire standard basis for !!f 2 .
This gives
S = { l + x, 1 - x, 1 , x, x 2 }
Now S is linearly dependent, by Theorem 6. IO(a), so we need to throw away some
vectors-in this case, two. Which ones? We use Theorem 6.l O(f), starting with the
first vector that was added, 1. Since 1 = i ( l + x) + i ( l - x), the set {l + x, 1 - x, l }
is linearly dependent, so we throw away 1 . Similarly, x = i ( l + x) - i ( l - x), so
{l + x, 1 - x, x} is linearly dependent also. Finally, we check that {l + x, 1 - x, x 2 }
is linearly independent. (Can you see a quick way to tell this?) Therefore, { l + x,
1 - x, x2 } is a basis for l!f 2 that extends { l + x, 1 - x}.
In Example 6.42, the vector space W of symmetric 2 X 2 matrices is a subspace of
the vector space M22 of all 2 X 2 matrices. As we showed, dim W = 3 ::=::: 4 = dim M22 .
This is an example of a general result, as the final theorem of this section shows.
Theorem 6 . 1 1
Let W be a subspace of a finite-dimensional vector space V. Then:
a. W is finite-dimensional and dim W ::=::: dim V.
b. dim W = dim V if and only if W = V.
Proof
(a) Let dim V = n. If W = {O}, then dim ( W ) = 0 ::=::: n = dim V. If W is
nonzero, then any basis B for V (containing n vectors) certainly spans W, since W is
contained in V. But B can be reduced to a basis B' for W (containing at most n vec­
tors), by Theorem 6. I O (f) . Hence, W is finite-dimensional and dim( W ) ::=::: n = dim V.
(b) If W = V, then certainly dim W = dim V. On the other hand, if dim W = dim
V = n, then any basis B for W consists of exactly n vectors. But these are then n lin­
early independent vectors in V and, hence, a basis for V, by Theorem 6. I O (c). There­
fore, V = span (B) = W.
I
Exercises 6 . 2
In Exercises 1 -4, test the sets of matrices for linear indepen­
dence in M22. For those that are linearly dependent, express
one of the matrices as a linear combination of the others.
4·
{ [� � l [ � �l [� � l [ � �] }
In Exercises 5-9, test the sets ofpolynomials for linear inde­
pendence. For those that are linearly dependent, express one
of the poly nomials as a linear combination of the others.
5. {x, 1 + x} in l!f
6. { l + x, 1 + x 2 , 1 - x + x 2 } in l!f 2
7. {x, 2x - x 2 , 3x + 2x 2 } in l!f 2
1
Section Linear Independence, Basis, and Dimension
6.2
8. {2x, x - x 2 , 1 + x 3 , 2 - x 2 + x 3 } in <;5} 3
9. { l - 2x, 3x + x 2 - x 3 , 1 + x 2 + 2x 3 , 3 + 2x + 3 x 3 } in <;5} 3
In Exercises 1 0-14, test the sets offunctions for linear in­
dependence in <;IF, For those that are linearly dependent,
express one of the functions as a linear combination of the
others.
1 1 . {l , sin2x, cos2x}
10. { l , sin x, cos x}
13. { l , ln (2x), ln (x 2 ) }
12. {eX, e - x }
14. {sin x, sin 2x, sin 3x}
� 15. Iff and g are in C(b (J J , the vector space of all functions
with continuous derivatives, then the determinant
I
I
f(x) g(x)
j'(x) g'(x)
is called the Wronskian off and g [named after the
W(x) =
Polish-French mathematician J 6sef Maria Hoene­
Wronski ( 1 776- 1 853), who worked on the theory of
determinants and the philosophy of mathematics] .
Show that f and g are linearly independent if their
Wronskian is not identically zero (that is, if there is
some x such that W(x) * O ).
� 16. In general, the Wronskian of f1 , • • • Jn in C(b ( n - i l is the
determinant
fz (x)
j{(x)
W(x) =
and f1 , . . . ,fn are linearly independent, provided W(x)
is not identically zero. Repeat Exercises 10-14 using
the Wronskian test.
17. Let {u, v, w} be a linearly independent set of vectors in
a vector space V.
(a) Is {u + v, v + w, u + w} linearly independent?
Either prove that it is or give a counterexample
to show that it is not.
(b) Is {u v, v w, u - w} linearly independent?
Either prove that it is or give a counterexample
to show that it is not.
- -
In Exercises 18-25, determine whether the set B is a basis
for the vector space V
18. V = M22 , B =
l9. V = M22 , B =
{ [� �l [ � �l [ _ � - � ] }
{ [ � � ] ' [ � -� ] ' [ � � ] ' [ � - � ] }
451
20. V = Mzz ,
21. V = M22 ,
B=
{ [ � � l [ � � l [ _ � �l [ � �l [� � ] }
-
22. V = <;IP 2 , B = {x, 1 + x, x x 2 }
23. V = <;IP 2 , B = { l - x, 1 - x 2 , x x 2 }
24. V = <;IP 2 , B = { l , 1 + 2x + 3x2 }
25. V = <;IP 2 , B = { l , 2 - x, 3 - x2 , x + 2x 2 }
-
[� !]
[� !]
{ [� �l [� �l [ � �l [ � � ] }
26. Find the coordinate vector of A =
with
respect to the basis B = {E22 , E21 , Ew E 1 1 } of M22 •
27. Find the coordinate vector of A =
to the basis B =
of M22 .
with respect
28. Find the coordinate vector of p (x) = 1 + 2x + 3x 2
with respect to the basis B = { l + x, 1 - x, x 2 } of <;JP 2 .
29. Find the coordinate vector of p (x) = 2 - x + 3x 2 with
respect to the basis B = { l , 1 + x, - 1 + x 2 } of <;JP 2 .
30. Let B be a set of vectors in a vector space V with
the property that every vector in V can be written
uniquely as a linear combination of the vectors in B.
Prove that B is a basis for V.
31. Let B be a basis for a vector space V, let u 1 , . . . , uk
be vectors in V, and let c 1 , . . . , ck be scalars. Show that
[ c 1U1 + . . . + ckud s = C 1 [ u 1 l s + . . . + cd ud s ·
32. Finish the proof of Theorem 6.7 by showing that if
{ [ u 1 ] 8, . . . , [ uk ] 8} is linearly independent in IR " then
{ u 1 , . . . , uk } is linearly independent in V.
33. Let { u 1 , . . . , um } be a set of vectors in an
n-dimensional vector space V and let B be a basis for V.
Let S = { [ u 1 ] 8, . . . , [ um ] 8} be the set of coordinate
vectors of {u 1 , . . . , um } with respect to B. Prove that
span ( u 1 , • . • , um ) = V if and only if span(S ) = !R n .
In Exercises 34-39, find the dimension of the vector space V
and give a basis for V
34. V = {p (x) in <;JP 2 : p (O) = O}
35. V = {p ( x) in <;JP 2 : p ( l ) = O}
� 36. V = {p ( x) in <;IP 2 : xp ' ( x) = p (x) }
Chapter Vector Spaces
6
458
37. V = {A in M22 : A is upper triangular}
38. V = {A in M22 : A is skew-symmetric}
39. V = {A in M22 : AB = BA}, where B =
[ 01 ]
1
1
40. Find a formula for the dimension of the vector space
of symmetric n X n matrices.
41. Find a formula for the dimension of the vector space
of skew-symmetric n X n matrices.
42. Let U and W be subspaces of a finite-dimensional
vector space V. Prove Grassmann's Identity:
dim ( U + W) = dim U + dim W - dim( U n W)
[Hint: The subspace U + W is defined in Exercise 48
of Section 6. 1 . Let B = {v1 , . . . , vk} be a basis for
U n W. Extend B to a basis C of U and a basis D of W.
Prove that C U D is a basis for U + W.]
43. Let U and V be finite-dimensional vector spaces.
(a) Find a formula for dim( U X V) in terms of dim U
and dim V. (See Exercise 49 in Section 6. 1 . )
(b) I f W is a subspace o f V, show that dim Li =
dim W, where Li = { ( w, w) : w is in W}.
44. Prove that the vector space i!f is infinite-dimensional.
[Hint: Suppose it has a finite basis. Show that there is
some polynomial that is not a linear combination of
this basis.]
45. Extend { l + x, 1 + x + x 2 } to a basis for llf 2 .
46. Extend
47. Extend
48. Extend
{ [� � l [� � ] }
{ [ � � l [ � �l [ � � ] }
{ [ � � ], [ � � ] }
to a basis for M22 •
-
to a basis for M22 •
to a basis for the vector
space of symmetric 2 X 2 matrices.
49. Find a basis for span ( l , 1 + x, 2x) in llf 1 .
50. Find a basis for span ( l - 2x, 2x - x 2 , 1 - x 2 , 1 + x 2 )
in i!f 2 .
5 1 . Find a basis for span ( l - x, x - x 2 , 1 - x 2 , 1 - 2x +
x 2 ) in i!f 2 •
O O l
52. Find a basis for span
,
,
1 1 0
1
[ � -�])
_
in M22 .
( [� J [
] [-1
53. Find a basis for span(sin2x, cos 2x, cos 2x) in '!fa.
54. Let S = {v1 , . . . , vn } be a linearly independent set in
a vector space V. Show that if v is a vector in V that is
not in span(S ), then S' = {v1 , • • • , vn , v} is still linearly
independent.
55. Let S = {v 1 , • • • , vn } be a spanning set for a vector
space V. Show that if vn is in span (v1 , . . . , vn _ 1 ), then
S' = {v1 , • . • , vn _ 1 } is still a spanning set for V.
56. Prove Theorem 6.l O(f) .
57. Let {v1 , • • • , vJ be a basis fo r a vector space V
and let c 1 , • . . , cn be nonzero scalars. Prove that
{c 1 V1 , . . . , C nvn } is also a basis for V.
58. Let {v1 , • • • , vJ be a basis for a vector space V. Prove
that
{vi , V1 + V2 , V1 + Vz + V3 , . . . , V1 + . . . + vJ
is also a basis for V.
Let a0, a 1 , • • • , a n be n + 1 distinct real numbers. Define
polynomials p0(x), p 1 ( x) , . . . , Pn ( x) by
( x - a0) · • · ( x - a ; _ 1 ) (x - a; 1 ) · • · ( x - a n )
P; ( x) = (a - a0) • • · (a ; - a ; _ ) (a ; - a ;+ ) • • · (a ; - a n )
;
1
+1
These are called the Lagrange polynomials associated
with a0, a 1 , . . . , aw [Joseph-Louis Lagrange (1 736- 1 8 1 3)
was born in Italy but spent most of his life in Germany and
France. He made important contributions to such fields as
number theory, algebra, astronomy, mechanics, and the
calculus of variations. In 1 773, Lagrange was the first to give
the volume interpretation of a determinant (see Chapter 4).]
59. ( a) Compute the Lagrange polynomials associated
with a0 = 1, a 1 = 2, a 2 = 3.
(b) Show, in general, that
p ; (a) =
{o
if i * j
1 if i = j
60. (a) Prove that the set B = {p0(x), p 1 ( x) , . . . , P n ( x) }
of Lagrange polynomials is linearly independent
in i!f w [Hint: Set c0p0(x) +
+ c np n (x) = 0 and
use Exercise 59(b) .]
(b) Deduce that B is a basis for i!f w
61. If q(x) is an arbitrary polynomial in i!f n ' it follows from
Exercise 60(b) that
q ( x) = CoPo ( x) + . . . + cnp n (x)
( 1)
for some scalars c0, . . . , cw
( a) Show that C; = q (a ; ) for i = 0, . . . , n, and deduce
that q ( x) = q(a0)p0(x) +
+ q(a n )P n (x) is the
unique representation of q(x) with respect to the
basis B.
·
·
·
·
·
·
Section Linear Independence, Basis, and Dimension
6.2
,
(b) Show that for any n + 1 points ( a0, c0 ) , ( a1, c1 ) ,
( a n , en ) with distinct first components, the func­
.
.
•
tion q(x) defined by Equation ( 1 ) is the unique
polynomial of degree at most n that passes
through all of the points. This formula is known
as the Lagrange interpolation formula. (Com­
pare this formula with Problem 19 in Explora­
tion: Geometric Applications of Determinants in
Chapter 4.)
(c) Use the Lagrange interpolation formula to find
the polynomial of degree at most 2 that passes
through the points
459
(i) ( 1 , 6), (2, - 1 ), and (3, - 2)
(ii) ( - 1 , 10), (0, 5), and (3, 2)
62. Use the Lagrange interpolation formula to show that
if a polynomial in i!f n has n + 1 zeros, then it must be
the zero polynomial.
63. Find a formula for the number of invertible matrices
in Mnn ( !f_p ) · [Hint: This is the same as determining the
number of different bases for z;. (Why?) Count the
number of ways to construct a basis for z;, one vector
at a time.]
Exp loration
Magic S qu a r e s
The engraving shown o n page 46 1 i s Albrecht Durer's Melancholia I ( 1 5 14). Among
the many mathematical artifacts in this engraving is the chart of numbers that hangs
on the wall in the upper right-hand corner. (It is enlarged in the detail shown.) Such
an array of numbers is known as a magic square. We can think of it as a 4 X 4 matrix
3 2
10 1 1
6 7
1 5 14
Observe that the numbers in each row, in each column, and in both diagonals have
the same sum: 34. Observe further that the entries are the integers 1 , 2, . . . , 16. (Note
that Durer cleverly placed the 1 5 and 14 adjacent to each other in the last row, giving
the date of the engraving.) These observations lead to the following definition.
An n x n matrix M is called a magic square if the sum of the
entries is the same in each row, each column, and both diagonals. This common
sum is called the weight of M, denoted wt(M). If M is an n X n magic square that
contains each of the entries 1 , 2, . . . , n 2 exactly once, then M is called a classical
D e fi n it i o n
magic square.
1 . If M is a classical n X n magic square, show that
n(_n_2_+_l_)
wt ( M) = _
[Hint: Use Exercise 5 1 in Section 2.4.]
2
2. Find a classical 3 X 3 magic square. Find a different one. Are your two ex­
amples related in any way?
460
3. Clearly, the 3 X 3 matrix with all entries equal to t is a magic square with
weight 1 . Using your answer to Problem 2, find a 3 X 3 magic square with weight 1 ,
all of whose entries are different. Describe a method fo r constructing a 3 X 3 magic
square with distinct entries and weight w for any real number w.
Let Magn denote the set of all n X n magic squares, and let Mag� denote the set of all
n X n magic squares of weight 0.
4. (a) Prove that Mag3 is a subspace of M33 .
(b) Prove that Mag� is a subspace of Mag3 .
5. Use Problems 3 and 4 to show that if M is a 3 X 3 magic square with weight
w, then we can write M as
M = M0 + kJ
where M0 is a 3 X 3 magic square of weight 0, J is the 3 X 3 matrix consisting entirely
of ones, and k is a scalar. What must k be? [Hint: Show that M - kJ is in Maij for an
appropriate value of k.]
Let's try to find a way of describing all 3 X 3 magic squares. Let
be a magic square with weight 0. The conditions on the rows, columns, and diag­
onals give rise to a system of eight homogeneous linear equations in the variables a,
b, . . . , i.
6. Write out this system of equations and solve it. [Note: Using a CAS will
facilitate the calculations.]
461
7. Find the dimension of Mag�. Hint: By doing a substitution, if necessary, use
your solution to Problem 6 to show that M can be written in the form
M=
[
s
-s - t
t
s- t
0
-s + t
-t
s+ t
-s
l
8. Find the dimension of Mag3 • [Hint: Combine the results of Problems 5 and 7.]
9. Can you find a direct way of showing that the ( 2, 2) entry of a 3 X 3 magic
square with weight w must be w/3? [Hint: Add and subtract certain rows, columns,
and diagonals to leave a multiple of the central entry.]
10. Let M be a 3 X 3 magic square of weight 0, obtained from a classical 3 X 3
magic square as in Problem 5. If M has the form given in Problem 7, write out an
equation for the sum of the squares of the entries of M. Show that this is the equation
of a circle in the variables s and t, and carefully plot it. Show that there are exactly
eight points (s, t) on this circle with both s and t integers. Using Problem 8, show that
these eight points give rise to eight classical 3 X 3 magic squares. How are these magic
squares related to one another?
462
Section 6.3 Change of Basis
463
C h a n g e o f Basis
In many applications, a problem described using one coordinate system may be
solved more easily by switching to a new coordinate system. This switch is usually
accomplished by performing a change of variables, a process that you have prob­
ably encountered in other mathematics courses. In linear algebra, a basis provides
us with a coordinate system for a vector space, via the notion of coordinate vectors.
Choosing the right basis will often greatly simplify a particular problem. For example,
consider the molecular structure of zinc, shown in Figure 6.3(a). A scientist studying
zinc might wish to measure the lengths of the bonds between the atoms, the angles
between these bonds, and so on. Such an analysis will be greatly facilitated by intro­
ducing coordinates and making use of the tools of linear algebra. The standard basis
and the associated standard xyz coordinate axes are not always the best choice. As
Figure 6.3(b) shows, in this case {u, v, w} is probably a better choice of basis for IR 3
than the standard basis, since these vectors align nicely with the bonds between the
atoms of zinc.
v
(a)
(b)
figure 6 . 3
Chan ge-of-Basis Matrices
Figure 6.4 shows two different coordinate systems for IR 2 , each arising from a different
basis. Figure 6.4(a) shows the coordinate system related to the basis !3 = { u , , u2 } ,
while Figure 6.4(b) arises from the basis C = {v1 , vz } , where
The same vector x is shown relative to each coordinate system. It is clear from the
diagrams that the coordinate vectors of x with respect to !3 and C are
[x]8 =
[�]
and [ x ] c =
[ _�]
respectively. It turns out that there is a direct connection between the two coordinate
vectors. One way to find the relationship is to use [ x] 8 to calculate
464
Chapter 6 Vector Spaces
y
y
2
-2
x
-2
-4
-4
(a)
(b)
Figure 6 . 4
Then we can find [ x ] c by writing x as a linear combination of v1 and v2 • However,
there is a better way to proceed-one that will provide us with a general mechanism
for such problems. We illustrate this approach in the next example.
Exa m p l e 6 . 4 5
Using the bases B and C above, find [ x ] c, given that [ x ] 8 =
1
.
3
Solulion
Since x = u 1 + 3u2 , writing u 1 and u2 in terms ofv 1 and v2 will give us the
required coordinates of x with respect to C. We find that
u1 =
and
so
[ -�] -3 [�] 2 [ � ]
[ �] [�] [ � ]
u2 =
+
_
=3
This gives
[x] c =
in agreement with Figure 6.4(b) .
2
= - 3v1 + v2
= 3v1 - V2
[ _�]
This method may not look any easier than the one suggested prior to Example 6.45,
but it has one big advantage: We can now find [ y ] c from [ y ] 8 for any vector y in IR 2
Section 6.3 Change of Basis
465
with very little additional work. Let's look at the calculations in Example 6.45 from a
different point of view. From x = u1 + 3u2, we have
by Theorem 6.6. Thus,
[� ]
[ -� -�] [�]
[ x ] c = [ [ u 1 J d u2 l c l
=
where P is the matrix whose columns are [ u 1 ] c and [ u2 ] c· This procedure generalizes
very nicely.
Let B = { u 1 , . . . , u"} and C = {v1 , . . . , v"} be bases for a vector
space V. The n X n matrix whose columns are the coordinate vectors [u1 ] C• . . . ,
[un l c of the vectors in B with respect to C is denoted by Pc<- B and is called the
change-of-basis matrix from B to C. That is,
D e fi n it i o n
Pc<- B = [ [ u 1 J d u2 l c · · · [ un l c l
Think of B as the "old" basis and C as the "new" basis. Then the columns of Pc<- B
are just the coordinate vectors obtained by writing the old basis vectors in terms of
the new ones. Theorem 6.12 shows that Example 6.45 is a special case of a general
result.
Theorem 6 . 1 2
Let B = {u 1 , . . . , uJ and C = {v1 , . . . , vn } be bases for a vector space V and let
Pc<- B be the change-of-basis matrix from B to C. Then
a. Pc,_ s [ x] 8 = [x] c for all x in V.
b. PC <- 8 is the unique matrix P with the property that P [ x ] 8 = [ x ] c for all x in V.
c. Pc<- B is invertible and (Pc ,_ 8) - 1 = Ps <- c·
Proof
(a) Let x be in V and let
[ x ] c = [ C 1 U 1 + . . . + C n un J c
tl
= C1 [ u 1 J c + . . . + Cn [ un J c
�
I [ u. J c
= Pc<-s [ x ] s
[ u" ] '
466
Chapter 6 Vector Spaces
(b) Suppose that P is an n X n matrix with the property that P [ x] 8 = [ x] c fo r all x
in V. Taking x = U;, the ith basis vector in B, we see that [x] 8 = [ u;] 8 = e;, so the ith
column of P is
P; = Pe ; = P [ u; ] B = [ U; ] c
which is the ith column of Pc ,_ 8, by definition. It follows that P = Pc.-B·
( c) Since {u1, . . . , un } is linearly independent in V, the set { [u1 J c, . . . , [ un l d is linearly
independent in ll�r, by Theorem 6.7. Hence, Pc .-8 = [ [ u 1 J c
[ u ] c is invert­
ible, by the Fundamental Theorem.
For all x in V, we have PC<- B [ x ] 8 = [ x ] c· Solving for [ x l t» we find that
·
·
·
[ x ] 8 = (Pc <-B ) - 1 [ x ] c
for all x in V. Therefore, (Pc .-8) - 1 is a matrix that changes bases from C to B. Thus, by
the uniqueness property (b), we must have (Pc .-8) - 1 = P8.- c .
R e m a rks
You may find it helpful to think of change of basis as a transformation (indeed,
it is a linear transformation) from !R n to itself that simply switches from one coordi­
nate system to another. The transformation corresponding to Pc<-B accepts [ x ] 8 as
input and returns [x]c as output; (Pc ,_ 8) - 1 = p8 ,_ c does just the opposite. Figure 6.5
gives a schematic representation of the process.
•
/
� [ ]B
/ Multiplication �
[ Jc
[x]• c
Change of basis
Figure 6 . 5
v
x•
by Pc� B
Multiplication
[x]• B
by PB� c = (Pc� B ) - 1
The columns of Pc.-8 are the coordinate vectors of one basis with respect to
the other basis. To remember which basis is which, think of the notation C +--- B as
saying "B in terms of C:' It is also helpful to remember that Pc <-t3 [ x] 8 is a linear com­
bination of the columns of Pc <-B· But since the result of this combination is [ xJ c, the
columns of Pc .-8 must themselves be coordinate vectors with respect to C.
•
Exa m p l e 6 . 4 6
Find the change-of-basis matrices Pc.-8 and P8.- c for the bases B = { 1 , x, x 2 } and C =
{ I + x, x + x 2 , 1 + x 2 } of \]]> 2 . Then find the coordinate vector of p(x) = 1 + 2x - x 2
with respect to C.
Changing to a standard basis is easy, so we find PB <- c first. Observe that the
coordinate vectors for C in terms of B are
Solution
Section 6.3 Change of Basis
461
(Look back at the Remark following Example 6.26.) It follows that
To find PC<- l3• we could express each vector in l3 as a linear combination of the vec­
tors in C (do this), but it is much easier to use the fact that Pc,_13 = (P13,_c) - 1 , by
Theorem 6.1 2(c). We find that
Pc<-- 13 = (P13,_c) - 1 =
It now follows that
[ -�: -�: -�::l
2
1
:-
2
2
1
1
:-
-
[p(x) ] c = Pc,_13 [p(x) ] 13
which agrees with Example 6.37.
Ifwe do not need Pc,_13 explicitly, we can find [p(x) l e from [p(x) ] 13 and
P13,_c using Gaussian elimination. Row reduction produces
Remark
[I I [p(x) l c l
(See the next section on using Gauss-Jordan elimination.)
It is worth repeating the observation in Example 6.46: Changing to a standard
basis is easy. If £ is the standard basis for a vector space V and l3 is any other basis,
then the columns of PE<- l3 are the coordinate vectors of l3 with respect to £, and these
are usually "visible:' We make use of this observation again in the next example.
Exa m p l e 6 . 4 1
In M22 , let l3 be the basis {E 11, E 2 1, E1 2 , E 22 } and let C be the basis {A, B, C, D}, where
A-
[ � ! ].
[ J [ ] [
1 O
1 1
1
CB'
'
O O
1
o o
�]
Find the change-of-basis matrix Pc,_13 and verify that [XJ c = Pc,_13 [X] 13 for X =
468
Chapter 6 Vector Spaces
To solve this problem directly, we must find the coordinate vectors of l3
with respect to C. This involves solving four linear combination problems of the form
X = aA bB cC dD, where X is in l3 and we must find a, b, c, and d. However,
here we are lucky, since we can find the required coefficients by inspection.
Clearly, E11 = A, E2 1 = - B C, E1 2 = -A B, and E22 = - C D. Thus,
+ + +
Solulion 1
[E u l c �
rn
+
+
+
m [ I} [ � l [ -!]
:
�
[ � -!]
-
[ £,, ] , �
-
[E , , ] , �
[E,, J c �
Pc�,; � [ [E u l c [E,. ] , [E ,, J c [E,, J c ]
If X =
[� !l
then
[X] s �
Pc +- 13 [ X ] 13 =
and
[i
0
-1
1
0
rn
-1
1
0
0
-!][ ;] [ =]
This is the coordinate vector with respect to C of the matrix
[� �] - [� �] - [ � �] + 4 [ � � ]
[� !]
- A - B - C + 4D = -
= x
as it should be.
We can compute Pc..-13 in a different way, as follows. As you will be
asked to prove in Exercise 2 1 , if £ is another basis for M22 > then Pc ..-13 = Pc ..-EPE..-13 =
(PE..-c) - 1 pE+- l3· If £ is the standard basis, then PE+-l3 and PE+-c can be found by inspec­
tion. We have
Solulion 2
Section 6.3 Change of Basis
�
469
(Do you see why?) Therefore,
Pc +- B = (Pt:+- c) - 1 Pt:+- B
[j J[j �]
[j �m �]
� [j
-� ]
1
0 1
0 0
0
-1
1 -1
1
0
0
0
0 -1
1
-1
1
0
0
0
0
0
1
0
-
0
1
0
0
0 0
0 1
1 0
0 0
which agrees with the first solution.
.+
The second method has the advantage of not requiring the computa­
tion of any linear combinations. It has the disadvantage of requiring that we find a
matrix inverse. However, using a CAS will facilitate finding a matrix inverse, so in
general the second method is preferable to the first. For certain problems, though,
the first method may be just as easy to use. In any event, we are about to describe yet
a third approach, which you may find best of all.
Remark
The Gauss-Jordan Melhod for Compuling a Change-of-Basis Malrix
Finding the change-of-basis matrix to a standard basis is easy and can be done
by inspection. Finding the change-of-basis matrix from a standard basis is almost
as easy, but requires the calculation of a matrix inverse, as in Example 6.46. If we do
it by hand, then (except for the 2 X 2 case) we will usually find the necessary inverse
by Gauss-Jordan elimination. We now look at a modification of the Gauss-Jordan
method that can be used to find the change-of-basis matrix between two nonstandard
bases, as in Example 6.47.
Suppose B = {u 1 , . . . , u"} and C = {v1 , , vn } are bases for a vector space V
and Pc +-B is the change-of-basis matrix from B to C. The ith column of P is
.
•
.
so U; = p 1;v1 + · · · + P n ivn - If £ is any basis for V, then
[ u; ]t; = [ P1 ;V 1 + · · · + Pnivn l t: = P1d v1 J t: + · · · + Pnd vn l t:
This can be rewritten in matrix form as
410
Chapter 6 Vector Spaces
which we can solve by applying Gauss-Jordan elimination to the augmented matrix
[ [v 1 L�·
· · · [vn l t: I [ u; ] EJ
There are n such systems of equations to be solved, one for each column of Pc,_ 8,
but the coefficient matrix [ [ v1 ] E • • • [ vn ] E ] is the same in each case. Hence, we can
solve all the systems simultaneously by row reducing the n X 2n augmented matrix
[ [v1 J t: · · · [ vn l t: l [ u 1 J t: · · ·
[ un l t: l = [ C I B J
Since {V1 ' . . . ' vn } is linearly independent, so is { [ v I l E> ' [ vn l E} ' by Theorem 6. 7.
Therefore, the matrix C whose columns are [ v1 ] E> , [ vn ] E has the n X n identity
matrix I for its reduced row echelon form, by the Fundamental Theorem. It follows
that Gauss-Jordan elimination will necessarily produce
•
•
•
•
•
•
[ C I B J ---+ [ I I P J
where P = Pc<-- 8.
We have proved the following theorem.
Theorem 6 . 1 3
Let l3 = {u 1 , , un } and C = {v1 , . . . , vn } be bases for a vector space V. Let
B = [ [ u 1 l t: . . . [ un l t: l and C = [ [v 1 ] ,<: . . . [vn l t: L where £ is any basis for V.
Then row reduction applied to the n X 2n augmented matrix [ C I B ] produces
[ C I B J ---+ [ I I Pc.-al
.
•
.
If £ i s a standard basis, this method i s particularly easy t o use, since i n that case
B = PE<-- B and C = PE<-- C · We illustrate this method by reworking the problem in
Example 6.47.
Exa m p l e 6 . 4 8
Rework Example 6.47 using the Gauss-Jordan method.
Taking £ to be the standard basis for M22 , we see that
Solution
B = PE<-- B =
[l f l
0 0
0
1 0
0 0
Row reduction produces
IC I BJ
�
�
[l
1
1 1
1
1 0
0 1
0
0 0 1 0
0
0
1
0
0
1
0
0
and C = PE.-c =
[
f: l
�
(Verify this row reduction.) It follows that
as we found before.
P
,�,
� [l
0 -1
1
-1
1 0
0 0
[l : :
0 1
0 0
0 0 0 1 0 -1
0 0 0 -1
1
0 1 0 0 1 0
0 0
0 0 0
-�]
-�]
I
Section Change of Basis
6.3
Exercises 6 . 3
[x] 8 [x]c x
[x]
In Exercises 1 -4:
(a) Find the coordinate vectors
and
of with
respect to the bases B and C, respectively.
(b) Find the change-of-basis matrix Pc,_8from B to C.
(c) Use your answer to part (b) to compute c, and
compare your answer with the one found in part (a).
(d) Find the change-of-basis matrix P8,_ c from C to B.
(e) Use your answers to parts (c) and (d) to compute
and compare your answer with the one found in part (a).
x = [ � ], = { [ � ], [ � ] } ,
= { [�l [ �] }
x = [ �l = { [�l [ � ] }
= { [�l [�] }
x � [ H � un m. [�J l.
c � m H:WJ l
4. x � [; J s � m J m . [ m
c � m rn rn J l
.
x
== = = = == - = =
1.
[ x ] 8,
B
C
2.
in � 2
_
_
B
,
in � 2
C
3.
B
_
in � '
in � '
In Exercises 5-8, follow the instructions for Exercises 1 -4
using p ( x) instead of
5. p ( x) 2 - x, B { l , x}, C {x, 1 + x} in 0\
6. p(x) 1 + 3x, B = { l + x, 1 x},
C = {2x, 4} in 0 \
7. p(x) 1 + x 2, B { l + x + x 2, x + x 2, x 2},
C
411
{ l , x, x 2} in rzf 2
8. p(x) 4 2x - x 2, B
C = { l , 1 + x, x 2} in rzf 2
{x, 1 + x2 , x + x 2},
x.
= [ � � ], =
c = { [ � - � l [ � � ], [ � � ], [ � � ] }
= [l 1]
= { [ � � l [ � �l [ � �l [ � � ] } ,
= { [ � � l [ � �l [ � � l [ � � ] }
.
x
== - =
=
=
In Exercises 9 and 1 0, follow the instructions for
Exercises 1 -4 using A instead of
9. A
B
_
10. A
1 1
the standard basis,
in M22
,
B
C
in M22
In Exercises 1 1 and 12, follow the instructions for
Exercises 1 -4 usingf(x) instead of
1 1 . f(x) 2 sin x 3 cos x, B {sin x + cos x, cos x},
C {sin x + cos x, sin x - cos x} in span (sin x, cos x)
12. f(x) = sin x, B = {sin x + cos x, cos x},
C {cos x - sin x, sin x + cos x} in span (sin x, cos x)
13. Rotate the xy-axes in the plane counterclockwise
through an angle () 60° to obtain new x' y' -axes.
Use the methods of this section to find (a) the
x'y'-coordinates of the point whose xy-coordinates
are (3, 2) and (b) the xy-coordinates of the point
whose x'y' -coordinates are (4, - 4).
14. Repeat Exercise 1 3 with ()
=
1 35°.
= { [ � ] [�] }
[ � -�]
15. Let B and C be bases for � 2 . If C
,
the change-of-basis matrix from B to C is
Pc<-B =
_
find B.
16. Let B and C be bases for rzf 2 . If B = {x, 1 + x,
1 x + x2 } and the change-of-basis matrix
from B to C is
-
find C.
and
412
Chapter 6 Vector Spaces
In calculus, you learn that a Taylor polynomial of degree n
about a is a poly nomial of the form
p(x) = a0 + a 1 (x - a) + a 2 (x - a) 2 + · · · + a " (x - a) "
where a n * 0. In other words, it is a polynomial that has
been expanded in terms ofpowers of x - a instead ofpow­
ers of x. Taylor poly nomials are very useful for approximat­
ingfunctions that are "well behaved" near x = a.
The set B = { l , x - a, (x - a) 2 , , (x - a) " } is a basis
for rtl'n for any real number a. (Do you see a quick way to
show this? Try using Theorem 6. 7.) This fact allows us to use
the techniques of this section to rewrite a poly nomial as a
Taylor poly nomial about a given a.
17. Express p (x) = 1 + 2x - 5x2 as a Taylor polynomial
about a = 1 .
•
•
•
18. Express p ( x) = 1 + 2 x - sx2 as a Taylor polynomial
about a = - 2 .
19. Express p (x) = x3 as a Taylor polynomial about a = - 1 .
20. Express p ( x) = x3 as a Taylor polynomial about a = t .
21. Let B, C, and V be bases for a finite-dimensional vec­
tor space V. Prove that
Pv +-cPc+-B = Pv +- B
22. Let V be an n-dimensional vector space with basis
B = {v 1 ,
, vn }. Let P be an invertible n X n matrix
.
•
.
and set
fo r i = 1 , . . . , n. Prove that C = {u 1 ,
for V and show that P = Ps+-c·
•
.
•
, uJ is a basis
linear Tra n st o r m a l i o n s
We encountered linear transformations in Section 3 . 6 in the context o f matrix trans­
formations from IR " to !R m . In this section, we extend this concept to linear transfor­
mations between arbitrary vector spaces.
A linear transformation from a vector space V to a vector space
W is a mapping T : V ---+ W such that, for all u and v in V and for all scalars c,
D e fi n it i o n
1. T(u + v) = T(u) + T(v)
2. T(cu) = cT(u)
It is straightforward to show that this definition is equivalent to the requirement
that T preserve all linear combinations. That is,
T : V ---+ W is a linear transformation if and only if
T(c 1 v1 + c2v2 + · · · + ckvk ) = c 1 T(v1 ) + c2 T(v2 ) + · · · + ck T(vk )
for all v1 , . . . , vk in V and scalars c 1 , . . . , ck .
Exa m p l e 6 . 4 9
Every matrix transformation is a linear transformation. That is, if A is an m X n
matrix, then the transformation TA : IR " ---+ !R m defined by
TA ( x) = Ax for x in IR "
is a linear transformation. This is a restatement of Theorem 3.30.
Section 6.4 Linear Transformations
Exa m p l e 6 . 5 0
413
Define T : Mnn ---+ Mnn by T(A) = A T. Show that T is a linear transformation.
Solution
We check that, for A and B in Mnn and scalars c,
T(A + B) = (A + Bf = A T + B T = T(A) + T(B)
T(cA) = (cAf = cA T = cT(A)
and
Therefore, T is a linear transformation.
Exa m p l e 6 . 5 1
Let D be the differential operator D : 0J ---+ '2F defined by D(j) = f' . Show that D is a
linear transformation.
Let f and g be differentiable functions and let c be a scalar. Then, from
calculus, we know that
Solution
D(f + g) = (j + g) ' = f' + g' = D (j) + D(g)
and
D(cj) = (cf) ' = cf' = cD(j)
Hence, D is a linear transformation.
In calculus, you learn that every continuous function on [a, b] is integrable. The
next example shows that integration is a linear transformation.
Exa m p l e 6 . 5 2
Define S : � [ a, b ] ---+
Solution
IR:
by S(j) = J: f (x) dx. Show that S is a linear transformation.
Let f and g be in � [a, b] . Then
S(f + g) =
=
r(j + g) (x) dx
r(j(x) + g(x)) dx
rf(x) dx + rg(x) dx
a
a
=
a
= S(f) + S(g)
and
S(cf) =
=
r(cj)(x) dx
fcf(x) dx
cff(x) dx
a
a
=
a
= cS(f)
It follows that S is linear.
a
414
Chapter 6 Vector Spaces
Exa m p l e 6 . 5 3
Show that none of the following transformations is linear:
(a) T : M22 ---+ IR defined by T(A) = <let A
(b) T : IR ---+ IR defined by T(x) = 2x
(c) T : IR ---+ IR defined by T(x) = x + 1
In each case, we give a specific counterexample to show that one of the
properties of a linear transformation fails to hold.
Solution
(a) Let A =
[� �]
and B =
[ � � ].
[� �l
I � �I
I� � I I � � I
Then A + B =
T(A + B ) = <let (A + B ) =
But
T(A ) + T( B ) = detA + detB =
so
= 1
+
=0+0=0
so T(A + B) * T(A) + T(B) and T is not linear.
(b) Let x = 1 and y = 2. Then
T ( x + y) = T ( 3 ) = 2 3 = 8 * 6 = 2 1 + 2 2 = T ( x) + T (y)
so T is not linear.
(c) Let x = 1 and y = 2. Then
T( x + y) = T( 3 ) = 3 + 1 = 4 * 5 = ( 1 + 1) + (2 + 1) = T ( x) + T(y)
Therefore, T is not linear.
Example 6.53(c) shows that you need to be careful when you encounter
the word "linear:' As a function, T(x) = x + 1 is linear, since its graph is a straight
line. However, it is not a linear transformation from the vector space IR to itself, since
it fails to satisfy the definition. (Which linear functions from IR to IR will also be linear
transformations?)
Remark
�
4
There are two special linear transformations that deserve to b e singled out.
Exa m p l e 6 . 5 4
(a) For any vector spaces V and W, the transformation T0 : V ---+ W that maps every
vector in V to the zero vector in W is called the zero transformation. That is,
T0 ( v) = 0 for all v in V
(b) For any vector space V, the transformation I : V ---+ V that maps every vector in V
to itself is called the identity transformation. That is,
J ( v) = v for all v in V
(If it is important to identify the vector space V, we may write Iv for clarity.) The
proofs that the zero and identity transformations are linear are left as easy exercises.
4
Section 6.4 Linear Transformations
415
Properties of linear Tra nsformations
_.,
Theorem 6 . 1 4
In Chapter 3, all linear transformations were matrix transformations, and their
properties were directly related to properties of the matrices involved. The fol­
lowing theorem is easy to prove for matrix transformations. (Do it! ) The full
proof for linear transformations in general takes a bit more care, but it is still
straightforward.
Let T : V ---+ W b e a linear transformation. Then:
a. T (O) = 0
b. T ( -v) = - T(v) for all v in V.
c. T (u - v) = T (u) - T (v) for all u and v in V.
We prove properties (a) and (c) and leave the proof of property (b) for
Exercise 2 1 .
Proof
_.,
(a) Let v be any vector in V. Then T (O) = T (Ov) = O T (v) = 0, as required. (Can you
give a reason for each step?)
(c) T(u - v) = T(u + ( - l )v) = T(u) + ( - l ) T(v) = T(u) - T(v)
Remark
Property (a) can be useful in showing that certain transformations are
not linear. As an illustration, consider Example 6.53(b ). If T(x) = 2x, then T(O) = 2 ° =
1 * 0, so T is not linear, by Theorem 6. 14(a) . Be warned, however, that there are lots
of transformations that do map the zero vector to the zero vector but that are still not
linear. Example 6.53(a) is a case in point: The zero vector is the 2 X 2 zero matrix 0,
so T( 0) = det 0 = 0, but we have seen that T(A) = det A is not linear.
The most important property of a linear transformation T : V ---+ W is that T is
completely determined by its effect on a basis for V. The next example shows what
this means.
Exa m p l e 6 . 5 5
Suppose T is a linear transformation from IR 2 to <!J' 2 such that
[�]
[ � J [�J
r
Find r
_.,
Solution
-
and r
Since B =
span (B) . Solving
= 2 - 3x + x2
and r
[�]
= 1 - x2
.
{ [ � ] , [ � ] } is a basis for IR2 (why?), every vector in IR2 is in
416
Chapter 6 Vector Spaces
we find that c 1 = - 7 and c2 = 3. Therefore,
= - 7r
[ � ] [�]
+ 3T
= - 7 ( 2 - 3x + x 2 ) + 3 ( 1 - x 2 )
= - 1 1 + 21x - 1 0x 2
Similarly, we discover that
[ :]
so
r
= ( 3a - 2 b)
[:J (
[�J
[ �J
[�]
+ ( b - a)
= r ( 3 a - 2b )
+ ( b - a)
= ( 3 a - 2b ) r
+ ( b - a) r
[�]
[�] )
[�J
= ( 3 a - 2b ) ( 2 - 3x + x 2 ) + ( b - a ) ( l - x 2 )
= ( Sa - 3b) + ( - 9a + 6b ) x + ( 4a - 3b ) x 2
II-
(Note tha\by setting a = - 1 and b = 2, we recover the solution r
2lx - lOx
.)
[ -�]
= -11 +
4
The proof of the general theorem is quite straightforward.
Theorem 6 . 1 5
Let T : V -+ W be a linear transformation and let l3 = {v 1 , , vn } be a spanning
set for V. Then T ( B ) = { T ( v1 ) , , T ( vn ) } spans the range of T.
•
•
•
•
•
•
The range of T is the set of all vectors in W that are of the form T(v), where
v is in V. Let T(v) be in the range of T. Since l3 spans V, there are scalars c 1 ,
, en
such that
Proof
.
.
•
Applying T and using the fact that it is a linear transformation, we see that
T ( v) = T ( c 1 v 1 + · · · + c nvn ) = c 1 T ( v1 ) + · · · + cn T ( vn )
In other words, T(v) is in span ( T(B)), as required.
Theorem 6. 1 5 applies, in particular, when l3 is a basis for V. You might guess that,
in this case, T(B) would then be a basis for the range of T. Unfortunately, this is not
always the case. We will address this issue in Section 6.5.
Composition of linear Tra nsformations
In Section 3.6, we defined the composition of matrix transformations. The definition
extends to general linear transformations in an obvious way.
Section 6.4 Linear Transformations
S T
0
is read of
"S
T:'
411
D e fi n it i o n If T : U ---+ V and S : V ---+ W are linear transformations, then the
composition of S with T is the mapping S 0 T, defined by
(S 0 T) (u) = S( T(u))
where u is in U.
Observe that S 0 T is a mapping from U to W (see Figure 6.6) . Notice also that for
the definition to make sense, the range of T must be contained in the domain of S.
u
U•
T
•
T(u)
v
s
--+
w
S(T(u)) = (S T)(u)
0
•
Composition of linear transformations
Figure 6 . 6
Exa m p l e 6 . 5 6
Let T : IR 2 ---+ <!/' 1 and S : <!/' 1 ---+ <!/' 2 be the linear transformations defined by
Find (S 0 T)
Solution
[ �]
_
r[ :]
= a + (a + b )x
and (S 0 T)
and S(p(x)) = xp (x)
[ :].
We compute
(S 0 T)
[ _� ] s( r[ _� ] )
=
= S(3 + (3 - 2)x) = S(3 + x) = x(3 + x)
= 3x + x 2
and
(S 0
T) [ :J ( r [ :] )
= s
= S(a + (a + b)x) = x(a + (a + b)x)
= ax + (a + b)x 2
Chapter 3 showed that the composition of two matrix transformations was
another matrix transformation. In general, we have the following theorem.
Theorem 6 . 1 6
If T : U ---+ V and S : V ---+ W are linear transformations, then S 0 T : U ---+ W is a
linear transformation.
418
Chapter 6 Vector Spaces
Proof
Let u and v be in U and let c be a scalar. Then
(S T) (u + v) = S( T(u + v))
= S( T(u) + T(v))
= S( T(u)) + S(T(v))
= (S T)(u) + (S T) (v)
0
0
and
0
(S T)(cu) = S( T(cu))
= S(cT(u))
= cS( T(u))
= c(S T) (u)
0
0
since Tis linear
since is linear
S
since Tis linear
since is linear
S
Therefore, S T is a linear transformation.
0
The algebraic properties of linear transformations mirror those of matrix trans­
formations, which, in turn, are related to the algebraic properties of matrices. For
example, composition of linear transformations is associative. That is, if R, S, and T
are linear transformations, then
provided these compositions make sense. The proof of this property is identical to
that given in Section 3.6.
The next example gives another useful (but not surprising) property of linear
transformations.
Exa m p l e 6 . 5 1
Let S : U � V and T : V � W be linear transformations and let I : V � V be the iden­
tity transformation. Then for every v in V, we have
( T J)(v) = T(I(v)) = T(v)
Since T I and T have the same value at every v in their domain, it follows that
T I = T. Similarly, I S = S.
0
0
o
o
Remark
The method of Example 6.57 is worth noting. Suppose we want to show
that two linear transformations Ti and T2 (both from V to W ) are equal. It suffices to
show that Ti ( v) = T2 (v) for every v in V.
Further properties of linear transformations are explored in the exercises.
Inverses or Linear Transrormalions
D e fi n il i O D A linear transformation T : V � W is invertible if there is a linear
transformation T' : W � V such that
T' T = Iv and T T ' = Iw
In this case, T' is called an inverse for T.
o
0
Section 6.4 Linear Transformations
419
R e m a rks
The domain V and codomain W of T do not have to be the same, as they do in
the case of invertible matrix transformations. However, we will see in the next section
that V and W must be very closely related.
•
The requirement that T' be linear could have been omitted from this definition.
For, as we will see in Theorem 6.24, if T' is
mapping from W to V such that
T' 0 T = Iv and T 0 T' = Iw, then T' is forced to be linear as well.
•
If T' is an inverse for T, then the definition implies that T is an inverse for T'.
Hence, T' is invertible too.
•
any
Exa m p l e 6 . 5 8
Verify that the mappings T : IR 2 ---+ r!J> 1 and T' : r!J> 1 ---+ IR 2 defined by
r [ : ] a + (a +
=
b ) x and T ' ( c
are inverses.
and
dx) =
We compute
Solution
(T'
+ [d � J
o
r) [ :J r' (r [ :] ) T' (a + (a + [ (a + :) _ a ] [ :]
+
+ r[ ] + +
+
=
( T 0 T ') (c
dx) = T ( T ' ( c
b ) x) =
=
dx)) =
c
d-c
= c
(c
( d - c )) x = c
dx
Hence, T ' T = IIR2 and T 0 T ' = I21' 1 • Therefore, T and T' are inverses of each
other.
o
4
As was the case for invertible matrices, inverses of linear transformations are
unique if they exist. The following theorem is the analogue of Theorem 3.6.
Theorem 6 . 1 1
If T is an invertible linear transformation, then its inverse is unique.
The proof is the same as that of Theorem 3.6, with products of matrices re­
placed by compositions of linear transformations. (You are asked to complete this
proof in Exercise 3 1 .)
Proof
Thanks to Theorem 6. 1 7, if T is invertible, we can refer to the inverse of T. It will
be denoted by r - 1 (pronounced " T inverse"). In the next two sections, we will ad­
dress the issue of determining when a given linear transformation is invertible and
finding its inverse when it exists.
480
..
I
Chapter 6 Vector Spaces
Exercises 6 . 4
In Exercises 1 - 12, determine whether T is a linear
transformation.
1. T : M22 M22 defined by
a b
a+b
=
r
c d
0
c+d
2. T : M22 M22 defined by
1
w-z
w x
r
1
x-y
y z
3. T : Mnn Mnn defined by T (A ) = AB, where B is a
fixed n X n matrix
4. T : Mnn Mnn defined by T (A ) = AB - BA, where B
is a fixed n X n matrix
5. T : Mnn IR defined by T (A ) = tr (A )
6. T : Mnn IR defined by T(A) = a ll a 22 a nn
7. T : Mnn IR defined by T (A ) = rank (A )
8. T : <JP2 <!P2 defined by T(a + bx + cx 2 ) = (a + 1 ) +
(b + l) x + (c + l ) x 2
9. T : <!P2 <!P2 defined by T(a + bx + cx 2 ) = a +
b(x + 1 ) + b(x + 1 ) 2
10. T : gjf ---,l> gji defined by T (j) = f (x 2 )
1 1 . T : gji gji defined by T (j) = (j(x) ) 2
12. T : gji IR defined by T (j) = j (c) , where c is a fixed
OJ
--,lo
[ ] [
[ ] [
]
--,lo
--,lo
16. Let T : <JP2 <JP2 be a linear transformation for which
T( l) = 3 - 2x, T(x) = 4x - x2 , and T(x2 ) = 2 + 2x2
Find T(6 + x - 4x2 ) and T(a + bx + cx 2 ).
17. Let T : <!P2 <!P2 be a linear transformation for which
T( l + x) = 1 + x 2 , T(x + x 2 ) = x - x 2 ,
T( 1 + x 2) = 1 + x + x 2
--,lo
--,lo
Find T ( 4 - x + 3x 2 ) and T(a + bx + cx 2 ).
18. Let T : M22 --,lo IR be a linear transformation for which
--,lo
--,lo
--,lo
•
•
•
--,lo
--,lo
--,lo
--,lo
--,lo
scalar
13. Show that the transformations S and T in Exam­
ple 6.56 are both linear.
14. Let T : IR 2 --,lo IR 3 be a linear transformation for which
[� J [�J
[ �]
[�]
[ - �] [ �J.
Find r
15. Let T : IR
T
Find r
and r
2
--,lo
.
<JP2 be a linear transformation for which
= 1 - 2x
and r
and T
_
= x + 2x 2
Find r
[� OJ [� ]
[� ] [� �]
[� � ] [ : �].
r
0
r
1
0
= 1'
r
= 3,
r
1
0
= 2'
= 4
and r
19. Let T : M22
--,lo IR be a linear transformation. Show that
there are scalars a, b, c, and d such that
[ ; :J
[ ; :J
r
for all
= aw + bx + cy + dz
in M22 .
20. Show that there is no linear transformation T : IR 3
such that
{]
� 1 + x,
t! J
r
[�]
--,lo
� 2 - x + x' .
� - 2 + 2x'
21. Prove Theorem 6. 14(b) .
22. Let {v1 , , vn } be a basis for a vector space V and
•
.
•
<JP 2
let T : V --,lo V be a linear transformation. Prove that if
T (v1) = V1, T (v2) = V2 . . . , T ( vn ) = vn , then T is the
identity transformation on V.
� 23. Let T : <JP n --,lo <JP n be a linear transformation such that
T(x k ) = kx k - I for k = 0, 1 , . . . , n. Show that T must
be the differential operator D.
Section 6.5 The Kernel and Range of a Linear Transformation
24. Let v1 , • • • , v" be vectors in a vector space V and let
T : V ---+ W be a linear transformation.
( a ) If { T(v1 ), . . . , T(vn )} is linearly independent in W,
show that {v1 , • • • , vJ is linearly independent in V.
(b) Show that the converse of part (a) is false.
That is, it is not necessarily true that if
{v1 , . • . , vn } is linearly independent in V, then
{T(v1 ), • • • , T(v)} is linearly independent in W.
Illustrate this with an example T : IR 2 ---+ IR 2 •
25. Define linear transformations S : IR 2 ---+ M22 and
T : IR 2 ---+ IR 2 by
s[ � ] [ :
a
=
b
Compute (S T)
0
compute ( T S)
0
]
[] [ :]
[;].
� b and T �
[�]
[;]?
a
and (S T)
0
2c d
_
Can you
If so, compute it.
26. Define linear transformations S : <!P 1 ---+ <!P 2 and
T : <!P 2 ---+ <!P 1 by
S(a + bx) = a + (a + b )x + 2bx 2
and
T(a + bx + cx 2 ) = b + 2cx
Compute (S T)(3 + 2x - x 2 ) and
(S T)(a + bx + cx 2 ) . Can you compute
( T S) (a + bx)? If so, compute it.
� 27. Define linear transformations S : <;if " <;if " and
T : <!P n -+ <!P n by
S(p(x)) = p(x + 1 ) and T(p(x)) = p ' (x)
0
0
0
---+
Find (S T) (p(x)) and ( T S) (p(x)). [Hint: Remember
the Chain Rule.]
� 28. Define linear transformations S : <;if n ---+ <;if n and
T : <;if n -+ <;if n by
0
0
S(p (x)) = p (x + 1 ) and T(p(x)) = xp ' (x)
Find (S T)(p(x)) and ( T S)(p(x)).
0
0
s[x] [ ]
[] [ ]
In Exercises 29 and 30, verify that S and T are inverses.
4x + y
=
29. S : IR 2 -+ IR 2 defined by
and T: IR 2 -+ IR 2
3x + y
y
x = x-y
defined by T
y
- 3x + 4y
30. S : <!P 1 ---+ <!P 1 defined by S(a + bx) =
( - 4a + b) + 2ax and T : <!P 1 ---+ <!P 1 defined by
T(a + bx) = b/2 + (a + 2b)x
31. Prove Theorem 6. 17.
32. Let T : V ---+ V be a linear transformation such that
T o T = I.
(a) Show that {v, T(v)} is linearly dependent if and
only if T(v) = ±v.
(b) Give an example of such a linear transformation
with V = IR 2 •
33. Let T : V ---+ V be a linear transformation such that
T T = T.
( a) Show that {v, T(v)} is linearly dependent if and
only if T(v) = v or T(v) = 0.
0
(b) Give an example of such a linear transformation
with V = IR 2 .
The set of all linear transformations from a vector space V
to a vector space W is denoted by ;£', ( V, W ) . If S and T are
in ;£', ( V, W ) , we can define the sum S + T of S and T by
(S + T) (v) = S(v) + T(v)
for all v in V If c is a scalar, we define the scalar multiple
cT of T by c to be
(cT)(v) = cT(v)
for all v in V Then S + T and cT are both transformations
from V to W
34. Prove that S + T and cT are linear transformations.
35. Prove that H', ( V, W ) is a vector space with this addi­
tion and scalar multiplication.
36. Let R, S, and T be linear transformations such that the
following operations make sense. Prove that:
( a) R o (S + T) = R o S + R o T
(b) c(R S) = (cR) S = R (cS) for any scalar c
0
II
481
0
0
T h e Kernel a n d R a n g e o f a l i n e a r Transfo r m a t i o n
The null space and column space are two o f the fundamental subspaces associated
with a matrix. In this section, we extend these notions to the kernel and range of a
linear transformation.
482
Chapter 6 Vector Spaces
wordEnglish word
is derived froma
theThe
Old
form oforthe"graiwordn:' Like ameaning
"seed"
kerneltrans­of
corn,
the
kernel
of
a
linear
formation
is itist"core"
orinforma­
"seed" in
thetion
sense
that
carries
about ofmanythe transformation.
of the important
properties
kernel
cyrnel,
corn,
Exa m p l e 6 . 5 9
D e fi n it i o n Let T: V---+ W be a linear transformation. The kernel of T, denoted
ker ( T), is the set of all vectors in V that are mapped by T to 0 in W. That is,
ker ( T) = {v in V: T ( v) = O}
The range of T, denoted range(T), is the set of all vectors in W that are images of
vectors in V under T. That is,
range ( T ) = { T ( v) : v in V}
= {w in W : w = T ( v) for some v in V}
Let A be an m X n matrix and let T = TA be the corresponding matrix transformation
from !R n to !R m defined by T(v) = Av. Then, as we saw in Chapter 3, the range of T is
the column space of A.
The kernel of T is
ker ( T) = {v in !R n : T (v) = O}
= {v in !R n : Av = O}
= null (A )
In words, the kernel of a matrix transformation is just the null space of the corre­
sponding matrix.
Exa m p l e 6 . 6 0
Find the kernel and range of the differential operator D : <;if 3 ---+ <!P 2 defined by
D (p(x)) = p'(x).
Solulion
Since D (a + bx + cx 2 + dx 3 ) = b + 2cx + 3dx 2 , we have
ker ( D ) = {a + bx + cx 2 + dx 3 : D(a + bx + cx 2 + dx 3 ) = O}
= {a + bx + cx 2 + dx 3 : b + 2cx + 3dx 2 = O}
But b + 2cx + 3dx 2 = 0 if and only if b = 2c = 3d = 0, which implies that b = c =
d = 0. Therefore,
ker ( D ) = {a + bx + cx 2 + dx 3 : b = c = d = O}
= {a : a in IR}
In other words, the kernel of D is the set of constant polynomials.
The range of D is all of<!P 2 , since every polynomial in <!P 2 is the image under D (i.e.,
the derivative) of some polynomial in <!P 3 . To be specific, if a + bx + cx 2 is in <!P 2 , then
Section 6.5 The Kernel and Range of a Linear Transformation
Exa m p l e 6 . 6 1
483
Let S : <!/' 1 � IR be the linear transformation defined by
p x rp (x)dx
S ( ( )) =
Find the kernel and range of S.
Solution
0
In detail, we have
S(a + bx) r (a + bx)dx
= [ ax + !x I
= ( a + %) - o = a + %
a + bx: S(a + bx)
{ a + bx : a + % 0 }
{ a + bx : a - !}
{ - !+ bx }
y
=
0
2
b
2
ker ( S ) = {
Therefore,
1
2
= O}
=
b
2
=
Ify b + bx
then rydx
Figure 6 . 1
--
=
2
0
=
'
0
Geometrically, ker(S ) consists of all those linear polynomials whose graphs have the
property that the area between the line and the x-axis is equally distributed above and
below the axis on the interval [O, l ] (see Figure 6.7).
The range of S is IR, since every real number can be obtained as the image under
S of some polynomial in <!/' 1 . For example, if is an arbitrary real number, then
so
Exa m p l e 6 . 6 2
a a
{ dx [ x
0
= S( ) .
a
=
a
a n= a
-o a
=
Let T : M22 � M22 b e the linear transformation defined by taking transposes:
T(A) = A T. Find the kernel and range of T.
Solution
We see that
ker ( T) = {A in M22 : T (A) = O}
= {A in M22 : AT = O}
But if A T = 0, then A = (A T ) T = o T = 0. It follows that ker (T) = { O}.
Since, for any matrix A in M22 , we have A = (A T ) T = T(A T ) (and A T is in M22 ),
we deduce that range(T) = M22 .
In all of these examples, the kernel and range of a linear transformation are sub­
spaces of the domain and co domain, respectively, of the transformation. Since we are
generalizing the null space and column space of a matrix, this is perhaps not surpris­
ing. Nevertheless, we should not take anything for granted, so we need to prove that
it is not a coincidence.
484
Chapter 6 Vector Spaces
Theorem 6 . 1 8
Let T : V ---+ W b e a linear transformation. Then:
a. The kernel of T is a subspace of V.
b. The range of T is a subspace of W.
(a) Since T (O) = 0, the zero vector of V is in ker(T), so ker(T) is nonempty.
Let u and v be in ker ( T ) and let c be a scalar. Then T (u) = T (v) = 0, so
Proof
T(u + v) = T(u) + T(v) = 0 + 0 = 0
and
T(cu) = cT(u) = c O = 0
Therefore, u + v and c u are in ker( T), and ker(T) is a subspace of V.
(b) Since 0 = T(O), the zero vector of W is in range(T), so range(T) is nonempty.
Let T (u) and T (v) be in the range of T and let c be a scalar. Then T(u) + T(v) =
T(u + v) is the image of the vector u + v. Since u and v are in V, so is u + v, and
hence T(u) + T(v) is in range (T). Similarly, cT(u) = T(c u) . Since u is in V, so is cu,
and hence cT(u) is in range(T). Therefore, range(T) is a nonempty subset of W that is
closed under addition and scalar multiplication, and thus it is a subspace of W.
Figure 6.8 gives a schematic representation of the kernel and range of a linear
transformation.
rne
a g (T)
v
�
0
T
The kernel and range of
•o
W
Figure 6 . 8
T : V --+ W
In Chapter 3, we defined the rank of a matrix to be the dimension of its column
space and the nullity of a matrix to be the dimension of its null space. We now extend
these definitions to linear transformations.
Let T : V ---+ W be a linear transformation. The rank of T is the
dimension of the range of T and is denoted by rank( T). The nullity of T is
the dimension of the kernel of T and is denoted by nullity( T).
Definition
Exa m p l e 6 . 6 3
If A is a matrix and T = TA is the matrix transformation defined by T (v) = Av, then
the range and kernel of T are the column space and the null space of A, respectively,
by Example 6.59. Hence, from Section 3.5, we have
rank ( T) = rank (A ) and nullity ( T) = nullity (A )
Section 6.5 The Kernel and Range of a Linear Transformation
Exa m p l e 6 . 6 4
485
Find the rank and the nullity of the linear transformation D : l!P 3 � <!J 2 defined by
D(p(x)) = p'(x).
Solution
In Example 6.60, we computed range ( D ) = <!J 2 , so
rank ( D ) = dim <!f 2 = 3
The kernel of D is the set of all constant polynomials: ker (D) = {a : a in IR} = {a · 1 : a
in IR}. Hence, { 1 } is a basis for ker (D), so
nullity ( D ) = dim ( ker ( D )) = 1
Exa m p l e 6 . 6 5
Find the rank and the nullity of the linear transformation S : <!J 1 � IR defined by
S(p(x)) =
Solution
r
0
p (x) dx
From Example 6.6 1 , range(S ) = IR and rank(S ) = dim IR = 1. Also,
ker ( S ) =
{�
-
+ bx : b in IR
}
= {b ( - t + x) : b in IR}
= span ( - t + x)
so { -t + x} is a basis for ker(S ). Therefore, nullity(S ) = dim(ker (S )) = 1 .
Exa m p l e 6 . 6 6
Find the rank and the nullity of the linear transformation T : M22 � M22 defined by
T (A) = A T.
Solution
In Example 6.62, we found that range( T ) = M22 and ker( T) = {O}. Hence,
rank ( T ) = dim M22 = 4 and nullity ( T ) = dim{O} = 0
In Chapter 3, we saw that the rank and nullity of an m X n matrix A are related
by the formula rank(A) + nullity(A) = n. This is the Rank Theorem (Theorem 3.26).
Since the matrix transformation T = TA has !R n as its domain, we could rewrite the
relationship as
rank (A ) + nullity (A ) = dim !R n
This version of the Rank Theorem extends very nicely to general linear transforma­
tions, as you can see from the last three examples:
rank ( D ) + nullity ( D ) = 3 + 1 = 4 = dim l!P3
rank ( S ) + nullity ( S) = 1 + 1 = 2 = dim <!J 1
rank ( T ) + nullity ( T ) = 4 + 0 = 4 = dim M22
Example 6.64
Example 6.65
Example 6.66
486
Chapter 6 Vector Spaces
Theorem 6 . 1 9
The Rank Theorem
Let T : V ---+ W be a linear transformation from a finite-dimensional vector space
V into a vector space W. Then
rank ( T ) + nullity ( T ) = dim V
In the next section, you will see how to adapt the proof of Theorem 3.26 to prove
this version of the result. For now, we give an alternative proof that does not use
matrices.
Let dim V = n and let {v1, . . . , vk } be a basis for ker (T) [so that nullity( T) =
dim(ker( T)) = k] . Since {v1, . . . , vd is a linearly independent set, it can be extended
to a basis for V, by Theorem 6.28. Let B = {v1 , . . . , vk > vk+ 1 , . . . , vn } be such a basis.
If we can show that the set C = { T (vk + 1), . . . , T (vn ) } is a basis for range( T), then we
will have rank( T) = dim(range( T)) = n - k and thus
Proof
rank ( T ) + nullity ( T ) = k + (n - k) = n = dim V
as required.
Certainly C is contained in the range of T. To show that C spans the range of T, let
T (v) be a vector in the range of T. Then v is in V, and since B is a basis for V, we can
find scalars c1, . . . , en such that
Since v1,
.
•
.
, vk are in the kernel of T, we have T (v1 ) = · · · = T( vk ) = 0, so
T ( v) = T(c 1 V 1 + . . . + c kvk + Ck+1 Vk+1 + . . . + c nvn )
= c 1 T(v1 ) + · · · + ck T(vk ) + c k+1 T(vk+1 ) + · · · + c n T(vn )
= Ck+ 1 T(vk+1 ) + . . . + c n T(vn )
This shows that the range of T is spanned by C.
To show that C is linearly independent, suppose that there are scalars ck + 1, . . . , en
such that
Then T ( ck + I Vk + I + + cnvn ) = 0, which means that Ck + ! Vk + l + + cnvn is in the
kernel of T and is, hence, expressible as a linear combination of the basis vectors
v1, . . . , vk of ker ( T)-say,
· · ·
· · ·
But now
and the linear independence of B forces c1 = = en = 0. In particular, ck + 1 = =
en = 0, which means C is linearly independent.
We have shown that C is a basis for the range of T, so, by our comments above, the
proof is complete.
· · ·
· · ·
We have verified the Rank Theorem for Examples 6.64, 6.65, and 6.66. In practice,
this theorem allows us to find the rank and nullity of a linear transformation with
only half the work. The following examples illustrate the process.
Section 6.5 The Kernel and Range of a Linear Transformation
Exa m p l e 6 . 6 1
�
481
Find the rank and nullity of the linear transformation T : <!/' 2 � <!/' 3 defined by
T(p(x)) = xp(x) . (Check that T really is linear.)
In detail, we have
T ( a + bx + cx 2 ) = ax + bx 2 + cx 3
It follows that
ker ( T ) = {a + bx + cx 2 : T ( a + bx + cx 2 ) = O}
= {a + bx + cx 2 : ax + bx 2 + cx 3 = O}
= {a + bx + cx 2 : a = b = c = o}
= {o}
so we have nullity(T) = dim(ker( T)) = The Rank Theorem implies that
Solution
0.
rank ( T ) = dim <!/' 2 - nullity ( T ) = 3 -
0
= 3
Remarll In Example 6.67, it would be just as easy to find the rank of T first, since
{x, x2 , x3 } is easily seen to be a basis for the range of T. Usually, though, one of the two
(the rank or the nullity of a linear transformation) will be easier to compute; the Rank
Theorem can then be used to find the other. With practice, you will become better at
knowing which way to proceed.
Exa m p l e 6 . 6 8
Let W be the vector space of all symmetric 2 X 2 matrices. Define a linear transfor­
mation T : W � <!/' 2 by
r [ : �]
iJ)-'"--
= ( a - b) + ( b - c)x + ( c - a )x 2
(Check that T is linear.) Find the rank and nullity of T.
Solution
as follows:
The nullity of T is easier to compute directly than the rank, so we proceed
{ [ : �] [ : �] 0 }
{ [ : �]
{ [ : �] :(a
{ [ : �] }
{ [ � �] } ( [ � � ] )
{ [� �] }
ker ( T) =
:T
=
: ( a - b) + ( b - c ) x + ( c - a ) x 2 =
- b) = ( b - c) = ( c - a ) =
0}
O}
:a = b = c
= span
Therefore,
is a basis for the kernel of T, so nullity( T) = dim (ker ( T)) = 1 .
The Rank Theorem and Example 6.42 tell us that rank( T) = dim W - nullity( T) =
3 - 1 = 2.
488
Chapter 6 Vector Spaces
one-to-one and Onto Linear Tra nsformations
We now investigate criteria for a linear transformation to be invertible. The keys to
the discussion are the very important properties one-to-one and onto.
A linear transformation T : V ---+ W is called one-to-one if T maps
distinct vectors in V to distinct vectors in W. If range( T) = W, then T is called onto.
Definition
R e m a rks
•
The definition of one-to-one may be written more formally as follows:
T : V ---+ W is one-to-one if, for all u and v in V,
u * v implies that T ( u ) * T ( v)
The above statement is equivalent to the following:
T : V ---+ W is one-to-one if, for all u and v in V,
T ( u ) = T ( v) implies that u = v
Figure 6.9 illustrates these two statements.
v
(a) T is
Fioure 6 . 9
•
one-to-one
w
v
(b) T is
not one-to-one
w
Another way to write the definition of onto is as follows:
T : V ---+ W is onto if, for all w in W, there is at least one v in V such that
w = T ( v)
In other words, given w in W, does there exist some v in V such that w = T(v) ? If,
for an arbitrary w, we can solve this equation for v, then T is onto (see Figure 6. 10).
Section 6.5 The Kernel and Range of a Linear Transformation
v
(a) T is
Figure 6 . 1 0
Exa m p l e 6 . 6 9
onto
w
v
(b) T is
not onto
489
w
r [;] [ � ]
Which of the following linear transformations are one-to-one? onto?
(') T !I ' -> !I ' defined by
�
x
y
(b) D : <!/' 3 ---+ <!1' 2 defined by D(p(x)) = p' (x)
(c) T : M22 ---+ M22 defined by T (A) = A T
Solution
(a) Let r
[;: J [;: J
= r
[;: ] [;: ],
[; l m
. Then
so 2x 1 = 2x2 and x 1 - y 1 = x2 - y2 • Solving these equations, we see that x 1 = x2 and
y 1 = Yz . Hence,
It-"-
[; l
so T is one-to-one.
=
T is not onto, since its range is not all of IR 3 . To be specific, there is no vector
in !I ' 'uch th" r
�
(Why not?)
(b) In Example 6.60, we showed that range(D) = <!/' 2 , s o D is onto. D is not one­
to-one, since distinct polynomials in <!/' 3 can have the same derivative. For example,
x 3 -=F x 3 + 1 , but D(x3 ) = 3x2 = D(x3 + 1 ) .
(c) Let A and B b e in M22 , with T (A) = T(B). Then A T = B T, s o A = (A T ) T =
(B T l = B. Hence, T is one-to-one. In Example 6.62, we showed that range(T) = M22 •
Hence, T is onto.
4
It turns out that there is a very simple criterion for determining whether a linear
transformation is one-to-one.
Theorem 6 . 2 0
A linear transformation T : V ---+ W is one-to-one if and only if ker (T) = {O}.
490
Chapter 6 Vector Spaces
Proof
Assume that T is one-to-one. If v is in the kernel of T, then T(v) = 0. But
we also know that T (O) = 0, so T (v) = T (O). Since T is one-to-one, this implies that
v = 0, so the only vector in the kernel of T is the zero vector.
Conversely, assume that ker (T) = {O}. To show that T is one-to-one, let u and v be
in V with T (u) = T(v) . Then T (u - v) = T(u) - T(v) = 0, which implies that u - v
is in the kernel of T. But ker ( T ) = {O}, so we must have u - v = 0 or, equivalently,
u = v. This proves that T is one-to-one.
Exa m p l e 6 . 10
Show that the linear transformation T : IR 2 ---+ rtP 1 defined by
is one-to-one and onto.
Solulion
If
[�]
r[�]
= a + ( a + b)x
is in the kernel of T, then
0=
r[�]
= a + ( a + b)x
It follows that a = 0 and a + b = 0. Hence, b = 0, and therefore
quently, ker ( T) =
{ [�] }
[ � ] [ � ].
Conse-
, and T is one-to-one, by Theorem 6.20.
By the Rank Theorem,
rank ( T ) = dim IR 2 - nullity ( T ) = 2 - 0 = 2
Therefore, the range of T is a two-dimensional subspace of IR 2 , and hence
range(T) = IR 2 . It follows that T is onto.
�
Theorem 6 . 2 1
For linear transformations between two n-dimensional vector spaces, the proper­
ties of one-to-one and onto are closely related. Observe first that for a linear trans­
formation T : V ---+ W, ker (T) = {O} if and only if nullity(T) = 0, and T is onto if and
only if rank( T) = dim W. (Why?) The proof of the next theorem essentially uses the
method of Example 6.70.
Let dim V = dim W = n . Then a linear transformation T : V ---+ W is one-to-one
if and only if it is onto.
Proof
Assume that T is one-to-one. Then nullity( T) = 0 by Theorem 6.20 and the
remark preceding Theorem 6.2 1 . The Rank Theorem implies that
rank ( T ) = dim V - nullity ( T ) = n - 0 = n
Therefore, T is onto.
Conversely, assume that T is onto. Then rank(T) = dim W = n. By the Rank
Theorem,
nullity ( T ) = dim V - rank ( T ) = n - n = 0
Hence, ker( T ) = {O}, and T is one-to-one.
Section 6.5 The Kernel and Range of a Linear Transformation
491
In Section 6.4, we pointed out that if T : V ---+ W is a linear transformation, then
the image of a basis for V under T need not be a basis for the range of T. We can now
give a condition that ensures that a basis for V will be mapped by T to a basis for W.
Theorem 6 . 2 2
Let T : V ---+ W be a one-to-one linear transformation. If S = {v1 , . . . , vk} is a lin­
early independent set in V, then T (S ) = { T (v1 ), . . . , T(vk) } is a linearly indepen­
dent set in W.
Let c 1 , . . . , ck be scalars such that
Proof
Then T(c 1 v1 + · · · + ckvk) = 0, which implies that c 1 v1 + · · · + ckvk is in the kernel of
T. But, since T is one-to-one, ker ( T ) = {O}, by Theorem 6.20. Hence,
c 1 v1 + · · · + ckvk = 0
But, since {v1 , • • • , vd is linearly independent, all of the scalars C; must be 0. Therefore,
{ T (v1 ), . . . , T (vk) } is linearly independent.
c o r o n a rv 6 . 2 3
Let dim V = dim W = n. Then a one-to-one linear transformation T : V ---+ W
maps a basis for V to a basis for W.
Let B = {v1 , . • . , vn } be a basis for V. By Theorem 6.22, T ( B) = { T (v 1 ), • • • ,
T ( vn ) } is a linearly independent set in W, so we need only show that T ( B ) spans
W. But, by Theorem 6. 15, T ( B ) spans the range of T. Moreover, T is onto, by Theo­
rem 6.2 1, so range(T) = W. Therefore, T ( B ) spans W, which completes the proof.
Proof
Exa m p l e 6 . 1 1
Let T : IR 2 ---+ \fP 1 be the linear transformation from Example 6.70, defined by
r
=
[ : J = a + (a + b)x
Then, by Corollary 6.23, the standard basis [
T ( [) { T ( e 1 ) , T ( e2 ) } of \fP 1 • We find that
T( e1 ) = r
= { e1 , e2 } for IR 2 is mapped to a basis
[�] 1 + x and T( e2) = r [ �] = x
=
It follows that { 1 + x, x} is a basis for \fP 1 .
We can now determine which linear transformations T : V ---+ W are invertible.
Theorem 6 . 2 4
A linear transformation T : V ---+ W is invertible if and only if it is one-to-one
and onto.
492
Chapter 6 Vector Spaces
Assume that T is invertible. Then there exists a linear transformation T - 1 :
W � V such that
Proof
T - 1 0 T = Iv and T 0 T - 1 = Iw
To show that T is one-to-one, let v be in the kernel of T. Then T (v) = 0. Therefore,
Y - 1 ( T (v)) = Y - 1 ( 0 ) =:> ( T - 1 0 T ) ( v) = 0
=:> I(v) = 0
=:> v = O
which establishes that ker ( T ) = { O}. Therefore, T is one-to-one, by Theorem 6.20.
To show that T is onto, let w be in W and let v = T - 1 (w) . Then
T ( v) = T ( T - 1 ( w))
= ( T o T - 1 ) ( w)
= I ( w)
=w
which shows that w is the image of v under T. Since v is in V, this shows that T is onto.
Conversely, assume that T is one-to-one and onto. This means that nullity(T) = 0
and rank( T) = dim W. We need to show that there exists a linear transformation
T ' : W � V such that T' 0 T = Iv and T 0 T' = Iw.
Let w be in W. Since T is onto, there exists some vector v in V such that T(v) = w.
There is only one such vector v, since, if v' is another vector in V such that T ( v' ) = w,
then T(v) = T (v' ); the fact that T is one-to-one then implies that v = v' . It therefore
makes sense to define a mapping T' : W � V by setting T'(w) = v.
It follows that
( T ' o T ) ( v) = T ' ( T ( v)) = T ' ( w) = v
and
( T 0 T ' ) (w) = T ( T ' ( w)) = T ( v) = w
It then follows that T' 0 T = Iv and T 0 T' = Iw. Now we must show that T' is a linear
transformation.
To this end, let w 1 and w2 be in W and let c 1 and c2 be scalars. As above, let
T (v1 ) = w 1 and T(v2 ) = w2 • Then v1 = T ' (w1 ) and v2 = T' (w2 ) and
T ' ( c 1 w 1 + c2w2 ) = T ' ( c 1 T ( v1 ) + c2 T ( v2 ))
= T ' ( T ( c 1 v 1 + c2v2 ))
= I ( c 1 V 1 + CzVz )
Consequently, T' is linear, so, by Theorem 6. 17, T' = T - 1 •
Section 6. 5 The Kernel and Range of a Linear Transformation
The words are derived from
and
the"equalGreek
words meaning
meaning
;
'
and
"shape:'isomorphic
Thus, figurati
velspaces
y speak­have
ing,
vector
"equal shapes:'
isomorphism
isomorphic
isos,
morph,
Exa m p l e 6 . 12
493
Isomorphisms of Vector Spaces
We now are in a position to describe, in concrete terms, what it means for two vector
spaces to be "essentially the same:'
A linear transformation T : V � W is called an isomorphism if it
is one-to-one and onto. If V and W are two vector spaces such that there is an iso­
morphism from V to W, then we say that V is isomorphic to W and write V = W.
Definition
Show that <!P n - I and !R n are isomorphic.
Solution The process of forming the coordinate vector of a polynomial provides us
with one possible isomorphism (as we observed already in Section 6.2, although we did
not use the term isomorphism there). Specifically, define T : <;JP n - i � !R n by T(p(x) ) =
[p(x) ], 0, where £ = { 1 , x, . . . , x" - 1 } is the standard basis for <;JP n - i That is,
·
Theorem 6.6 shows that T is a linear transformation. If p(x) = a0 + a1x + · · · +
a n - 1X n- l is in the kernel of T, then
Hence, a0 = a1 = = a n - i = 0, so p(x) = 0. Therefore, ker ( T ) = {O}, and T is one­
to-one. Since dim <!P n- i = dim IR " = n, T is also onto, by Theorem 6.2 1 . Thus, T is
an isomorphism, and <!P n - i = IR " .
· · ·
Exa m p l e 6 . 13
Show that Mm n and !R m " are isomorphic.
Once again, the coordinate mapping from Mm n to !R m " (as in Example 6.36 )
is an isomorphism. The details of the proof are left as an exercise.
Solution
In fact, the easiest way to tell if two vector spaces are isomorphic is simply to
check their dimensions, as the next theorem shows.
494
Chapter 6 Vector Spaces
Theorem 6 . 2 5
Let V and W b e two finite-dimensional vector spaces (over the same field of scalars).
Then V is isomorphic to W if and only if dim V = dim W.
Let n = dim V. If V is isomorphic to W, then there is an isomorphism
T : V ---+ W. Since T is one-to-one, nullity( T) = 0. The Rank Theorem then implies
that
Proof
rank ( T ) = dim V - nullity ( T ) = n - 0 = n
Therefore, the range of T is an n-dimensional subspace of W. But, since T is onto,
W = range(T), so dim W = n, as we wished to show.
Conversely, assume that Vand Whave the same dimension, n. Let B = {v1 , , vn }
be a basis for V and let C = {w1, . . . , wn } be a basis for W. We will define a linear
transformation T : V ---+ W and then show that T is one-to-one and onto. An arbitrary
vector v in V can be written uniquely as a linear combination of the vectors in the
basis B-say,
.
•
.
We define T by
.-..
It is straightforward to check that T is linear. (Do so.) To see that T is one-to-one,
suppose v is in the kernel of T. Then
and the linear independence of C forces c1 = · · · = en = 0. But then
so ker( T) = {O}, meaning that T is one-to-one. Since dim V = dim W, T is also onto,
by Theorem 6.21 . Therefore, T is an isomorphism, and V = W.
Exa m p l e 6 . 14
Show that u;g n and <lP n are not isomorphic.
Solution
Since dim u;g n = n * n + 1 = dim <lP n' u;g n and <lP n are not isomorphic, by
Theorem 6.25.
Exa m p l e 6 . 15
Let W be the vector space of all symmetric 2 X 2 matrices. Show that W is isomorphic
to IFR 3 .
In Example 6.42, we showed that dim W = 3. Hence, dim W = dim IFR 3 ,
so W = IFR 3 , by Theorem 6.25. (There is an obvious candidate for an isomorphism
T : W ---+ IFR 3 . What is it?)
Solution
Section 6.5 The Kernel and Range of a Linear Transformation
..
I
495
Remark
Our examples have all been real vector spaces, but the theorems we
have proved are true for vector spaces over the complex numbers C or ZP ' where p is
prime. For example, the vector space M22 (Z 2 ) of all 2 X 2 matrices with entries from
2 2 has dimension 4 as a vector space over 2 2 , and hence M22 (Z 2 ) � Zi.
Exercises 6 . 5
j]h 4. Let T : '!!' 2 ---+ '!!' 2 be the linear transformation defined by
T (p ( x)) = xp ' (x).
( a) Which, if any, of the following polynomials are in
1. Let T : M22 ---+ M22 be the linear transformation
defined by
1
(a) Which, if any, of the following matrices are in
ker( T)?
(ii)
[� �]
(iii)
[3 J
O
0 -3
ker( T)?
(i)
(ii) x
(iii) x 2
(b) Which, if any, of the polynomials in part (a) are in
range(T)?
( c ) Describe ker( T ) and range(T).
2. Let T : M22 ---+ IR be the linear transformation defined
In Exercises 5-8, find bases for the kernel and range of the
linear transformations T in the indicated exercises. In each
case, state the nullity and rank of T and verify the Rank
Theorem.
6. Exercise 2
5. Exercise
8. Exercise 4
7. Exercise 3
[1 -1]
In Exercises 9- 1 4, find either the nullity or the rank of T
and then use the Rank Theorem to find the other.
a b = a-b
9. T : M22 ---+ IR 2 defined by T
c d
c d
(b) Which, if any, of the matrices in part (a) are in
range(T)?
(c) Describe ker( T) and range(T).
by T (A) = tr (A ) .
( a) Which, if any, of the following matrices are in
ker ( T ) ?
(i)
[ - 11 ]
2
3
(ii)
[� �]
(iii)
0
3
(b) Which, if any, of the following scalars are in
range(T)?
(i) 0
(ii) 2
(iii) Vl/2
( c) Describe ker( T ) and range(T).
3. Let T : '1P 2 ---+ IR 2 be the linear transformation
defined by
[b + c]
(a) Which, if any, of the following polynomials are in
1
ker (T)?
(i) + x
(ii) x - x 2
(iii) + x - x 2
(b) Which, if any, of the following vectors are in
range(T)?
(i)
(ii)
[�]
(iii)
( c) Describe ker ( T ) and range(T).
[�]
[ ] [ ]
_
10. T : r;; 2 ---+ 1R 2 defined by TCpCx)) =
[ - 11 - 11 ]
[� � ]
[� � � n
1 1 . T : M22 ---+ M22 defined by T (A ) = AB, where
B=
12. T : M22 ---+ M22 defined by T(A) = AB - BA, where
a-b
T ( a + bx + cx 2 ) =
1
[�]
1
B=
J]h 13. T : '1P 2 ---+ IR defined by T(p(x)) = p '(O)
14. T : M33 ---+ M33 defined by T (A ) = A - Ar
In Exercises 1 5-20, determine whether the linear transfor­
mation T is (a) one-to-one and (b) onto.
x
2x - y
15. T : IR 2 ---+ IR 2 defined by T =
y
x + 2y
[] [
]
Chapter 6 Vector Spaces
496
16. T : IR 2 ---+ <.5P 2 defined by
r[ :J
3 1 . Show that 'ii; [ O, l ] = � [ 0, 2 ] .
32. Show that 'ii; [ a, b ] = 'ii; [ c, d ] for all a < b and c < d.
33. Let S : V---+ W and T : U---+ V be linear transformations.
( a ) Prove that if S and T are both one-to-one, so is
S T.
(b) Prove that if S and T are both onto, so is S T.
= (a - 2b ) + ( 3a + b ) x + ( a + b ) x2
[� ]
17. T : <.5P 2 ---+ IR 3 defined by
-b
2
T ( a + bx + cx ) = a b - 3c
c-a
18. T : <.5P 2 ---+ IR 2 defined by T (p ( x)) =
[]
r[�l
19. T : IR 3 ---+ M 22 defined by r
20. 1" ' nl' --+ w defined by
[
]
0
0
[� � � ; ]
34. Let S : V---+ W and T : U---+ V be linear
transformations.
( a ) Prove that if S T is one-to-one, so is T.
(b) Prove that if S T is onto, so is S.
0
: = [ aa +- bb bb -+ cc ]
c
�
a + b + c b - 2c
, where W is the vector space of
b - 2c
a-c
0
35. Let T : V---+ W be a linear transformation between two
finite-dimensional vector spaces.
( a) Prove that if dim V < dim W, then T cannot be
onto.
(b) Prove that if dim V > dim W, then T cannot be
one-to-one.
36. Let a 0, a 1 , , a n be n + 1 distinct real numbers.
Define T : <.5P n ---+ !R n+ I by
•
•
•
all symmetric 2 X 2 matrices
In Exercises 21 -26, determine whether V and W are
isomorphic. If they are, give an explicit isomorphism
T : V ---+ W.
2 1 . V = D 3 (diagonal 3 X 3 matrices), W = IR 3
22. V = S 3 (symmetric 3 X 3 matrices), W = U3 (upper
triangular 3 X 3 matrices)
23. V = S 3 (symmetric 3 x 3 matrices), W = S� (skew­
symmetric 3 X 3 matrices)
24. V = <.5P 2 , W = {p ( x) in <.5P 3 : p ( O ) = O}
F+"S7 25. V = C, W = IR 2
26. V = {A in M22 : tr (A ) = O}, W = IR 2
� 27. Show that T : <.5Pn ---+ <.5Pn defined by T(p(x) ) = p(x) +
p' (x) is an isomorphism.
28. Show that T : <.5P n ---+ <.5P n defined by T(p(x) ) = p(x - 2)
is an isomorphism.
(�)
29. � how_ that T : <.IP_n ---+ <.5P n defined by T (p ( x)) = x np
1s an 1Somorph1sm.
30. (a) Show that 'ii; [O, l] = 'ii; [2, 3 ] . [Hint: Define T :
� [O, l ] ---+ � [2, 3] by letting T (j) be the function
whose value at x is ( T(j)) ( x) = j ( x - 2 ) for x in
[2, 3 ] . ]
(b) Show that � [ O , l ] = <'.{;; [ a , a + l ] for all a.
Prove that T is an isomorphism.
37. If V is a finite-dimensional vector space and T : V ---+ V is
a linear transformation such that rank( T) = rank( T 2 ),
prove that range(T) n ker( T) = {O}. [Hint: T 2 denotes
T T. Use the Rank Theorem to help show that the
kernels of T and T 2 are the same.]
38. Let U and W be subspaces of a finite-dimensional
vector space V. Define T : U X W ---+ V by T(u, w) =
u - w.
( a) Prove that T is a linear transformation.
(b) Show that range(T) = U + W.
(c) Show that ker(T) = U n W. [Hint: See Exercise 50
in Section 6. 1 . ]
(d) Prove Grassmann's Identity:
0
dim ( U + W) = dim U + dim W - dim( U n W)
[Hint: Apply the Rank Theorem, using results
(a) and (b) and Exercise 43(b) in Section 6.2.]
Section 6.6 The Matrix of a Linear Transformation
491
T h e Matrix 01 a l i n e a r Transto r m a t i o n
Theorem 6. 15 showed that a linear transformation T : V � W is completely deter­
mined by its effect on a spanning set for V. In particular, if we know how T acts on a
basis for V, then we can compute T (v) for any vector v in V. Example 6.55 illustrated
the process. We implicitly used this important property of linear transformations
in Theorem 3.31 to help us compute the standard matrix of a linear transformation
T : !Rn � !Rm. In this section, we will show that every linear transformation between
finite-dimensional vector spaces can be represented as a matrix transformation.
Suppose that V is an n-dimensional vector space, W is an m-dimensional vector
space, and T : V � W is a linear transformation. Let B and C be bases for V and W,
respectively. Then the coordinate vector mapping R (v) = [ v ] 6 defines an isomor­
phism R : V � !Rn. At the same time, we have an isomorphism S : W � !Rm given by
S(w) = [wJ c, which allows us to associate the image T (v) with the vector [ T (v) J c in
!Rm. Figure 6. 1 1 illustrates the relationships.
v
v
•
R- 1
U
T (v)
T
•
ls
R
.
[Rn
- - - - - - -
[v] s S 0 T 0 R - l
w
[Rnl
•
�
[T(v) J c
figure 6 . 1 1
Since R is an isomorphism, it is invertible, so we may form the composite mapping
S o T o R- 1 : [Rn � [R m
which maps [v] 6 to [ T (v) J c. Since this mapping goes from !Rn to !Rm, we know
from Chapter 3 that it is a matrix transformation. What, then, is the standard
matrix of S 0 T 0 R - 1 ? We would like to find the m X n matrix A such that A [ v J 6
( S 0 T 0 R- 1 ) ( [v] 6 ) . 0r, since (S 0 T 0 R- 1 ) ( [v] 6) = [ T(v) ] c, we require
A [v ] s = [ T ( v) ] c
It turns out to be surprisingly easy to find. The basic idea is that of Theorem 3.3 1 . The
columns of A are the images of the standard basis vectors for !Rn under S 0 T 0 R - 1 •
But, if B = {v 1 , • • • , vn } is a basis for V, then
R (v; ) = [v; ] 6
0
+-- ith entry
0
498
Chapter 6 Vector Spaces
s o R - 1 (e;) = v;. Therefore, the ith column of the matrix A we seek is given by
(S T R - 1 ) ( e; ) = S ( T ( R - 1 ( e; )))
= S ( T ( v; ))
= [ T ( v; ) l e
which is the coordinate vector of T (v;) with respect to the basis C of W.
We summarize this discussion as a theorem.
0
Theorem 6 . 2 6
0
Let V and W be two finite-dimensional vector spaces with bases B and C, respec­
tively, where B = {v1 , , vJ. If T : V ---+ W is a linear transformation, then the
m X n matrix A defined by
A = [ [ T (v 1 ) l c ! [ T (v2 ) l c ! · · · ! [ T(vn ) l c l
satisfies
A [v] 8 = [ T ( v) J c
•
•
•
for every vector v in V.
The matrix A in Theorem 6.26 is called the matrix of T with respect to the bases B
and C. The relationship is illustrated below. (Recall that TA denotes multiplication
by A.)
T
v -----+
T ( v)
J,
J,
TA
[v] 8 -----+ A [v] 8 = [ T ( v) J c
The matrix of a linear transformation T with respect to bases B and C is some­
times denoted by [ T] c<-- B · Note the direction of the arrow: right-to-left (not left-to­
right, as for T: V---+ W ) . With this notation, the final equation in Theorem 6.26 becomes
R e m a rks
•
[ T J c <-- B [v] s = [ T ( v) J c
Observe that the Bs in the subscripts appear side by side and appear to "cancel" each
other. In words, this equation says, "The matrix for T times the coordinate vector for
v gives the coordinate vector for T (v) :'
In the special case where V = W and B = C, we write [ T] 8 (instead of [ T] B <-- B) .
Theorem 6.26 then states that
[ T J s [vJ s = [ T ( v) J s
•
The matrix of a linear transformation with respect to given bases is unique.
That is, for every vector v in V, there is only one matrix A with the property specified
by Theorem 6.26-namely,
A [v ] 8 = [ T ( v) J c
(You are asked to prove this in Exercise 39. )
Section 6.6 The Matrix of a Linear Transformation
499
The diagram that follows Theorem 6.26 is sometimes called a commutative
diagram because we can start in the upper left-hand corner with the vector v and get
•
to [T(v) J c in the lower right-hand corner in two different, but equivalent, ways. If, as
before, we denote the coordinate mappings that map v to [ v J 8 and w to [w]c by R and
S, respectively, then we can summarize this "commutativity" by
s 0
T = TA R
0
The reason for the term commutative becomes clearer when V = W and B = C, for
then R = S too, and we have
R T = TA R
0
0
suggesting that the coordinate mapping R commutes with the linear transformation T
(provided we use the matrix version of T-namely, TA = Tl Tl 8-where it is required).
•
The matrix [ TJ c<-- B depends on the order of the vectors in the bases B and
C. Rearranging the vectors within either basis will affect the matrix [ TJ c ._ 8. [See
Example 6.77(b) .]
Exa m p l e 6 . 16
2
Let T : IR 3 ---+ IR be the linear transformation defined by
and let B = { e 1 , e2 , e J and C = { e2 , ei } be bases for IR 3 and !R 2 , respectively. Find the
mote ix of T whh '"P"' to B ond C and vecify Themem 6.26 foe v �
Solution
First, we compute
Next, we need their coordinate vectors with respect to C. Since
we have
Therefore, the matrix of T with respect to B and C is
=
[�
-2
-
�]
[ _� l
·
500
Chapter 6 Vector Spaces
To verify Theorem 6.26 for v, we first compute
T ( v) �
Then
[ T ( v) l e =
=
(Check these.)
Using all of these facts, we confirm that
A [v] 8 �
Exa m p l e 6 . 11
�
[v i s �
and
_.,..
T [ J [ �� ]
u[ �� L [ � � ]
[: - � -�1 [ J [ � � ]
�
� [ T ( v) J ,
Let D : l!P3 � <!P2 be the differential operator D(p(x)) = p' (x). Let B = { l , x, x 2 , x 3 }
and C = { l , x, x 2 } be bases for l!P3 and l!P2 , respectively.
(a) Find the matrix A of D with respect to B and C.
(b) Find the matrix A ' of D with respect to B ' and C, where B ' = {x 3 , x 2 , x, l}.
(c) Using part (a), compute D(S
Theorem 6.26.
Solulion
-
x + 2x3 ) and D(a + bx + ex 2 + dx 3 ) to verify
First note that D(a + bx + cx 2 + dx 3 ) = b + 2cx + 3dx 2 • (See Example 6.60.)
(a) Since the images of the basis B under D are D( l ) = 0, D(x) = 1, D(x 2 ) = 2x, and
D (x 3 ) = 3x 2 , their coordinate vectors with respect to C are
[D(l) J, �
[H
[ D (x) J , �
Consequently,
[O
[H
[ D (x ' ) J , �
OJ
[H
[ D (x ' ) J , �
A = [ D Jc.- s = [ [ D ( l ) l e ! [ D (x) l e ! [ D (x2 ) l e ! [ D (x3 ) l e l
1 0
= 0 0 2 0
0 0 0 3
(b) Since the basis B ' is just B in the reverse order, we see that
A ' = [ D b- s · = [ [ D (x3 ) l e ! [ D (x2 ) l e ! [ D (x) l e ! [ D ( l ) l e l
=
[� � � �i
3 0 0 0
m
Section 6.6 The Matrix of a Linear Transformation
501
(This shows that the order of the vectors in the bases l3 and C affects the matrix of a
transformation with respect to these bases.)
( c) First we compute D(S x + 2x 3 ) = - 1 + 6x 2 directly, getting the coordinate
vector
-
[D (S
-
-
x + 2x') ] , � [ 1 + 6x' ] , �
[ -�]
On the other hand,
so
[D (S
-
x + 2x 3 ) l e
which agrees with Theorem 6.26. We leave proof of the general case as an exercise .
.+
Since the linear transformation in Example 6. 77 is easy to use directly, there is re­
ally no advantage to using the matrix of this transformation to do calculations. How­
ever, in other examples-especially large ones-the matrix approach may be simpler,
as it is very well-suited to computer implementation. Example 6.78 illustrates the
basic idea behind this indirect approach.
Exa m p l e 6 . 18
Let T : <!J' 2 � <!J' 2 be the linear transformation defined by
T (p (x)) = p(2x - 1 )
(a) Find the matrix of T with respect to £ = { l , x, x2 }.
(b) Compute T(3 + 2x - x 2 ) indirectly, using part (a).
(a) We see that
Solution
T ( l ) = 1 , T ( x) = 2x
so the coordinate vectors are
Therefore,
-
1, T(x 2 ) = (2x
-
1 )2 = 1
-
4x + 4x 2
502
Chapter 6 Vector Spaces
(b) We apply Theorem 6.26 as follows: The coordinate vector of p(x) = 3 + 2x - x 2
with respect to E is
[p (x) ] , �
Therefore, by Theorem 6.26,
[J
[ T(3 + 2x - x 2 ) ] £ = [ T(p(x)) ] £
[ T ] £ l p(x) ] £
[ � � -: J [ J U l
It follows that T(3 + 2x - x 2 ) = 0 · 1 + 8 · x - 4 · x 2 = 8x - 4x 2 . [Verify this by
computing T(3 + 2x - x 2 ) = 3 + 2(2x - 1 ) - (2x - 1 ) 2 directly.]
The matrix of a linear transformation can sometimes be used in surprising ways.
Example 6.79 shows its application to a traditional calculus problem.
Exa m p l e 6 . 19
Let <f/J be the vector space of all differentiable functions. Consider the subspace W of
<f/J given by W = span(e 3x, xe 3X, x 2 e 3x ) . Since the set B = {e 3X, xe 3X , x 2 e 3x} is linearly
independent (why?), it is a basis for W.
(a) Show that the differential operator D maps W into itself.
(b) Find the matrix of D with respect to B.
(c) Compute the derivative of 5e 3x + 2xe 3x - x 2 e 3x indirectly, using Theorem 6.26,
and verify it using part (a) .
(a) Applying D to a general element of W, we see that
D ( ae 3x + bxe 3x + cx 2 e 3x) = ( 3a + b ) e 3x + ( 3b + 2c ) xe 3x + 3cx 2 e 3x
Solulion
�
(check this), which is again in W.
(b) Using the formula in part (a), we see that
D (e 3x ) = 3e 3X , D ( xe 3x ) = e 3x + 3xe 3X, D ( x 2 e 3x) = 2xe 3x + 3x 2 e 3x
so
It follows that
Section 6.6 The Matrix of a Linear Transformation
503
(c) Forf(x) = 5e 3x + 2xe 3x - x 2 e 3X, we see by inspection that
V (x) ] a �
Hence, by Theorem 6.26, we have
[ D (j (x)) l s = [ D l s [f(x) l s =
[J
[� � �i [ �i [ :]
=
1
0 0 3 -1
-3
which, in turn, implies thatf'(x) = D(j(x)) = 1 7e 3x + 4xe 3x - 3x 2 e 3x, in agreement
4
with the formula in part (a) .
Remark The point of Example 6. 79 is not that this method is easier than direct
differentiation. Indeed, once the formula in part (a) has been established, there is
little to do. What is significant is that matrix methods can be used at all in what
appears, on the surface, to be a calculus problem. We will explore this idea further in
Example 6.83.
Exa m p l e 6 . 8 0
Let V be an n-dimensional vector space and let I be the identity transformation on
V. What is the matrix of I with respect to bases B and C of V if B = C (including the
order of the basis vectors) ? What if B * C?
Solution
Let B = {v 1 , . . . , vJ. Then J(v1 ) = v1 , . . . , J(vn ) = vn , so
and, if B = C,
[ I l s = [ [ J(v1 ) l s ! [ J(vz ) l s ! · · · ! [ J(vn ) l s l
= [ e 1 ! ez ! · · · ! e n l
= In
the n X n identity matrix. (This is what you expected, isn't it?)
In the case B * C, we have
[ J ( v1 ) l e = [vi l e ,
, [ J ( vn ) l e = [vn l e
so
[ IJ e ..- s = [ [vi l e ! · · · ! [vn l e l
= Pe ..- s
the change-of-basis matrix from B to C.
·
·
.
Malrices of Composite and Inverse linear Tra nsformations
We now generalize Theorems 3.32 and 3.33 to get a theorem that will allow us to
easily find the inverse of a linear transformation between finite-dimensional vector
spaces (if it exists) .
504
Chapter 6 Vector Spaces
Theorem 6 . 2 1
Let U, V, and W b e finite-dimensional vector spaces with bases B, C, and D,
respectively. Let T : U � V and S : V � W be linear transformations. Then
R e m a rks
In words, this theorem says, "The matrix of the composite is the product of
the matrices:'
•
Notice how the "inner subscripts" C must match and appear to cancel each
other out, leaving the "outer subscripts" in the form D +--- B.
•
We will show that corresponding columns of the matrices [ S T] v<---- 6 and
[ S J v<-- c l T] c<---- 6 are the same. Let V; be the ith basis vector in B. Then the ith column
of [ S T ] v<---- 6 is
Proof
0
0
[ ( S T) (v; ) ] v = [ S ( T( v; ) J v
[ S l v<-- c l T(v;) ] c
= [ s l v<-- c l T l C<-- 6 [ v; ] 6
0
by two applications of Theorem 6.26. But [ v; ] 6 = e ; (why?), so
is the ith column of the matrix [ S l v<-- c l T l c<---- 6 . Therefore, the ith columns of
[ S T l v<---- 6 and [ S l v<-- c [ T] C<-- 6 are the same, as we wished to prove.
0
Exa m p l e 6 . 8 1
Use matrix methods to compute ( S T)
0
Example 6.56.
Solution
[:]
for the linear transformations S and T of
Recall that T : IR 2 � <J/' 1 and S : <JP , � <!P 2 are defined by
r[:]
= a + ( a + b)x
and S ( a + bx) = ax + bx 2
Choosing the standard bases £, [', and £" for IR 2 , <JP 1 , and <JP 2 , respectively, we see that
�
(Verify these.) By Theorem 6.27, the matrix of S T with respect to [ and £" is
0
Section 6.6 The Matrix of a Linear Transformation
505
Thus, by Theorem 6.26,
Consequently, ( S T)
0
Example 6.56.
[ �]
= ax + (a + b )x 2 , which agrees with the solution to
In Theorem 6.24, we proved that a linear transformation is invertible if and only
if it is one-to-one and onto (i.e., if it is an isomorphism) . When the vector spaces in­
volved are finite-dimensional, we can use the matrix methods we have developed to
find the inverse of such a linear transformation.
Theorem 6 . 2 8
Let T : V ---+ W be a linear transformation between n-dimensional vector spaces V
and W and let B and C be bases for V and W, respectively. Then T is invertible if
and only if the matrix [ T] C +- B is invertible. In this case,
Observe that the matrices of T and r - 1 (if T is invertible) are n X n. If T is
invertible, then r - 1 T = Iv. Applying Theorem 6.27, we have
In = [ Iv l s = [ Y- 1 TJ s
[ Y - 1 J B+- c [ T] C+- B
This shows that [ T ] c.- B is invertible and that ( [ T] c..-8 ) - 1 = [ r- 1 ] B+- C·
Conversely, assume that A = [ T J c..- B is invertible. To show that T is invertible, it
is enough to show that ker (T) = {O}. (Why?) To this end, let v be in the kernel of T.
Then T(v) = 0 , so
Proof
0
0
�
A [v ] a = [ T l c..- a [v] a = [ T(v) J c = [ O l e = 0
which means that [ v] 8 is in the null space of the invertible matrix A. By the Fundamen­
tal Theorem, this implies that [ v J 8 = 0, which, in turn, implies that v = 0, as required.
Exa m p l e 6 . 8 2
In Example 6.70, the linear transformation T : IR 2 ---+ !Jl 1 defined by
r [�]
= a + (a + b)x
was shown to be one-to-one and onto and hence invertible. Find r - 1 .
506
Chapter 6 Vector Spaces
Solulion
In Example 6.8 1 , we found the matrix of T with respect to the standard
bases £ and £' for IR 2 and <!P 1 , respectively, to be
[ T] £' +- £ =
[ � �]
By Theorem 6.28, it follows that the matrix of r - 1 with respect to £' and [ is
[ ] [ OJ
l 0 -1
=
[ Y - 1 1 £+- £' = ( [ T] £' +- £) - 1 =
1
By Theorem 6.26,
1
1
-1
1
[ _� � ][�]
[b : a]
This means that
(Note that the choice of the standard basis makes this last calculation virtually
irrelevant.)
4
The next example, a continuation of Example 6.79, shows that matrices can be
used in certain integration problems in calculus. The specific integral we consider is
usually evaluated in a calculus course by means of two applications of integration by
parts. Contrast this approach with our method.
Exa m p l e 6 . 8 3
Show that the differential operator, restricted to the subspace W = span (e 3x, xe3x,
x 2 e 3x) of '20 , is invertible, and use this fact to find the integral
I x 2 e 3x dx
In Example 6.79, we found the matrix of D with respect to the basis
B = { e 3X, xe 3X , x 2 e 3x} of W to be
Solulion
l [ t �:i
By Theorem 6.28, therefore, D is invertible on W, and the matrix of D - 1 is
o -1
2
3
= 0
0
Section 6.6 The Matrix of a Linear Transformation
501
Since integration is antidifferentiation, this is the matrix corresponding to integration
on W. We want to integrate the function x 2 e 3x whose coordinate vector is
Consequently, by Theorem 6.26,
[�
It follows that
(To be fully correct, we need to add a constant of integration. It does not show up here
because we are working with linear transformations, which must send zero vectors to
zero vectors, forcing the constant of integration to be zero as well.)
Warning
In general, differentiation is not an invertible transformation. (See
Exercise 22.) What the preceding example shows is that, suitably restricted, it some­
times is. Exercises 27-30 explore this idea further.
Change of Basis and Simila rilv
Suppose T : V --+ V is a linear transformation and 13 and C are two different bases for
V. It is natural to wonder how, if at all, the matrices [ T ] 8 and [ T ] c are related. It turns
out that the answer to this question is quite satisfying and relates to some questions
we first considered in Chapter 4.
Figure 6. 1 2 suggests one way to address this problem. Chasing the arrows around
the diagram from the upper left-hand corner to the lower right-hand corner in two
different, but equivalent, ways shows that I T = T I, something we already knew,
since both are equal to T. However, if the "upper" version of T is with respect to the
0
v
•V
i
I
•V
Figure 6 . 1 2
Jo T = ToJ
v
0
v
T
-
T
-
• T(v)
i1
• T(v)
v
}
basis C
}
basis 13
508
Chapter 6 Vector Spaces
basis C and the "lower" version is with respect to B, then T = I T = T I is with
respect to C in its domain and with respect to B in its codomain. Thus, the matrix of
T in this case is [ T] B<- C · But
0
0
[ T] a.-- c = [I T] a.-- c = [IJ a.-- d T l c.-- c
0
and
[ T] a.-c = [ T I] a.-c = [ T] a.-- a [I] a.-- c
0
Therefore, [I l a.-- d T] c.-c = [ T] a.-- a [I] a.-c·
From Example 6.80, we know that [I] 8 .-c = P8 .-c, the (invertible) change-of­
basis matrix from C to B. If we denote this matrix by P, then we also have
p - 1 = (P .-- c) - 1 = Pc.-- a
5
With this notation,
so
Thus, the matrices [ T] 8 and [ T] c are similar, in the terminology of Section 4.4.
We summarize the foregoing discussion as a theorem.
Theorem 6 . 2 9
Let V be a finite-dimensional vector space with bases B and C and let T : V � V
be a linear transformation. Then
where P is the change-of-basis matrix from C to B.
Remark As an aid in remembering that P must be the change-of-basis matrix
from C to B, and not B to C, it is instructive to look at what Theorem 6.29 says when
written in full detail. As shown below, the "inner subscripts" must be the same (all Bs)
and must appear to cancel, leaving the "outer subscripts;' which are both Cs.
S ame
Theorem 6.29 is often used when we are trying to find a basis with respect to
which the matrix of a linear transformation is particularly simple. For example, we
can ask whether there is a basis C of V such that the matrix [ T J c of T : V � V is a
diagonal matrix. Example 6.84 illustrates this application.
Exa m p l e 6 . 8 4
Let T :
IR:2 IR:2
�
be defined by
If possible, find a basis C for
IR:2
T
x
x + 3y
[] [ ]
y
=
2x + 2y
such that the matrix of T with respect to C is diagonal.
Section 6.6 The Matrix of a Linear Transformation
Solution
509
The matrix of T with respect to the standard basis E is
[ T] E =
[� �]
This matrix is diagonalizable, as we saw in Example 4.24. Indeed, if
p=
[ � -�J
and D =
[0 0]
4
-1
then P - 1 [ T] Ep = D. If we let C be the basis of IR 2 consisting of the columns of P, then
P is the change-of-basis matrix PE.- c from C to E. By Theorem 6.29,
[ T] c = P - 1 [ T] EP = D
so the matrix of T with respect to the basis C =
{ [ � ] , [ _�] }
is diagonal.
R e m a rks
It is easy to check that the solution above is correct by computing [ T ] c directly.
We find that
•
r[ � ] [! ] 4 [ � ] o [ _ � J r[ _ � J [ - � ] o [ � ] [ _ � J
[ r[ � J L [�] [ r[ _ � J L [ _ �J
=
+
and
=
Thus, the coordinate vectors that form the columns of [ T ] c are
and
in agreement with our solution above.
•
The general procedure for a problem like Example 6.84 is to take the stan­
dard matrix [ T] E and determine whether it is diagonalizable by finding bases for its
eigenspaces, as in Chapter 4. The solution then proceeds exactly as in the preceding
example.
Example 6.84 motivates the following definition.
D e f i n i t i o n Let V be a finite-dimensional vector space and let T : V ---+ V be a
linear transformation. Then T is called diagonalizable if there is a basis C for V
such that the matrix [ T ] c is a diagonal matrix.
It is not hard to show that if B is any basis for V, then T is diagonalizable if and
only if the matrix [ T] 8 is diagonalizable. This is essentially what we did, for a special
case, in the last example. You are asked to prove this result in general in Exercise 42.
Sometimes it is easiest to write down the matrix of a linear transformation with
respect to a "nonstandard" basis. We can then reverse the process of Example 6.84
to find the standard matrix. We illustrate this idea by revisiting Example 3.59.
Exa m p l e 6 . 8 5
Let e be the line through the origin in IR 2 with direction vector d =
standard matrix of the projection onto e.
[ �: ] .
Find the
510
Chapter 6 Vector Spaces
df d
Let T denote the projection. There is no harm in assuming that d is
a unit vector (i.e.,
+ � = 1), since any nonzero multiple of d can serve as a
Solulion
direction vector for
-€. Let d' [ -d�2 ] so that d and d' are orthogonal. Since
=
d' is also a unit vector, the set D = {d, d'} is an orthonormal basis for IR 2 .
As Figure 6. 1 3 shows, T(d) = d and T(d') = 0. Therefore,
[ T(d) J v =
[�]
and [ T(d') J v =
[�]
y
Projection onto -€
Figure 6 . 1 3
so
[ T] v =
[� �]
The change-of-basis matrix from D to the standard basis [ is
so the change-of-basis matrix from E to D is
By Theorem 6.29, then, the standard matrix of T is
which agrees with part (b) of Example 3.59.
Section 6.6 The Matrix of a Linear Transformation
Exa m p l e 6 . 8 6
- 1)
Let T : <!P 2 ---+ <!P 2 be the linear transformation defined by
T(p(x)) = p ( 2x
511
1
(a) Find the matrix of T with respect to the basis B = { l + x, - x, x 2 } of <!f 2 .
(b) Show that T is diagonalizable and find a basis C for <!P 2 such that [ T ] c is a diago­
nal matrix.
(a) In Example 6.78, we found that the matrix of T with respect to the
standard basis £ = { l , x, x 2 } is
Solution
[ T] ,
[ � � -:l
11 -� o�
[0 l
�
The change-of-basis matrix from B to £ is
p = PE<-B =
It follows that the matrix of T with respect to B is
-1 - [
u -! � m 0 : J i i � ]
[ - 0� �0 -:i
1,
[ T] B = p- 1 [ T] EP
2
-
4
�
�
(Check this.)
(b) The eigenvalues of [ T] E are 2, and 4 (why?), so we know from Theorem 4.25
that [ T] E is diagonalizable. Eigenvectors corresponding to these eigenvalues are
[ � JTJ H l
[� � :l [� � �l
respectively. Therefore, setting
P�
-
-
and D �
we have p- i [ T] Ep = D. Furthermore, P is the change-of-basis matrix from a basis
C to £, and the columns of P are thus the coordinate vectors of C in terms of £. It
follows that
C = {l,
+ x, - 2x + x 2 }
and [ T l c = D.
-1 1
Chapter 6 Vector Spaces
512
The preceding ideas can be generalized to relate the matrices [ T ] C<-- B and
[ T] c '<-- B ' of a linear transformation T : V ---+ W, where 13 and 13 ' are bases for V and C
and C' are bases for W. (See Exercise 44.)
We conclude this section by revisiting the Fundamental Theorem of Invertible
Matrices and incorporating some results from this chapter.
Theorem 6 . 3 0
The Fundamental Theorem of Invertible Matrices: Version 4
Let A be an n X n matrix and let T : V ---+ W be a linear transformation whose
matrix [ T] C<-- B with respect to bases 13 and C of V and W, respectively, is A. The
following statements are equivalent:
a. A is invertible.
b. Ax = b has a unique solution for every b in !R n .
c. Ax = 0 has only the trivial solution.
d. The reduced row echelon form of A is Iw
e. A is a product of elementary matrices.
f. rank(A) = n
g. nullity(A) = 0
h. The column vectors of A are linearly independent.
i. The column vectors of A span !R n .
j. The column vectors of A form a basis for !R n .
k. The row vectors of A are linearly independent.
1. The row vectors of A span !R n .
m. The row vectors of A form a basis for !R n .
n. det A * 0
o. 0 is not an eigenvalue of A.
p. T is invertible.
q. T is one-to-one.
r. T is onto.
s. ker ( T) = {O}
t. range(T) = W
The equivalence (q) -¢:? (s) is Theorem 6.20, and (r) -¢:? (t) is the definition of
onto. Since A is n X n, we must have dim V = dim W = n. From Theorems 6.2 1 and
6.24, we get (p) -¢:? ( q) -¢:? ( r) . Finally, we connect the last five statements to the others
by Theorem 6.28, which implies that (a) -¢:? (p ).
Proof
..
I
Exercises 6 . 6
In Exercises 1 - 12, find the matrix [ T] C<-- B of the linear
transformation T : V ---+ W with respect to the bases 13 and C
of V and W, respectively. Verify Theorem 6.26 for the vector v
by computing T(v) directly and using the theorem.
1 . T : <!P 1 ---+ <!P 1 defined by T ( a + bx) = b - ax,
13 = C = { l , x}, v = p(x) = 4 + 2x
..
2. T : <!P 1 ---+ <!P 1 defined by T(a + bx) = b - ax,
13 = { l + x, 1 - x}, C = {l, x}, v = p(x) = 4 + 2x
3. T : <!P 2 ---+ <!P 2 defined by T(p(x) ) = p(x + 2),
13 = { l , x, x 2 }, C = { l , x + 2, (x + 2 ) 2 },
v = p(x) = a + bx + cx 2
Section 6.6 The Matrix of a Linear Transformation
4. T : <J/l 2 ---+ <J/l 2 defined by T(p(x) ) = p(x + 2),
B = { l , x + 2, (x + 2 ) 2 }, C = { l , x, x 2 },
v = p(x) = a + bx + cx2
5. T : <J/l 2 ---+ IR 2 defined by T(p(x) ) =
B = { l , x, x 2 }, C = {e 1 , ez },
[� � � � l
v = p(x) = a + bx + cx 2
6. T : <J/l 2 ---+ IR 2 defined by T(p(x) ) =
B= {x2, x, l },C = { [ � ] , [ � ] } ,
[� � � � l
v = p(x) = a + bx + cx 2
7. T : IR 2 ---+ IR 3 defined by
a 2b
B
a
r[:J [ � l � WH - � ] } .
c � m H J [: l l · v � n
[ �] .
[ : �]
8. Repeat Exercise 7 with v =
9. T : M22 ---+ M22 defined by T(A) = A r, B = C =
{E 1 1 , E 1 2 , E 2 1 , E 22 }, v = A =
10. Repeat Exercise 9 with B = {E w E 2 1 , E 1 2 > E 11 } and
C = {E 1 2 , E 2 1 > E w E 11 }.
11. T: M22 ---+ M22 defined by T (A) = AB - BA, where
B= [
l
-1
v=A =
]
[ : �]
-l
, B = C = {E 11 , E 1 2 > E2 1 , E22 },
1
12. T : M22 ---+ M22 defined by T(A) = A - A r, B =
C = {E 11 , E 1 2 > E2 1 , E22 }, v = A =
[ : �]
� 13. Consider the subspace W of '.2ll , given by
W = span (sin x, cos x).
(a) Show that the differential operator D maps W into
513
ilili_ 14. Consider the subspace W of '.2ll , given by
W = span ( e 2x, e-2x) .
(a) Show that the differential operator D maps W into
itself.
(b) Find the matrix of D with respect to B = { e 2 , e - zx}.
(c) Compute the derivative of f ( x) = e 2x - 3e - zx
indirectly, using Theorem 6.26, and verify that it
agrees with j'( x) as computed directly.
ilili_ 15. Consider the subspace W of '.2ll , given by W = span ( e 2 ,
e 2x cos x, e 2x sin x).
( a) Find the matrix of D with respect to B = {e 2x,
e 2x cos x, e 2x sin x}.
(b) Compute the derivative off(x) = 3 e 2x - e 2x cos x +
2e 2x sin x indirectly, using Theorem 6.26, and
verify that it agrees with f' (x) as computed directly.
ilili_ 16. Consider the subspace W of '.2ll , given by
W = span (cos x, sin x, x cos x, x sin x).
( a) Find the matrix of D with respect to B = {cos x,
sin x, x cos x, x sin x}.
(b) Compute the derivative off( x) = cos x + 2x cos x
indirectly, using Theorem 6.26, and verify that it
agrees with f' (x) as computed directly.
X
X
In Exercises 1 7 and 1 8, T : U ---+ V and S : V ---+ W are linear
transformations and B, C, and D are bases for U, V, and
W, respectively. Compute [S T l v +- B in two ways: (a) by
finding S T directly and then computing its matrix and
(b) by finding the matrices of S and T separately and using
Theorem 6.27.
p(O)
17. T : <J/l 1 ---+ IR 2 defined by T(p(x)) =
, S : IR 2 ---+ IR 2
p( l)
2
B = { l , x},
=
defined by s
a
C = D = {e 1 , ez }
18. T : <J/l 1 ---+ <J/l 2 defined by T(p(x) ) = p(x + 1 ) ,
S : <JP 2 ---+ <JP 2 defined by S(p(x) ) = p(x + 1 ) ,
B = { l , x}, C = D = { l , x, x2 }
0
0
[ �] [ ; -- � l
[ ]
In Exercises 1 9-26, determine whether the linear transfor­
mation T is invertible by considering its matrix with respect
to the standard bases. If T is invertible, use Theorem 6.28
itself.
and the method of Example 6.82 to find T - 1 •
(b) Find the matrix of D with respect to
20. T in Exercise 5
19. T in Exercise 1
B = {sin x, cos x}.
21. T in Exercise 3
(c) Compute the derivative off( x) = 3 sin x - 5 cos x
22. T : <Jfl 2 -+ <Jfl 2 defined by T(p (x) ) = p' (x)
indirectly, using Theorem 6.26, and verify that it
agrees withj' (x) as computed directly.
� 23. T : <J/l 2 ---+ <JP 2 defined by T(p(x) ) = p(x) + p' ( x)
514
Chapter 6 Vector Spaces
24. T : M22 � M22 defined by T (A ) = AB, where
[� �]
B=
25. T in Exercise 1 1
26. T in Exercise 1 2
� In Exercises 27-30, use the method of Example 6.83
to evaluate the given integral.
27. f (sin x - 3 cos x) dx. (See Exercise 13.)
28. f 5e - 2x dx. (See Exercise 14.)
29. f (e2x cos x - 2e2x sin x) dx. (See Exercise 15.)
30. f (x cos x + x sin x) dx. (See Exercise 16.)
In Exercises 31 -36, a linear transformation T : V � V is
given. Ifpossible, find a basis C for V such that the matrix
[ T] c of T with respect to C is diagonal.
[�] [ ]
r[ �] [ : : �]
- 4b
a + Sb
31. T : IR 2 � IR 2 defined by r
32. T : IR 2 � IR 2 defined by
=
33. T : <!/' 1 � <!/' 1 defined by T(a + bx) = (4a + 2b) +
(a + 3b)x
34. T : <!1' 2 � <!1' 2 defined by T(p(x)) = p(x + 1 )
� 35. T : <!/' 1 � <!/' 1 defined by T(p( x)) = p(x) + xp' (x)
36. T : <!1' 2 � <!1' 2 defined by T(p( x)) = p(3x + 2)
37. Let € be the line through the origin in IR 2 with direction
vector d =
[ �: ]
it to compute the orthogonal projection of v onto W,
where
. Use the method of Example 6.85 to
find the standard matrix of a reflection in e.
38. Let W be the plane in IR 3 with equation x - y +
2z = 0. Use the method of Example 6.85 to find
the standard matrix of an orthogonal projection
onto W. Verify that your answer is correct by using
Compare your answer with Example 5. 1 1 .
[Hint: Find an orthogonal decomposition of IR 3 as
IR 3 = W + WJ_ using an orthogonal basis for W. See
Example 5.3.]
39. Let T : V � W be a linear transformation between
finite-dimensional vector spaces and let l3 and C be
bases for V and W, respectively. Show that the matrix
of T with respect to l3 and C is unique. That is, if A is a
matrix such that A [ v ] 8 = [ T ( v) l e for all v in V, then
A = [ T l c +- B· [Hint: Find values of v that will show
this, one column at a time.]
In Exercises 40-45, let T : V � W be a linear transforma­
tion between finite-dimensional vector spaces V and W
Let l3 and C be bases for V and W, respectively, and let
A = [ T] c +- B·
40. Show that nullity( T) = nullity(A).
41. Show that rank( T) = rank(A) .
42. I f V = W and l3 = C , show that T is diagonalizable if
and only if A is diagonalizable.
43. Use the results of this section to give a matrix­
based proof of the Rank Theorem (Theorem 6. 19).
44. If !3 ' and C' are also bases for V and W, respectively,
what is the relationship between [ T] c +- B and [ T] c · +- B '?
Prove your assertion.
45. If dim V = n and dim W = m, prove that � ( V, W ) =
Mm n - (See the exercises for Section 6.4.) [Hint: Let l3
and C be bases for V and W, respectively. Show that the
mapping cp ( T) = [ T ] c._8, for T in � ( V, W ) , defines
a linear transformation <p : � ( V, W) � Mmn that is an
isomorphism.]
46. If V is a vector space, then the dual space of V is
the vector space V* = � ( V, IR). Prove that if V is
finite-dimensional, then V* = V.
Exp loration
Tilings , L att i c e s , and t h e
C rystall o gr ap h i c Restriction
Repeating patterns are frequently found in nature and i n art. The molecular struc­
ture of crystals often exhibits repetition, as do the tilings and mosaics found in the
artwork of many cultures. Tiling (or tessellation) is covering of a plane by shapes that
do not overlap and leave no gaps. The Dutch artist M. C. Escher ( 1 898- 1 972) pro­
duced many works in which he explored the possibility of tiling a plane using fanciful
shapes (Figure 6. 14).
M. C. Escher's "Symmetry Drawing El03"
Figure 6 . 1 4
515
translationDrawing El03"
M.Invariance
C. Escher'under
s "Symmetry
figure 6 . 1 5
•
•
•
•
•
•
"
•
•
A lattice
l
•
•
•
figure 6 . 1 6
•
•
"
•
u
•
•
•
•
•
•
•
M.Rotational
C. Escher'symmetry
s "Symmetry Drawing El03"
figure 6 . 1 1
In this exploration, we will be interested in patterns such as those in Figure 6. 14,
which we assume to be infinite and repeating in all directions of the plane. Such a
pattern has the property that it can be shifted (or translated) in at least two directions
(corresponding to two linearly independent vectors) so that it appears not to have
been moved at all. We say that the pattern is invariant under translations and has
translational symmetry in these directions. For example, the pattern in Figure 6. 14
has translational symmetry in the directions shown in Figure 6. 1 5 .
If a pattern has translational symmetry in two directions, it has translational sym­
metry in infinitely many directions.
1 . Let the two vectors shown in Figure 6. 1 5 be denoted by u and v. Show that the
pattern in Figure 6.14 is invariant under translation by any integer linear combination
of u and v-that is, by any vector of the form au + bv, where a and b are integers.
For any two linearly independent vectors u and v in IR 2 , the set of points deter­
mined by all integer linear combinations of u and v is called a lattice. Figure 6. 1 6
shows an example o f a lattice.
2. Draw the lattice corresponding to the vectors u and v of Figure 6.15.
Figure 6. 14 also exhibits rotational symmetry. That is, it is possible to rotate
the entire pattern about some point and have it appear unchanged. We say that it is
invariant under such a rotation. For example, the pattern of Figure 6. 14 is invariant
under a rotation of 1 20° about the point 0, as shown in Figure 6. 17. We call 0 a center
of rotational symmetry (or a rotation center) .
Note that if a pattern is based on an underlying lattice, then any symmetries of the
pattern must also be possessed by the lattice.
516
3. Explain why, if a point 0 is a rotation center through an angle e, then it is
a rotation center through every integer multiple of e. Deduce that if 0 < e :s 360°,
then 360 / e must be an integer. (If 360 / e = n, we say the pattern or lattice has n-fold
rotational symmetry.)
4. What is the smallest positive angle of rotational symmetry for the lattice in
Problem 2? Does the pattern in Figure 6.14 also have rotational symmetry through
this angle?
5. Take various values of e such that 0 < e :s 360° and 360/e is an integer. Try
to draw a lattice that has rotational symmetry through the angle e. In particular, can
you draw a lattice with eight-fold rotational symmetry?
We will show that values of e that are possible angles of rotational symmetry for a
lattice are severely restricted. The technique we will use is to consider rotation trans­
formations in terms of different bases. Accordingly, let Re denote a rotation about the
origin through an angle e and let £ be the standard basis for IR 2 . Then the standard
matrix of Re is
[
cos e - sin e
[ Re l E -
sm e
.
cos e
]
6. Referring to Problems 2 and 4, take the origin to be at the tails of u and v.
(a) What is the actual (i.e., numerical) value of [ R e ] E in this case?
(b) Let B be the basis { u, v}. Compute the matrix [ R 0 ] a7. In general, let u and v be any two linearly independent vectors in IR 2 and
suppose that the lattice determined by u and v is invariant under a rotation through
an angle e. If B = {u, v}, show that the matrix of R11 with respect to B must have the
form
where a, b, c, and d are integers.
8. In the terminology and notation of Problem 7, show that 2 cos e must be an
integer. [Hint: Use Exercise 35 in Section 4.4 and Theorem 6.29.]
9. Using Problem 8, make a list of all possible values of e, with 0 < e :s 360°,
that can be angles of rotational symmetry of a lattice. Record the corresponding val­
ues of n, where n = 360/e, to show that a lattice can have n-fold rotational symmetry
if and only if n = 1 , 2, 3, 4, or 6. This result, known as the crystallographic restriction,
was first proved by W Barlow in 1 894.
10. In the library or on the Internet, see whether you can find an Escher tiling
for each of the five possible types of rotational symmetry-that is, where the smallest
angle of rotational symmetry of the pattern is one of those specified by the crystal­
lographic restriction.
511
518
Chapter 6 Vector Spaces
A p p l icati o n s
Jlh
Homogeneous Linear Diffe rential Equalions
In Exercises 69-72 in Section 4.6, we showed that if y = y(t) is a twice-differentiable
function that satisfies the differential equation
y " + ay ' + by = 0
(1)
then y is of the form
if ,\ 1 and ,\ 2 are distinct roots of the associated characteristic equation ,\ 2 + a,\ +
b = 0. (The case where ,\ 1 = ,\ 2 was left unresolved.) Example 6. 1 2 and Exercise 20
in this section show that the set of solutions to Equation ( 1 ) forms a subspace of 9F,
the vector space of functions. In this section, we pursue these ideas further, paying
particular attention to the role played by vector spaces, bases, and dimension.
To set the stage, we consider a simpler class of examples. A differential equation
of the form
y' + ay = 0
�
Theorem 6 . 3 1
(2)
is called a first-order, homogeneous, linear differential equation. ("First-order"
refers to the fact that the highest derivative that is involved is a first derivative, and
"homogeneous" means that the right-hand side is zero. Do you see why the equa­
tion is "linear"?) A solution to Equation (2) is a differentiable function y = y (t) that
satisfies Equation (2) for all values of t.
It is easy to check that one solution to Equation (2) is y = e at (Do it.) However,
we would like to describe all solutions-and this is where vector spaces come in. We
have the following theorem.
-
.
The set S of all solutions to y' + ay = 0 is a subspace of ?F.
Since the zero function certainly satisfies Equation (2), S is nonempty. Let x
and y be two differentiable functions of t that are in S and let c be a scalar. Then
Proof
x' + ax = 0 and y' + ay = 0
so, using rules for differentiation, we have
( x + y) ' + a (x + y) = x' + y' + ax + ay = (x' + ax) + (y ' + ay) = 0 + 0 = 0
and
(cy) ' + a (cy) = cy ' + c (ay) = c (y' + ay) = c · O = 0
Hence, x + y and cy are also in S, so S is a subspace of ?F.
Now we will show that S is a one-dimensional subspace of 9F and that { e - at } is a
basis. To this end, let x = x ( t) be in S. Then, for all t,
x' ( t) + ax (t) = 0 or x' ( t) = - ax ( t)
Section 6.7 Applications
519
Define a new function z (t) = x ( t) eat . Then, by the Chain Rule for differentiation,
z' ( t) = x ( t) ae at + x'(t)e at
= a x (t)eat - ax (t)ea t
=O
Since z ' is identically zero, z must be a constant function-say, z ( t) = k. But this
means that
x (t)e at = z ( t) = k for all t
so x ( t) = ke - a1 • Therefore, all solutions to Equation (2) are scalar multiples of the
single solution y = e - at . We have proved the following theorem.
Theorem 6 . 3 2
If S is the solution space of y' + ay = 0, then dim S = 1 and {e - a1 } is a basis for S.
One model for population growth assumes that the growth rate of the popula­
tion is proportional to the size of the population. This model works well if there are
few restrictions (such as limited space, food, or the like) on growth. If the size of the
population at time t is p(t), then the growth rate, or rate of change of the population,
is its derivative p' (t) . Our assumption that the growth rate of the population is pro­
portional to its size can be written as
p ' (t) = kp (t)
where k is the proportionality constant. Thus, p satisfies the differential equation
p' - kp = 0, so, by Theorem 6.32,
p (t) = ce k t
for some scalar c. The constants c and k are determined using experimental data.
Exa m p l e 6 . 8 1
in Michael
Crichton'is(Nmentioned
sewnovel
York:
Deln" inl, 1969),
alwasthough
the
"vi
l
a
i
that
novel
supposedl
y
an
alien
vi
r
us.
In
real
life,
contaminated
the
town
waterin 2000,
supplyresulting
of Walkerton,
Ontario,
in seven
deaths
and
causing
hundreds
people to become seriously ill.of
E. coli
The Andromeda
Strain
E. coli
The bacterium Escherichia coli (or E. coli, for short) is commonly found in the
intestines of humans and other mammals. It poses severe health risks if it escapes
into the environment. Under laboratory conditions, each cell of the bacterium divides
into two every 20 minutes. If we start with a single E. coli cell, how many will there
be after 1 day?
We do not need to use differential equations to solve this problem, but we
will, in order to illustrate the basic method.
To determine c and k, we use the data given in the statement of the problem. If
we take 1 unit of time to be 20 minutes, then we are given that p(O) = 1 and p(l) = 2.
Therefore,
Solution
c = c · 1 = ce k · o = 1 and 2 = ee k · ! = i
It follows that k = ln 2, so
p (t) = e t ln 2 = e ln 2 ' = 2 t
After 1 day, t = 72, so the number of bacteria cells will be p(72) = 2 7 2
(see Figure 6. 18).
=
4.72 X 1 0 2 1
520
Chapter 6 Vector Spaces
p (t)
5 x 1021
4.72 x 1021
4 x 1021
3 x 1021
I
I
I
I
I
I
I
I
I
I
2 x 1021
1 x 1021
0 '---+---+--I--+--.+---+�-+-'-- t
0 10 20 30 40 50 60 70"' 72
Exponential growth
Figure 6 . 1 8
Radioactive substances decay by emitting radiation. If m(t) denotes the mass of
the substance at time t, then the rate of decay is m '(t). Physicists have found that the
rate of decay of a substance is proportional to its mass; that is,
m '(t) = km (t) or m ' - km = 0
where k is a negative constant. Applying Theorem 6.32, we have
m (t) = ce k t
for some constant c. The time required for half of a radioactive substance to decay is
called its half-life.
Exa m p l e 6 . 8 8
After 5.5 days, a 1 00 mg sample of radon-222 decayed to 37 mg.
(a) Find a formula for m(t), the mass remaining after t days.
(b) What is the half-life of radon-222?
(c) When will only 1 0 mg remain?
Solution
(a) From m(t) = c/1, we have
100 = m (O) = ce k · o = c • 1 = c
so
m ( t) = l OOe k t
With time measured in days, we are given that m ( 5.5) = 37. Therefore,
100 e 5 · 5 k = 37
so
e 55k = 0.37
Solving for k, we find
5.5k = ln ( 0.37 )
so
1
Therefore, m ( t) = l OOe- 0 · 81.
k=
ln ( 0.37 )
5.5
=
-0.18
Section 6.7 Applications
521
m (t)
1 00
80
60
40
"so
I
I
I
I
I
I
I
I
20
2
3.85 :
""
4
6
8
10
Radioactive decay
Figure 6 . 1 9
(b) To find the half-life of radon-222, we need the value of t for which m(t) = 50.
Solving this equation, we find
so
Hence,
and
l00e - 0 · 1 8 t = 50
e - o . 1 s t = 0.50
- 0. 1 8t = ln ( t) = - ln 2
ln 2
t = - = 3.85
0. 1 8
Thus, radon-222 has a half-life of approximately 3.85 days. (See Figure 6. 1 9.)
(c) We need to determine the value of t such that m(t) = 10. That is, we must solve
the equation
IOOe - o . 1 s t = 1 0 or e - o . 1 s t
=
0.1
Taking the natural logarithm o f both sides yields - 0 . 1 8t = ln 0. 1 . Thus,
ln 0. 1
-0.18
s o 1 0 mg o f the sample will remain after approximately 12.79 days.
t = -- = 12.79
SeeFriedberg, A. J. Insel,by andS. H.L. E.
Spence
(Englewood
Prentice-Hall,
1979).Cliffs, NJ:
Linear Algebra
The solution set S of the second-order differential equation y " + ay' + by = 0
is also a subspace of ?F (Exercise 20), and it turns out that the dimension of S is 2.
Part (a) of Theorem 6.33, which extends Theorem 6.32, is implied by Theorem 4.40.
Our approach here is to use the power of vector spaces; doing so allows us to obtain
part (b) of Theorem 6.33 as well, a result that we could not obtain with our previous
methods.
Chapter 6 Vector Spaces
522
Theorem 6 . 3 3
Let S be the solution space of
y " + ay' + by = 0
and let A 1 and A 2 be the roots of the characteristic equation A 2 + a,\ + b = 0.
a. If A 1 * A 2 , then {e ;\ ' t, e ;\21} is a basis for S.
b. If A 1 = A 2 , then {e A1 1, te ;\ 1 1} is a basis for S.
R e m a rks
•
Observe that what the theorem says, in other words, is that the solutions of
y " + ay' + by = 0 are of the form
in the first case and
in the second case.
•
Compare Theorem 6.33 with Theorem 4.38. Linear differential equations
and linear recurrence relations have much in common. Although the former belong to
continuous mathematics and the latter to discrete mathematics, there are many parallels.
(a) We first show that {e A1 1, e A21} is contained in S. Let A be any root of the
characteristic equation and let f( t) = eA t . Then
Proof
J'(t) = Ae A t and J"(t) = A 2 e At
from which it follows that
2
f" + af' + bf = A e A t + aAe A t + be A t
= (A2 + a,\ + b)eA t
= 0 · e At = 0
Therefore, f is in S. But, since A 1 and A 2 are roots of the characteristic equation, this
means that e A, t and e A, t are in S.
The set {e A1 1, e A21} is also linearly independent, since if
then, setting t = 0, we have
y
r
Y = e
e A2
Next, we set t = 1 to obtain
c 1 + c2 = 0 or c2 = - c 1
------------
But e A , - e A2 * 0, since e A , - e A2 = 0 implies that e A , = e\ which is clearly im­
possible if A 1 * A 2 . (See Figure 6.20.) We deduce that c 1 = 0 and, hence, c2 = 0, so
{e A1 1, e A2t} is linearly independent.
Since dim S = 2, {e A , t, e A21} must be a basis for S.
Figure 6 . 2 0
(b) You are asked to prove this property in Exercise 2 1 .
Section 6.7 Applications
Exa m p l e 6 . 8 9
523
Find all solutions of y 11 - Sy' + 6y = 0.
The characteristic equation is A 2 - SA + 6 = (A - 2) (,\ - 3) = 0. Thus,
the roots are 2 and 3, so {e 2 1, e 3 1} is a basis for the solution space. It follows that the
solutions to the given equation are of the form
Solution
The constants c 1 and c2 can be determined if additional equations, called bound­
ary conditions, are specified.
Exa m p l e 6 . 9 0
Find the solution of y11 + 6y' + 9y = 0 that satisfies y(O) = 1, y'(O) = 0.
The characteristic equation is A 2 + 6A + 9 = (A + 3) 2 = 0, so - 3 is a
repeated root. Therefore, {e - 3 1, te - 3 1} is a basis for the solution space, and the general
solution is of the form
Solution
The first boundary condition gives
1 = y(O) = c 1 e - 3 · o + 0 = c 1
so y = e - 3 t + c2 te - 3 1. Differentiating, we have
y ' = - 3e - 3 t + c2 ( - 3 te- 3 t + e- 3 t)
so the second boundary condition gives
0 = y'(O) = - 3e _ 3 . 0 + c 2 (0 + e _ 3 . 0 ) = - 3 + c2
or
Therefore, the required solution is
y = e- 3 t + 3te - 3 1 = (1 + 3t) e- 3 1
Theorem 6.33 includes the case in which the roots of the characteristic equation
are complex. If A = p + qi is a complex root of the equation A 2 + a,\ + b = 0,