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Contemporary
Linear Algebra
Howard Anton
Drexel University
Robert C. Busby
Drexel University
John Wiley & Sons, Inc.
CONTENTS
CHAPTER 1
CHAPTER 7
1.1
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
1.2
1.3
Vectors 1
Vectors and Matrices in Engineering and
Mathematics; ra-Space 1
Dot Product and Orthogonality 15
Vector Equations of Lines and Planes 29
CHAPTER 2
2.1
2.2
2.3
Systems of Linear Equations 39
Introduction to Systems of Linear Equations 39
Solving Linear Systems by Row Reduction 48
Applications of Linear Systems 63
CHAPTER 3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Matrices and Matrix Algebra 79
Operations on Matrices 79
Inverses; Algebraic Properties of Matrices 94
Elementary Matrices; A Method for
Finding A~' 109
Subspaces and Linear Independence 123
The Geometry of Linear Systems 135
Matrices with Special Forms 143
Matrix Factorizations; LU-Decomposition 154
Partitioned Matrices and Parallel Processing 166
CHAPTER 4
4.1
4.2
43
4.4
CHAPTER 5
5.1
5.2
5.3
5.4
175
Matrix Models 225
Dynamical Systems and Markov Chains 225
Leontief Input-Output Models 235
Gauss-Seidel and Jacobi Iteration; Sparse Linear
Systems 241
The Power Method; Application to Internet Search
Engines 249
CHAPTER 6
; 6.1
L 6.2
Determinants
Determinants; Cofactor Expansion 175
Properties of Determinants 184
Cramer's Rule; Formula for A" 1 ; Applications of
Determinants 196
A First Look at Eigenvalues and Eigenvectors 210
Linear Transformations
265
Matrices as Transformations 265
Geometry of Linear Operators 280
Kernel and Range 296
t.6.4 Composition and Invertibility of Linear
x l Transformations 305
5 Computer Graphics 318
Dimension and Structure
329
Basis and Dimension 329
Properties of Bases 335
The Fundamental Spaces of a Matrix 342
The Dimension Theorem and Its Implications 352
The Rank Theorem and Its Implications 360
The Pivot Theorem and Its Implications 370
The Projection Theorem and Its Implications 379
Best Approximation and Least Squares 393
Orthonormal Bases and the Gram-Schmidt
Process 406
7.10 (^-Decomposition; Householder
Transformations 417
7.11 Coordinates with Respect to a Basis 428
CHAPTER 8
Diagonalization
443
8.1
Matrix Representations of Linear
Transformations 443
8.2 Similarity and Diagonalizability 456
8.3 Orthogonal Diagonalizability; Functions
of a Matrix 468
8.4 Quadratic Forms 481
8.5 Application of Quadratic Forms
to Optimization 495
8.6 Singular Value Decomposition 502
8.7 The Pseudoinverse 518
8.8 Complex Eigenvalues and Eigenvectors 525
8.9 Hermitian, Unitary, and Normal Matrices 535
8.10 Systems of Differential Equations 542
CHAPTER 9
9.1
9.2
9.3
General Vector Spaces 555
Vector Space Axioms 555
Inner Product Spaces; Fourier Series 569
General Linear Transformations;
Isomorphism 582
APPENDIX A
APPENDIX B
How to Read Theorems AI
Complex Numbers A3
ANSWERS TO ODD-NUMBERED EXERCISES
PHOTO CREDITS
INDEX
A9
C1
1-1
XI
Guide for the Instructor
15
4.3
4.4
% Chapter 5
Matrix Models
5.1
5.2
5.3
5.4
Chapter 6
16
17
18
19
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6.1
6.2
6.3
6.4
6.5
Chapter 7
20
21
22
23
24
25
26
27
28
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29
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
Chapter 8
30
31
32
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8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
Chapter 5
Linear Transformations
Chapter 6
Matrices as Transformations
Geometry of Linear Operators
Kernel and Range
Composition and Invertibility of Linear Transformations
Computer Graphics
15
16
17
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Dimension and Structure
Chapter 7
Basis and Dimension
Properties of Bases
The Fundamental Spaces of a Matrix
The Dimension Theorem and Its Implications
The Rank Theorem and Its Implications
The Pivot Theorem and Its Implications
The Projection Theorem and Its Implications
Best Approximation and Least Squares
Orthonormal Bases and the Gram-Schmidt Process
(^-Decomposition; Householder Transformations
Coordinates with Respect to a Basis
Matrix Representations of Linear Transformations
Similarity and Diagonalizability
Orthogonal Diagonalizability; Functions of a Matrix
Quadratic Forms
Application of Quadratic Forms to Optimization
Singular Value Decomposition
The Pseudoinverse
Complex Eigenvalues and Eigenvectors
Hermitian, Unitary, and Normal Matrices
Systems of Differential Equations
General Vector Spaces
9.1
9.2
9.3
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14
Dynamical Systems and Markov Chains
Leontief Input-Output Models
Gauss-Seidel and Jacobi Iteration; Sparse Linear Systems
The Power Method; Application to Internet Search Engines
Diagonalization
Chapter 9
33
34
35
Cramer's Rule; Formula for A '; Applications of Determinants
A First Look at Eigenvalues and Eigenvectors
Vector Space Axioms
Inner Product Spaces; Fourier Series
General Linear Transformations; Isomorphism
Appendices
Appendix A How to Read Theorems
Appendix B Complex Numbers
18
19
20
21
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22
23
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24
Chapter 8
25
26
27
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Chapter 9
28
29
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