I.
INTRODUCTION
1. Title of Module
Mathematics 4, Linear Algebra, Mr. Tendayi Chihaka, University of Zimbabwe
2. Prerequisite Courses or Knowledge
Unit 1: Matrices and linear transformations
Secondary school mathematics is prerequisite.
This is a level 1 course.
Unit 2: Applications of matrices
Linear Algebra 1 is prerequisite.
This is a level 2 course.
3. Time
120 hours
4. Material
The course materials for this module consist of:
Study materials (print, CD, on-line)
(pre-assessment materials contained within the study materials)
Two formative assessment activities per unit (always available but with specified
submission date). (CD, on-line)
References and Readings from open-source sources (CD, on-line)
ICT Activity files
Those which rely on copyright software
Those which rely on open source software
Those which stand alone
Video files
Audio files (with tape version)
Open source software installation files
Graphical calculators and licenced software where available
(Note: exact details to be specified when activities completed)
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5. Module Rationale
The study of the field of Linear Algebra will equip you with the requisite
background knowledge and understanding which will enable you to teach such
topics as simple linear equations and their solutions; vectors and operations on
vectors; matrices and operations on matrices. Furthermore, the study will help
you to realise the global connections between these topics and apply the
knowledge in teaching transformation geometry and mechanics.
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II.
CONTENT
6. Overview
Overview
The general layout of content in the units proceeds, wherever possible from the
concrete representation of the concepts to their abstract forms.
Unit 1 begins with a treatment of systems of linear equations and their solutions.
This is followed by a section that introduces vectors and matrices and dwells
quite a lot on operations on these and the theory and properties of determinants.
The relatively more abstract concept of vector spaces is treated next. The theory
and properties of Linear transformations closes this unit.
Unit 2 introduces the notions of eigenvalues and eigenvectors. The
diagonalisation property is demonstrated and proved.
Each unit has a maximum of four activities, one of which focuses on mathematics
education, pedagogics and didactics. This helps students not only to focus on
mathematical content, but also to focus on their goal as teachers of mathematics
in the secondary school.
Outline
Unit 1: Matrices and linear transformations (80 hours/35 hours)
Level 1. Priority A. No prerequisite.
Vector spaces over R. (12/5)
Vector subspaces. (10/4)
Linear independence. (8/3)
Basis and dimension. (8/3)
Matrices. (10/4)
Linear transformations and their matrices. (12/4)
Determinants. (8/3)
Systems of linear equations. (12/4)
Unit 2: Applications of matrices (40 hours/35 hours)
Level 2. Priority B. Linear Algebra 1 is prerequisite.
Eigenvalues and Eigenvectors. (8/7)
Minimal polynomials. (8/7)
Linear functionals. (8/7)
Bilinear and quadratic forms. (8/7)
Orthogonal matrices and operators. (8/7)
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Graphic Organiser
This diagram shows how the different sections of this module relate to each
other.
The central or core concept is in the centre of the diagram. (Shown in red).
Concepts that depend on each other are shown by a line.
For example: Vector Space is the central concept. The Vectors depend on the
idea of a Vector Space. The Eigenvalues and Eigenvectors depend on the
Vectors.
Linear
Equations
Vectors
Vector Space
Matrices
Linear
Transformation
Eigenvalues
& Eigenvectors
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7. General Objective(s)
You should be equipped with knowledge of vector spaces, matrices, linear
transformations and determinants and their appropriate applications, including
what is necessary to confidently teach these subjects at the secondary school
level.
You will acquire a secure knowledge of the content of school mathematics to
confidentially teach these subjects at the secondary school level.
You will acquire a knowledge of and can apply available ICT to improve the
teaching and learning of school mathematics.
8. Specific Learning Objectives (Instructional Objectives)
You should be able to:







Demonstrate an understanding of the concepts of vector spaces and
subspaces, systems of linear equations, matrices, linear transformations,
determinants and their applications, eigenvalues and eigenvectors, linear
functionals, bilinear and quadratic forms, orthogonal matrices and
orthogonal operators.
Determine the linear dependence or independence of a set of vectors in a
given vector space.
Find the basis and dimension of a vector space.
Identify and describe a linear transformation.
Operate with matrices.
Determine the conditions and nature of solutions of systems of linear
equations.
Find the determinant of square matrices
You should secure your knowledge of school mathematics in:
 Vectors.
 Systems of equations
 Matrices and their inverses (22)
 Determinants
 Geometric linear transformations
 Finding the matrix for a given linear transformation.
III.
TEACHING AND LEARNING ACTIVITIES
9. Pre-assessment
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Title of Pre-assessment : BASIC Linear Algebra Concepts
Rationale : This pretest aims to:
 Test the security of the students basic knowledge of Linear Algebra by
setting questions on linear algebra that are based on their previous
knowledge of secondary school mathematics.

Find out the strngths and weaknesses of the different students in Linear
Algebra so as to guide programme developers on the knowledge that the
students bring with them to the course.
_____________________________
QUESTIONS
. Linear algebra
Pretest
1. The solution to the simultaneous equations
x  y 1
2x  3y  7
Is:
A.
B.
C.
D.
(-5,4)
(4,-5)
(-4,5)
(-4,-5)
2. The system
x  2y  3
3x  6 y  9
Has
A. no solution
B. A unique solution
C. Many solutions
D. Zero solutions
3. The length of the vector
5 
 
 12 
Is
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A. 17 
B. 13
C. 7
D. 60
4. Which of the following has no solution
3 4
   
A.  4    0 
6 0
   
1 
 2  
B.     4 
3  2
 
 0  1 
   
C. 1    0 
0 0
   
 4  2
D.     
 6  3 
5. Which of the following has no solution
 2 3  1 2 2 
A.  2 1 X 0 3 7 

 

 4 5 1 4 5 
 7 1
B.  2 3 1   4 3


 2 1
a x   x 
C
 
c d   z 
D. None of the above
6. Which of the following matrices do not have a determinant
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a b 
A 

c d 
1
B. 
4
1
C. 
6
2
8 
3 5
4 1
0 8 7 
D. 7 1 9 


5 6 4 
7 The inverse of the matrix
 2 5
2


1
5

Under multiplication is
 1
A.  2

 5
5 

2

 2
B.  2

5
5 

1

0 0
C. 

0 0
D. Non existent
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8 The matrix representing the linear transformation for a 90 anticlockwise
rotation through the origin is
0 1 
A. 

1 0 
 1 0
B. 

 1 0
1 0 
C. 

0 1
1 1
D. 

 0 1
9. The matrix transformation representing a reflection in the line y=x is
0
A. 
1
1
B. 
0
1
0
0
1 
 1 1
C. 

 0 1
 Cos60 sin 60 
D. 

  sin 60 cos 60 
10. The matrix that reflects an enlargement factor of k, centre (0,0) is
k
A
k
k
k 
0 k 
B. 

k 0 
k 0
C. 

0 1
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1 0 
D. k 

0 1 
Title of Pre-assessment : Basic linear algebra
ANSWER KEY
1. C
2. C
3. B
4. B
5. D
6. C
7. A
8. A
9. A
10. D
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Title of Pre-assessment : Basic Linear Algebra
PEDAGOGICAL COMMENT FOR LEARNERS
(100-200 words)
This pretest aims to find out whether your knowledge of basic concepts of linear
algebra that you acquired during your primary and secondary school and you
should sail through the test!
Should you score a score of less than 8, then you are advised to seriously revisit
your secondary school materials on the topic and revise this with the help of a
friend or a local secondary school teacher in your area.
Ideally, though, the content of the test is so basic and elementary that a score of
less than 10 should be a cause for concern… GOOD LUCK!!!
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10. KEY CONCEPTS (GLOSSARY)
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Linear equation

Variables, Constants

Non-linear equation

System of linear equations

Homogeneous system

Consistent systems

Inconsistent systems

Trivial solutions

Non trivial solutions

Vector

Vector addition

Scalar multiplication

Triangle inequality

Matrix

Row operations

Gauss’ method

Gauss’ theorem

Echelon form

Free variables

Parameter

Parametrize

Matrix

Zero vector

Non-singular matrix

Dot product

Length of a vector

Triangle inequality theorem
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
Cauchy schwartz inequality theorem

Reduced echelon form

Row equivalence

Gauss-jordan reduction

Nonsingular

Determinant

Permutation expansion

Multilinear map

N-permutation

Signum

Parallelepiped

Laplace’s expansion

Adjoint

Cramer’s rule
11. COMPULSORY READINGS
Reading #1
Complete reference:
Linear Algebra by Jim Hefferon Mathematics, Saint Michael’s College,
Colchester, Vermont USA 05439, 2006
Web reference: http://joshua.smcvt.edu
Abstract/Rationale: A complete open-source text book in Linear Algebra. The
book completely covers all of the requirements o this course. Within the learning
activities students will be directed to specific page references for readings,
activities and exercises.
Note that there is only one compulsory refernce because it is a complete text book
providing coverage for the whole course.
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12. COMPULSORY RESOURCES
Resource #1
wxMaxima. This is Computer Algebra System (CAS). You should double click on
the Maxima_Setup file. Follow the prompts to install the software. Different
versions will be installed. We will always use the version called wxMaxima. Be
careful to choose the correct one. You will find a general introduction to maxima
in the Integrating ICT and Maths module. However, there is a complete manual
for the software available. To find it, run wxMaxima and choose Maxima help in
the Help menu. The web site for this software is http://maxima.sourceforge.net.
Look in activity 3 to see how to get started using mxMaxima for matrix
operations.
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13. USEFUL LINKS
The Linear Algebra Toolkit (visited 07.11.06)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=sys
 This site shows a full matrix solution to any system of linear equations that
you input.
 Choose the number of equations and numbers of unknowns, press NEXT.
 Enter the coefficients. Press NEXT.
 Check carefully through the method of solution.
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/LinearSystemofEquations.html
 Read this entry for Linear Systems.
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/System_of_linear_equations
 Read this entry for Systems of Linear Equations.
 Follow links to explain specific concepts as you need to.
MacTutor History of Mathematics (visited 07.11.06)
http://www-history.mcs.st-andrews.ac.uk/Indexes/HistoryTopics.html
 Search the history topics for Linear equations.
The Linear Algebra Toolkit (visited 07.11.06)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=det
 Explore determinants with this device in the toolkit.
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/Determinant.html
 Read this entry for Determinants.
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Determinant
 Read this entry for Determinants.
 Follow links to explain specific concepts as you need to.
MacTutor History of Mathematics (visited 07.11.06)
http://www-history.mcs.standrews.ac.uk/HistTopics/Matrices_and_determinants.html
 Explore the history of matrices and determinants
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14. LEARNING ACTIVITIES
Module 4: Linear Algebra
Unit 1: Matrices and linear transformations
Learning Activities
Activity 1: Linear Systems
1.0
Objectives
At the end of the activity, you should be able to:

Define a linear equation

Distinguish between a linear equation and a non-linear equation

Define a system of linear equation

Define a homogeneous system of linear equation

Use appropriate notation to represent a system of linear equations

Solve linear equations and distinguish between consistent and
inconsistent systems

Use Gaussian elimination to solve systems of linear equations

Define a vector

Perform vector addition and scalar multiplication of vectors

Find the length of a vector and prove the triangle inequality

State and prove the Cauchy-Schwartz inequality

Define a matrix

Represent linear equations in matrix form

Perform row operations on matrices

Solve linear equations using the Gauss-Jordan reduction method
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1.1
Glossary
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Linear equation

Variables, Constants

Non-linear equation

System of linear equations

Homogeneous system

Consistent systems

Inconsistent systems

Trivial solutions

Non trivial solutions

Vector

Vector addition

Scalar multiplication

Triangle inequality

Matrix

Row operations

Gauss’ method

Gauss’ theorem

Echelon form free variables

Parameter

Parametrize

Matrix

Zero vector

Non-singular matrix

Dot product

Length of a vector

Triangle inequality theorem
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
Cauchy schwartz inequality theorem reduced echelon form row
equivalence

Gauss-jordan reduction
1.2
Summary of the Learning Activity
In this activity we will explore the concepts and nature of linear equations, their
systems and properties, their solutions and the nature of their solutions. We go
on to introduce elementary matrix theory and its application in solving systems of
linear equations. The final section deals with an introduction to the theory of
determinants. This activity forms the basis for the further study of linear algebra.
1.3a Compulsory Reading
All of the readings for the module come one Open Source text book. This means
that the author has made them available for any to use them without charge. We
have provided a complete copy of the text on the CD accompanying this course.
The course text is:
Linear Algebra by Jim Hefferon
Mathematics, Saint Michael’s College, Colchester, Vermont USA 05439, 2006
Web reference: http://joshua.smcvt.edu
1.3b Internet and Software Resources
For the Linear Algebra course we have provided a copy of open source software.
You are free to use this software without charge. You should install the software
and make sure you have access to a computer in order to use them. The
software provides open tools to explore mathematics in general including
powerful linear algebra tools. You should use this software as often as possible,
so that you get used to how it works.
1. wxMaxima. This is Computer Algebra System (CAS). You should double
click on the Maxima_Setup file. Follow the prompts to install the software.
Different versions will be installed. We will always use the version called
wxMaxima. Be careful to choose the correct one. You will find a general
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introduction to maxima in the Integrating ICT and Maths module. However,
there is a complete manual for the software available. To find it, run
wxMaxima and choose Maxima help in the Help menu. The web site for
this software is http://maxima.sourceforge.net. Look in activity 3 to see
how to get started using mxMaxima for matrix operations.
Note: You must not use wxMaxima to answer exercise questions for you!
Instead, you should try different examples of calculations and operations to
make sure that you understand how they are done, so that you are better able
to do them without the support of the software.
Web References:
The Linear Algebra Toolkit (visited 07.11.06)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=sys
 This site shows a full matrix solution to any system of linear equations that
you input.
 Choose the number of equations and numbers of unknowns, press NEXT.
 Enter the coefficients. Press NEXT.
 Check carefully through the method of solution.
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/LinearSystemofEquations.html
 Read this entry for Linear Systems.
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/System_of_linear_equations
 Read this entry for Systems of Linear Equations.
 Follow links to explain specific concepts as you need to.
MacTutor History of Mathematics (visited 07.11.06)
http://www-history.mcs.st-andrews.ac.uk/Indexes/HistoryTopics.html
 Search the history topics for Linear equations.
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1.4
Introduction
Let us consider the following scenarios:
You are the village Mathematics teacher and are consulted by two people, Mrs.
Nhau who runs a restaurant and Mr. Kondo, who is the local herbalist.
Mr. Kondo produces two types of health potions, Rudo and Zwanamina. To
prepare a Kg of Rudo, Mr. Kondo spends 20 minutes on the pestle and mortar
and 16 minutes on the grinding stone. For a kilogramme of Zwanamina, he has
16 minutes on the pestle and mortar and 8 minutes on the grinding stone. On
one particular day, he has to borrow the pestle and mortar and grinding stone
from a fellow herbalist who only gives him 6 hours on the pestle and mortar and 4
hours on the grinding stone. How many kilogrammes of each type of potion
would you recommend him to fully utilize the borrowed equipment?
Mrs. Nhau owns a restaurant that produces a meal consisting of foods X, Y and
Z.
Each Kg of X contains 4 units of protein, 6 units of fat and 8 units of
carbohydrates.
Each Kg of Y contains 3 units of protein, 2 units of fat and 1 unit of
carbohydrates. Each Kg of Z contains 3 units of protein 3 units of fat and 2 units
of carbohydrates.
She has been told by health officials that the meal must provide exactly 25 units
of protein, 24 units of fat and 21 units of carbohydrates. Advise her on how many
kilogrammes of each type she must use to meet the stipulated health
requirements.
What immediately comes into your mind?
DO THIS: Let us consider Mr. Kondo’s and Mrs. Nhau’s cases.
1.
You had to find relationships and connections in the information
given.
What are these relationships?
2.
You had to find the mathematical representations to these connections and
advise the two entrepreneurs accordingly.
What advice did you give?
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3.
What if any, are the major differences between the two scenarios?
4.
What difficulties, if any, did you face in trying to advise the two?
5.
Did you require help from a friend, a colleague on any of the two problems?
Which
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and
why?
22
1.5 Equations
Since elementary (Primary School) we have been asked to solve for an unknown
or unknowns in given practical situations. You need to think here of the many
examples and encounters you had with unknowns. The “statements” you had to
solve for the unknowns were “called” equations and the unknowns were called
variables.
More specifically, we were also able to represent these equations by a line (or
lines) in the x-y plane in the form of
a1 x  a2 y  b
More generally, we define a Linear Equation in n variables x1 , x2 x3 ,......xn as one
that can be expressed as
a1 x1  a2 x2  ......  an xn  b
Where a1 , a 2 ,......, an and b are constants.
The following are examples of linear equations:
3x + 4y =8, y= 12 x  4 , 3x1  2 x2  4 x3  x4  7
DO THIS: Which of these would be similar to the equations, if any, derived from
the two problems above.
You will notice that the highest power of any of the variables in the equations is
1. That is; there is no product of any of the variables..
The following are not linear equations:
xy = 4, x 2  2 y  7, y  sin x  1, x1  x2  x3  6
Finding the value of the unknown, or solving the equation, means finding the
solution of the linear equation. Thus the solution of:
x1  2  7
Is the number 5 such that x1  5
More specifically, a solution of the equation
a1 x1  a2 x2  ......  an xn  b
is the set of numbers
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c1 , c2 ,......, cn such that
a1c1  a2 c2  ......  an cn  b
Examples here are:
Find the solution set of
(a) 3x  4 y  1,
(b)
x1  x2  7 x3  5
This is rather different from what you are used to isn’t it? Why? You may want
to discuss with a friend or colleague the solutions to the equations above.
1.5.1 Systems of Linear Equations.
You also met situations where you were supposed to solve, say, the following
equations
x+y=1
2x + y = 3
That is you were supposed to find the values of x and y that satisfied the two
equations.
When we have more than one linear equation to solve we say that we have a
system of linear equations or a linear system. More generally, a system of m
linear equations in n unknowns is written as
a11 x1  a12 x 2  ......a1n x n  b1
a 21 x1  a 22 x 2  ......a 2 n x n  b2
.
.
.
.a m1 x1  a m 2 x 2 ......  a mn x n  bn
DO THIS: Write down the 4th and the ith equation
A sequence of numbers s1 , s2 ,......s n is called a solution of the system if
x1  s1 , x2  s 2 ,......, xn  s n
for all the equations in the system.
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Not all systems have solutions. If the system has no solution, it is said to be
inconsistent If it has a solution it is called consistent.
If b1  b2  ......  bm , the system is called homogeneous system.
If x1  x2  ......  xn  0 is a solution to a homogeneous system, it is called the
trivial solution.
A solution to a homogeneous system in which one of the xi 's  0 is called a nontrivial
solution.
DO THIS: Now, l want you to investigate the solutions of the following pairs of
equations in two variables and give their geometrical interpretations.
Solve:
(a)
2x + 3y = 5

2x  y = 4
(b)
x + 2y = 3
(c)
2x + 4y
=5
2x  y
=1
4x  2y

Solve each pair of
equation using
algebra.
Then solve each pair
by drawing graphs.
=2
DO THIS: Which of the system of equations has one solution, no solution and
infinitely many solutions?
1.5.2 Equivalent Systems.
Two systems are said to be equivalent if they both have exactly the same
solutions.
Looks, for example, at the systems
2 x1  x 2  7
x1  3x 2  7
And
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3x1  2 x 2  0
8 x1  3x 2  7
5 x1  x 2  7
DO THIS: Are these systems equivalent? Verify this assertion.
DO THIS: You will need to verify the following assertions on equivalent systems
using the equations:
3x + 2y = 5
2x + y = 6
Given a system of equations,
(i)
interchanging two equations
(ii)
multiplication of an equation by a non-zero constant and
(iii)
adding a multiple of an equation to another equation produces an
equivalent system of equations. Principles of equivalent systems
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1.6
Gaussian Elimination
You will remember the many methods you used to solve systems of equations –
then called simultaneous equations. Chief among the methods was the method
of elimination in which you sought to eliminate one of the variables.
Try and describe in words the process you used.
Now try the method on the following system of equation
x  2y + 3z = 9…………….1
y + 3z = 5……………..2
z = 2……………...3
Easy! Why?
You did not have to use elimination as from 3 you already had the value of z and
all you needed to do was back substitution.
This system is in what is called row echelon form, which means that it follows a
stair–step pattern and has leading co-efficients of 1.
Not all systems, however, are expressed in this form.
1.6.1 Example
x – 2y + 3z = 9…………………………….. R1
x
+3y
= 4…………………………….. R2
Ri  row i
2x  5y + 5z =17 ……………………………. R3
This has to be reduced to its equivalent row-echelon form before it becomes
easy to solve. (Not here, though that substitution can also be used by reducing
the equations into those involving y and z.) We need to use a systematic process
that can easily be applied to systems with more variables.
Working from the left corner of the system and using R1 we have:
R1  R1
R2  R2  R1 ( R4 )
R3  R3
Read this:
“Row one goes into Row one”
“Row two goes into Row 2 plus row one and becomes Row four”
“Row here remains as row three”
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Thus:
x – 2y + 3z = 9……………………………. R1
y + 3z = 5……………………………. R4
2x  5y + 5z = 17…………………………... R3
R1  R1
R4  R4
R3  R3  (2 R1 )( R5) )
x  2y + 3z = 9…………………….. R1
y + 3z = 5…………………….. R4
y
– z = -1……………………. R5
R1  R1
R4  R4
R5  R5  R4 ( R6 )
x +2y  3z = 9………………………… R1
y + 3z = 5………………………….. R4
2z = 4…………………………. R6
R1  R1
R4  R4
R6 
1
R6 ( R7 )
2
x  2y + 3z = 9………………………….. R1
y + 3z = 5…………………………… R4
z = 2………………………….. R7
Which is the equivalent row echelon form to our system.
Back substitution gives us x = 1, y =1, z = 2
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HINT:
Since we can easily make errors in this method, it is prudent to confirm
that this is indeed the solution to the equation by substituting the values of x, y
and z into the original system.
1.6.2 Example
Solve:
x2  x3  0................................................R1
x1  3x3  1.............................................R2
 x1  3x2  1.............................................R3
R1  R2 ....( R4 )
R2  R1 ....( R5 )
R3  R 3
x1  3x3  1..................................R4
x2  x3  0.........................................R5
 x1  3x2  1....................................R3
R4  R4
R5  R5
R3  R3  R4 ....( R6 )
x1  3x3  1...................................R4
x2  x3  0........................................R5
3x2  3x3  0....................................R6
R4  R4
R5  R5
R6  R6  (3R5 )....( R7 )
x1  3x3  1.................................R4
x2  x3  0......................................R5
0  0....................................................R7
R7 becomes unnecessary and so we need to drop it!
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R4  R4
R5  R5
x1  3x3  1...............R4
x 2  x3  0.....................R5
We choose to let x3 be the free variable (independent)
Thus
x 2  x3
And x1  3x3  1
So x3 can take any real number say s and
x3  s
x2  s
x1  3s  1
This shows that this system has infinitely many solutions.
Again let us look at the following system of equations:
1.6.3 Example
Solve:
x1  3 x 2  x3  1.............R1
2 x1  x 2  2 x3  2............R 2
x1  2 x 2  3 x3  1.........R3
R1  R1
R 2  R 2  (2 R1)..............R 4
R3  R3
x1  3 x2  x3  1........................R1
5 x2  4 x3  0..............................R 4
x1  2 x2  3 x3  1......................R3
R1  R1
R4  R4
R3  R3  ( R1).............R5
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x1  3x2  x3  1.............R1
5 x2  4 x3  0..................R 4
5 x2  4 x3  2................R5
R1  R1
R4  R4
R5  R5  ( R 4)............R 6
x1  3 x 2  x3  1...............R1
5 x 2  4 x3  0...................R 4
0  2....................................R 6
1.6.4 Discussion
We have thus found an equivalent system that has been reduced to an absurdity
and we conclude that the system has no solution.
You will have noticed that in this activity there are certain prompts questions,
hints that one needs to keep in mind and asking themselves when mathematizing
problems. At the back of your mind, always be asking yourself why6 you are
doing certain things in the solution of problems. Why, for instance, do we have to
interchange rows during Gaussian elimination? Why do we transform only one
equation at a time? Is it not possible to pivot row one and transform all the others
at the same time? What are the consequences of doing this? And so on and so
on…………….
In the following activities much of the discussive narration has been left out and
instead, you will be referred to materials and readings that have been availed to
you. I hope that you will take the hints above and keep asking
yourselves
questions that will make you comprehend, understand and conceptualise
mathematics meaningfully. GOOD LUCK!
At this stage I will initiate you into the preceding paragraphs message by using
your core text - Linear Algebra by Jim Hefferon which is an open source
manuscript found on the Internet.
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You will notice that some of the ideas discussed above will be repeated in the
section below but these will be in the format of your core text. You are strongly
advised to familiarise yourself with the notation used in the text and be able to
relate it to other notations in the different sources you will meet in the course of
your study
1.7
Solving Linear Systems

Reading and examples pp1- 8

Gauss’ Method I.1 p2

Definition 1.1 p2
Reading and Activity from:
Linear Algebra by Jim
Hefferon
1.7.1 Linear equation, coefficients, constant, system of linear equations
Important Note:
Note the correction a1 ,.......an  , d  , and ( s1 , s2 .....sn ) 
n
in the definition.

Theorem 1.4 p3 Gauss’ method

Definition 1.5 p4; elementary reduction operations or row operations or
Gaussian operations. Swapping, pivoting, multiplying by a scalar or
Important Note:
Note the change from R1 for “row one” in the preceding section to the Greek
character 1 in the text and that you can carry out more than one row
operation if you can manage it!
rescaling.

Definition 1.9 p5 leading variable, echelon form
DO THIS:
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
Exercises 1.16- 1.17. Exercise 1.18 takes you back to the high school and
makes interesting reading.

Exercise 1.21, 1.25. Do exercises 1.26 -1.29, 1.35- 1.37 with a colleague.
(ANSWERS: P1-12)
1.7.2 Describing the solution set

Reading and examples p11-18
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
Note the use of the terms parameter and parametrize on p13 which will be
used extensively in the text

Definition 2.2 free variables

Definition 2.6 p13 nxn matrix

Definition 2.8 vector, column vector, row vector, components; 2.9; 2.10
vector sum; 2.11 scalar multiplication p15
DO THIS

Exercises 2.15- 2.21, 2.22, 2.26 pp 18-20
(ANSWERS: P12-16)
1.7.3 General = Particular + Homogeneous

Theorem 3.1 p21

Definition 3.2 p21 homogeneous equation

Definition 3.4 p22 zero vector

Lemma 3.7 p22

Lemma 3.8 p24

Corollary 3.11 p26 solution sets
Module Development Template
Reading and Activity from:
Linear Algebra by Jim
Hefferon
33
Important Note:
The table on page 27 is quite handy as it summarizes the factors affecting the
size of a general solution

Definition 3.12 p27 singular and non singular matrix
DO THIS

Exercise 3.15- 3.20 p29-31

Do 3.31-3.25 with a colleague
(ANSWERS P16-19)
1.7.4 Linear Geometry of n-Space

Reading and examples p32-37
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises1.1- 1.7 p37
(ANSWERS: P20-21)
1.7.5 Length and angle measures
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p39-42

Definition 2.1 p39 Length of a vector

Definition 2.3 p40 the dot product or inner product or scalar product of two
vectors

Theorem 2.5 p40 The triangle inequality

Corollary 2.6 p41 The Cauchy- Schwartz Inequality

Definition 2.7 The angle between two vectors
Important Note:
Take careful note of the definition of orthogonal vectors on p42
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DO THIS

Exercises 2.10, 2.12. p42-45

Do exercises 2.17, 2.28, 2.25, 2.32, 2.38, 2.39 with a colleague.
(ANSWERS: P22-26)
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1.8
Reduced Echelon Form
1.8.1 Gauss- Jordan reduction

Reading and examples p46-51

Definition 1.3 p47 Reduced Echelon Form

Lemma 1.4, 1.5 p50

Definition 1.6 p50 Row Equivalence of Matrices
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Complete exercises 1.7-1.13 p51 with a colleague
(ANSWERS: P27-29)
1.8.2 Row Equivalence

Reading and examples p52- 5.9

Definition 2.1 p52 Linear Combination

Lemma 2.2 p52 Linear Combination Lemma

Corollary 2.3 p53

Lemma 2.5 p55

Lemma 2.6 p56

Theorem 2.7 p57
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises 2.11, 2.19. p59

Do Exercises 2.24-2.28 with a colleague.
(ANSWERS: P29-33)
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1.9
Synthesis
In this activity you have learnt the definitions and properties of the basic concepts
of equations and systems of equations. You have been able to find and
categorise solutions using various methods, chief among which was Gauss’
method. You should by now also be familiar with basic notions of vectors and
matrices and the corresponding operations on these and you should be in a
position to apply these in the activities that follow.
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Activity 2 : Vector Spaces
2.0
Objectives
By the end of this activity you should be able to:

Define a vector and write down the general form of a vector.

Discuss conditions for the equality of two vectors.

add vectors and perform scalar multiplication on vectors

prove the theorems and laws that guide the two operations of vector
addition and scalar multiplication and solve related problems

Define and state what is meant by a vector space and all its axioms

determine whether a given set and two operations form a vector space

use the vector space axioms to show important characteristics of vector
spaces.

State and define what is meant by a subspace of a vector space

List and use the conditions for a subspace to determine whether a given
subset of a vector space is a subspace.

Find the basis and dimension of a given vector space

Define and determine the transpose of a given matrix

Find the direct sum of given subspaces
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2.1
Glossary
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Vector space

Trivial space

Subspace

Span

Linear closure linear independence

Linear dependence

Basis

Dimension

Row space

Row rank

Column space

Column rank

Transpose

Sum of subspaces

Concatenation

Direct sum
2.2
Introduction
In this activity we explore the concept of an important aspect of linear algebrathe vector space. The idea of the vector space is central to the course of linear
algebra as all that has gone on before this activity and will follow depends on a
full understanding of this concept. A clear and full understanding of the notion
and structure of a field, covered in the course Basic Algebra, essential in
order for you to fully benefit from this activity.
We further go on to explore the behaviour of subsets of vector spaces under the
same operations defined on the vector space which are vector spaces in their on
right and bring out the important structure called a subspace which as a
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consequence inherits the properties of a vector space. But first we start with a
story that introduces one of the fundamental components of a vector space.
You and your friends are presented with the following problems

A car leaves Nairobi travelling at an average speed of 120
km/h. Where is the car after 1 hour.

A plane leaves Nairobi Airport travelling at an average
speed of 600 km/h. What is the position of the plane after
1 hour.
1. Discuss and list all the information you would require in order for you to
solve the two problems.
2. What are the key words you require that are necessarily sufficient for
you to solve and answer the two
problems.
3. What , if any, are the fundamental
physical or conceptual differences
between the two problems?
In this activity we begin by defining and geometrically representing the concept of
a vector. This is followed by both algebraic and geometric representations of
vector addition and scalar multiplication. Basic properties of these operations
will be discussed. A vector has been defined in the dimensional plane as an
ordered pair of members or as a 2 x 1 matrix. Similarly a vector in space has
been described as an ordered triple of real numbers or as a 3 x 1 matrix. In ndimensional space or n-space a vector has been described as an n- tuple
Practically, in subjects such as physics, we have defined a vector as an entity
which has both magnitude and direction or as a directed line segment.
We thus have a number of different ways of conceptualising a vector and have
called all these a “vector”. The only thing that is of concern common in these
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conceptions is the behaviour of the vector. This activity deals with the rules and
properties that structure this behaviour.
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2.3
Internet and Software Resources
Web References:
The Linear Algebra Toolkit (visited 07.11.06)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
 This is the menu page. There are a number of tools for practicing with
vector spaces. Use these to help check your understanding in the
activities.
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/VectorSpace.html
 Read this entry for Vector Spaces.
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Vector_space
 Read this entry for Vector Spaces.
 Follow links to explain specific concepts as you need to.
MacTutor History of Mathematics (visited 07.11.06)
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces.html
 Read for interest, the history of the development of abstract linear spaces.
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Note: You will need your copy of Linear Algebra by Jim Hefferon throughout activity
2.
2.4
Vector Spaces
2.4.1 Definition of a Vector Space

Reading and related examples pg79-87.

Definition 1.1 p80 Vector space
Reading and Activity from:
Linear Algebra by Jim
Hefferon

You should carefully follow examples 1:3 through 1:15, that deal with how you
determine that a given set with two operations defined on it forms or does not
form a vector space.
Note, in particular, examples 1.11, 1.9, 1.8 which show you that the concept of
vector spaces is not only confined to the set of vectors and its related operations,
but to other sets as well.
.
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DO THIS:

You should, with a colleague, try and complete all the examples that have
no solutions in the section you have just been reading.

Exercise: p88-90
1.18 (c-d)
1.19 (b-c)
1.20 (b)
1.25
1.44 (a)
(ANSWERS: P39-42)
2.4.2 Subspaces and Spanning Sets

Reading and examples p91-97

Definition: 2.1 p91 Subspaces

Lemma 2.9 p93

Definition 2.13 p95 span or linear closure
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
In other texts that using strict mathematical symbols, the span of a vector
space, Sp(S) is defined as follows:
Let S  V and S  xi  then by the Span of S we mean y  SpS  such that
y   ai xi . You may need to familiarise yourself with such notation as you
are bound to meet it if your read other texts on the same subject.

Lemma 2.13 p95
DO THIS

Exercise 2.20 - 2.47 p 97
(ANSWERS: P42- 48)
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Commentary
Exercise 2.42 gives very interesting insights into constructions of subspaces or
operations on subspaces and here we prove and give answers to some of the
issues raised by the question:
2.42(a). If A and B are subspaces of a vector space V, must A  B be a
subspace of V? Always? Sometimes? Never?
Answer
If A and B are subspaces, then they are vector spaces in their own right, so
O  AandO  B, henceO  A  B.Ifv, w  A  Bthenv, w  Aandv, w  B.
Thus
any
linear combination v  w  Aandv  w  B since A and B are subspaces.
Thus v  w  A  B. It follows then that A  B is a subspace of V.
In general, the intersection of any number of subspaces of a vector space V is a
subspace.
Important Note:
You will have noticed here that we only used the conditions that O  A  B
and that any linear combination of two vectors in A and B is also in their
intersection to prove that the intersection is a vector space. These are
sufficient conditions to prove that a given subset of a vector space is a
subspace.
2.4.3 Linear Independence
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p101 – 108

Lemma 1.1 p101

Definition 1.3 p 103 linear independence, linear dependence

Lemma 1.4
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Important Note:
You should note that the Lemma in 1.4 is the one given as a definition of
linear independence in other texts. Also note that the differences between
Definition, Theorems and Lemma’s is only a question of choice. The definition
says take it as it is and use it as a guideline. Theorems and Lemmas take the
definition as a proposition and prove it.

Theorem 1.12 p104
DO THIS

Prove Lemma 1.14 p105
Important Note:
The table on page 107 is a useful tool that summarises the properties of
independence and dependence and the relations of subset and superset.
DO THIS

Work through exercises 1.18-1.41 p108-114
(ANSWERS: P48-55)
2.4.4
Basis and Dimension

Reading and examples p112- 122

Reading and examples p112-116

Definition 1.1 p112 Basis

Definition 1.5 p113 standard basis

Theorem 1.12 p114

Definition 1.13 p115 representation
Module Development Template
Reading and Activity from:
Linear Algebra by Jim
Hefferon
46
DO THIS

Work through exercises1.16 -1.34 p116-118
(ANSWERS: P55-59)
2.4.5 Dimension

Reading and examples p118-122

Definition 2.1 p119 finite dimensional space

Lemma 2.2 p119 Exchange lemma

Theorem 2.3 p119

Definition 2.4 p120 dimension

Corollary 2.8 p120

Corollary 2.11 p121
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises 2.14- 2.18 p122-123

Do exercises 2.22, 2.24, 2.31, 2.33, with a colleague.
(ANSWERS: 60-62)
2.4.6 Vector Spaces and Linear Systems

Reading and examples p123-128

Definition: 3.1 p124 Row space, row rank

Lemma 3.3, 3.4 p124

Definition 3.6 p125 Column Space, column rank

Definition 3.8 p126 Transpose of a matrix
Module Development Template
Reading and Activity from:
Linear Algebra by Jim
Hefferon
47
DO THIS
Discuss the transformation that maps the matrix A into Atrans with a colleague:
1 1 
 2 5
1 2 3 4
Hint: Try
trans

A= 
,
A


rotation!
 3 6
1 5 6 0


 4 0
Is it possible to do this from your transformation geometry knowledge?
Important Note:
Most texts write Atrans simplyasAT
Vector Spaces and Linear Systems (Continued)

Lemma 3.10

Theorem 3.11 Row Rank = column rank

Definition. 3.12 p127

Theorem 3.13 p128

Corollary 3.15 p128
DO THIS.

Exercise p129-130 3.16- 3. 21

Do 3.30, 3.34, 3.37, 3.38 with a colleague
(ANSWERS: P63-67)
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2.4.7 Combining Subspaces

Reading and examples p131-136

Definition 4.1 p131 sum of subspaces

Definition 4.7 concatenation of sequences

Lemma 4.8 p133

Definition 4.9, independent subspaces, 4.10 direct sum of subspaces,
p135

Corollary 4.13 p135

Definition 4.14 p135 Complements

Lemma 4.15 p135
DO THIS

Exercises 4.20, 4.25, 4.31 p136-139

Do 4.40, 4.38, 4.40, 4.43 with a colleague
(ANSWERS:P68-71)
2.5
Synthesis
This activity extended your basic notions of vectors and introduced you to a new
structure, the vector space and its requisite axioms. The subsets of vector
spaces, subspaces were also introduced and their properties explored. You
should now be familiar with properties of spaces such as basis, dimension and
use these to solve related problems and prove lemmas, theorems and their
corollaries.
Armed with this knowledge you should now be ready to explore mappings and
transformations within the vector space and between spaces.
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Activity 3: Linear Transformations, Maps Between Spaces
3.0
Objectives
By the end of this activity you should be able to:

State what is meant by a mapping between two spaces

Define and distinguish between an isomorphism and a homomorphism

Determine and show whether a given map is a linear mapping or a linear
transformation

prove that the range and kernel of a linear map are subspaces of the
respective vector spaces

Represent Linear maps with matrices

Calculate the coordinate vector of an element with respect to a given
basis of the vector space

Write down the matrix of the transformation with respect to a given basis

Find composite mappings between spaces and their matrix
representations

perform the operations of addition, subtraction, scalar multiplication and
matrix multiplication on matrices

Find the inverse of a given matrix
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3.1
Glossary
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Isomorphism

Homomorphism

Linear map

Linear transformation

Rangespace

Nullspace

Kernel

Nullity

Nonsingular map

Sums

Scalar product

Matrix multiplication

Unit matrix

Identity matrix

Diagonal matrix

Permutation matrix

Nonsingular matrix

Matrix equivalence

Orthogonal projection

Gram-schmidt orthogonalisation

Orthogonal basis

Orthogonal complement
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3.2
Introduction
In this activity we look at mappings on the vector spaces we explored in the last
activity. We will be interested in particular, with mappings that preserve the
operations on vector spaces, vector addition and scalar multiplication. These
mappings are called linear mappings or linear transformations.
In Module one you should have familiarized yourself with the concept of
mappings or functions between sets and the various ways of describing them
such as one to one (injective), onto (surjective), one to one correspondence
and so on. We extend these to isomorphisms and homomorphisms .
We go on to show that these linear mappings on finite dimensional vector spaces
are basically matrices meaning that the study of linear mappings is the study of
matrices.
A household consists of the following family members:




Father, Masenge
Mother, Maria
2 sons, Tendayi and Paul
3 daughters, Anesu, Memory and Rudo.
Discuss with a colleague all the relations that exist
between the members of the family. For example, “is a
mother of” is a relation between Maria and Memory.
The father gets a new job in another town and the
family has to move to the new town and a new house.
Although the new house may be lager than their
original one’; Might contain more bedrooms and might
have a bigger kitchen, the movement will preserve the
relations you have listed among the members of the
family. Discuss the analogy between this depiction and
the description of a linear map given above.
Note: You will need your copy of Linear Algebra by Jim Hefferon throughout
activity 3.
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3.3
Internet and Software Resources
Software
You can practice the full range of matrix operations using wxMaxima.
Run the software. The screen should look like this:
Type your
commands
in here
Be careful:

Don’t add extra spaces or
punctuation.

Make sure you choose the
correct brackets.

When you open a bracket,
the close bracket is
automatically entered.
Getting started with Matrices

Type:

Press RETURN on your keyboard

Type:

You should see this:
A:Matrix([3,0,0],[0,3,0],[0,0,3])
B:Matrix([1,2,3],[4,5,6],[7,8,9])
(If you get stuck and you want to start again,
choose Restart maxima in the maxima
menu).
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Now that you have entered two matrices, you can test some
matrix operations.

Type:
A+B and press RETURN

Type:
3*A and press RETURN

Type:
A.B (to calculate the dot product) and press
RETURN

Type:
determinant(A) and press RETURN

Type:
invert(A) and press RETURN

Type:
eigenvectors(A) and press RETURN

Type:
eigenvalues(A) and press RETURN
You should now practice with mxMaxima.
When you are ready to extend yourself, choose maxima help in the help menu
and choose item 26: Matrices and Linear Algebra. Look for matrix operations that
you need and follow the instructions to check that you understand how to use
them.
Web References:
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/LinearTransformation.html
 Read this entry for Linear Transformations
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Linear_transformations
 Read this entry for Linear Transformations.
 Follow links to explain specific concepts as you need to.
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3.4 Isomorphisms and Homomorphisms
3.4.1 Isomorphisms

Reading and examples p153-160

Definition 1.3 p155 isomorphism
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
We also have a new name for a map, morphism, and the meaning of
isomorphism on the same page.

Definition p157 automorphism

Lemma 1.8, 1.9 p159
Important Note:
For the proof of 1.9 you need to recall the skill of induction you learnt in unit 2
on Number Theory.
DO THIS

Exercise 1.10, 1.11, 1.13, 1.17,1.19, 1.25, 1.28, 1.34
(ANSWERS P77-85)
3.4.2 Dimension and isomorphism

Reading and examples p163-168.

Theorem 2.1 p163
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
It is important here to remember the definition of an equivalence relation.
That is the relation must be:
 Symmetric
 Reflexive
 Transitive.
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
Theorem 2.2 p164 Isomorphic vector spaces

Lemma 2.3, 2.4 p164

Corollary 2.6 p166
DO THIS:

Exercise p168-169: 2.8, 2.9, 2.11, 2.15, 2.17, 2.25.
(ANSWERS: P85-87)
3.4.3 Homomorphisms

Reading and examples p170- 175

Definition II.1 p170 homomorphism
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Notes:
1.
The definition is also the definition for a linear map or a linear
transformation (in other sources that you will encounter as explained in
remark 1.12 below). Also note example 1.4 which discusses the concept
of a zero homomorphism .
2.
Example 1.5 should be given special attention as it explains the concept
of a homomorphism as a linear map and also the method of disproof by
a counter example.

Lemma 1.6 p172

Lemma 1.7 p172
Important Notes:
Lemma 1.7 p172 (1) is a sufficient condition for definition II.1. For, if we take
c1  c2  1
in (1) we have the first condition for a homomorphism, and if we
take c2  0 in (1) we have the second condition for definition II.1. You will find
that Lemma 1.7 is often used to prove that a mapping is a linear
transformation since it’s shorter that the definition.
DO THIS

Discuss the use of (2) of the Lemma with a friend.

What adjustments do you have to make to (2) in order for you to transform it
to theorem II.1?
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Module Development Template
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Homomorphisms (Continued)

Theorem 1.9 p172

Theorem 1.9 leads to a very important definition of linear transformation
which is very peculiar to this particular source.

Definition 1.11 p173 linear transformation

Remark 1.12 p174 explains the statement made above and the departure
from other sources

Lemma 1.16 p174
DO THIS

Exercise 1.17, 1.18, 1.19

Discuss examples 1.20, 1.23, 1.26, 1.35, 1.38, with a colleague and come
up with the solutions.

ANSWERS: P87-92
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3.5
Linear Maps
3.5.1 Rangespace and Nullspace
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
The rangespace is called the image set or just the range in other sources
while the nullspace is called the kernel.

Reading and examples: p177-186.

Lemma 2.1 p178
Important Note:
The dimension of the rangespace is the map’s rank

Definition 2.2 p178. rangespace.

Lemma 2.10 p182.
Important Note:
You will need to recall the ideas of the inverse function and inverse image
for this section.

Definition 2.11 p182 nullspace or kernel.
Important Note:
The nullspace or kernel of the linear map h:V  W is sometimes denoted by
ker(h) and is the set of all elements of V that are mapped to 0 W .

Theorem 2.14 p183

Corollary 2.17 p184

Lemma 2.18

Definition 2.19 p184 Non-singular linear map
Important Note:
Notice here how the author has carefully linked the linear maps to matrices by
introducing the idea of a non-singular map.

Theorem 2.21 p185
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DO THIS

Exercise 2.22, 2.23, 2.24, 2.27

Discuss these with a colleague 2.35, 2.38, 2.40, 2.42. p186-188.
(ANSWERS: P93-97)
3.5.2 Computing Linear Maps

Reading and examples p189-203
Reading and Activity from:
Linear Algebra by Jim
Hefferon
3.5.3 Representing linear maps with matrices

Definition 1.2 p191 matrix representation

Theorem 1.4 p192
Important Note:
This theorem is basically defining the process of matrix multiplication ‘with
which you are already familiar

Definition 1.5 p 192 Matrix vector product
Important Note:
Examples 1.8 which takes us back to transformation geometry. You need to
revisit this topic every time you deal with linear maps and transformations and
also the vector dot product.
DO THIS

Exercise: 1.11, 1.12, 1.13, 1.15, 1.17, 1.26. p196-199

Work on 1.28 with a colleague.
(ANSWERS: P97-105)
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3.5.4 Any Matrix Represents a Linear Map
You will recall in an earlier activity that we introduced the concept of a matrix and
used row operations to solve linear equations. In this activity we look at the
concept of a matrix as a linear map. You, therefore must revisit the earlier activity
and familiarize yourself with the row operations that we performed there.

Reading and examples: p199-203

Theorem 2.1 p199.
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
The comment on the bottom of page 200 is very important and you should take
note of it.

Theorem 2.3 p201

Corollary 2.5 and 2.6 p202
DO THIS

Exercises 2.9, 2.10, 2.11, 2.14, 2.18 p203-205 (Note the definition of a
diagonal matrix in this exercise)

Do 2.20, 2.21, 2.22, with a colleague.
(ANSWERS: P105-109)
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3.6
Matrix Operations
Now, this is a section that should not give you any problems as you have been
performing operations on matrices since you were in secondary school! All you
have to do is pay particular attention to the notation and translate it to fit your
own previous experience or vice versa !
Have Fun !
3.6.1 Sums (addition of matrices) and Scalar products (Multiplication of a
matrix by a scalar)

Reading and examples p206-207

Definition 1.3 p207 sum and scalar multiple

Theorem 1.5 p207
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercise 1.7, 1.8, 1.9 (What does this exercise remind you of?)

Do 1.10,-1.16 with a colleague
(ANSWERS: P109-110)
3.6.2 Matrix multiplication

Reading and examples: p209-214

Lemma 2.1 p209

Definition 2.3 p 210
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
Example 2.5 should ring bells as this is what you have been doing all along!

Theorem 2.6 p210 Matrix multiplicative product
Important Note:
Example 2.9, 2.10, and remark 2.11 p212 bring up a very important concept of
matrix multiplication and the commutative law.

Theorem 2.12 p213 Associativity of Matrix multiplication, distributivity of
matrix multiplication over matrix addition
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Important Note:
The theory of functions have been used to prove this theorem. You will notice
that in other texts the proofs are based more on the theory of matrices and
the operations on matrices. The various ways of proving these are equally
acceptable.
DO THIS
Exercise 2.14 -2.17, 2.23, 2.24, p214-216
Do 2.26, 2.30, 2.34, 2.36, with a colleague.
ANSWERS: P111-115
3.6.3 Mechanics of Matrix Multiplication

Reading and examples p 216-2.23
Reading and Activity from:
Linear Algebra by Jim
Hefferon
This is the section that takes you back to what you are used to and you should
easily sail through it!

Definition 3.2 p217 unit matrix

Lemma 3.7 p218

Definition 3.8 p219 The main diagonal, principle diagonal or diagonal

Definition 3.9 p219 the identity matrix

Definition 3.12 p219 the diagonal matrix

Definition 3.14 p220 the permutation matrix

Definition 3.18 p221 the elementary reduction matrix

Lemma 3.19 p221
This section should clearly remind you of the row operations we discussed when
we solved systems of linear equations

Corollary 3.22
DO THIS

Exercise 3.23, 3.24, 3.25, 3.26
(Note: This exercise introduces the concepts of incidence
and symmetric
matrices).
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
Do 3.38, 3.39, 3.40, 3.43 , 3.44, 3.46 with a colleague and note the
concepts of trace, upper triangular, Markov matrix.
(ANSWERS: P115-123)
Module Development Template
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3.6.4 Inverses
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p225-230

Definition 4.2 p226 left and right inverse matrix, invertible matrix,
inverse matrix

Lemma 4.3 p336

Theorem 4.4 p226

Lemma 4.5 p226

Lemma 4.8 p228

Corollary 4.12 p229 definition of the inverse of a 2x2 matrix
Does
Ring
bell?
this
a
DO THIS

Exercises 4.14 -4.21

Do 4.48, 4.33, 4.34 with a colleague.
3.6.5 Change of basis

Reading and examples p232 -234
Reading and Activity from:
Linear Algebra by Jim
Hefferon
3.6.6 Changing representation of vectors

Definition 1.1 p 232 Change of basis matrix

Lemma 1.2 and 1.4 p233

Corollary 1.5 p234
DO THIS

Exercises 1.6 – 1.18 p233
ANSWERS P123-127
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3.6.7 Changing map representations

Reading and examples p236 – 241

Definition 2.3 p238 Matrix equivalence

Corollary 2.4 p 238

Theorem 2.6 p 239

Corollary 2.8 p240
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercise 2.10- 2.19

Do 2.22 – 2.27 with a colleague
ANSWERS: P127-130
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3.7

Projection
Reading and exercises p244 -247
Reading and Activity from:
Linear Algebra by Jim
Hefferon
3.7.1 Orthogonal projection to a line

Definition 1.1 p245 orthogonal projection
DO THIS

Exercise p247-248, 1.7- 1.10.

Do exercise 1.18- 1.21 with a colleague
(ANSWERS: P130-133)
3.7.2 Gram-Schmidt Orthogonalization

Reading and examples p249- 254

Definition 2.1, p249 mutually orthogonal vectors

Theorem 2.2 p249

Corollary 2.3 p249

Definition 2.5 p250 orthogonal basis

Theorem 2.7 Gram- Schmidt orthogonalisation
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises p252 – 254, 2.9- 2.15

Do exercise 2.18 - 2.23 with a colleague.
(ANSWERS:P133-139)
3.7.3 Projection into a Subspace
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p254-260

Definition 3.1 p254

Definition 3.4 p257 Orthogonal complement, orthogonal projection

Lemma 3.7 p258

Theorem 3.8 p259
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DO THIS

Exercise 3.10-3.13. p260

Do 3.18, 3.22, 3.23, 3.25 with a colleague.
ANSWERS: P140-146
3.8
Synthesis
One of the fundamental principles that you learnt in this activity is that a
discussion of linear transformations is in essence a discussion of the theory of
matrices and that operations on matrices and linear transformations are in fact
synonymous.
The concepts of matrix addition and multiplication have been put forward as
theorems of linear transformations and justified and this clearly shows that matrix
multiplication is not a mechanical thing but a logical consequence of linear
transformations.
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Module 4: Linear Algebra
Unit 2: Applications of matrices
Learning Activities
Activity 1: Determinants
1.0
Objectives
By the end of this exercise you will be able to:

Define a determinant

Develop a formula to determine whether a square matrix is non singular
or not

Calculate the determinants of given square matrices

Describe the properties of determinants

Use Cramer’s rule to solve systems of linear equations
1.1
Glossary
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Nonsingular

Determinant

Permutation expansion

Multilinear map

N-permutation

Signum

Parallelepiped

Laplace’s expansion

Adjoint

Cramer’s rule
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1.2
Introduction
You have during your school days met, calculated and used the determinant in
solving systems of linear equations in two variables. This activity takes these
ideas further and explores the concept of a determinant not only as a property of
a non singular matrix but also as a function that translates the spaces of square
matrices, onto the space of real numbers. All you need to do in this exercise is to
keep your previous knowledge and keep trying to match the theory in this
exercise to your background knowledge.
Just as in Unit 1, you will need your copy of Linear Algebra by Jim Hefferon
throughout this unit.
1.3
Internet and Software Resources
Software
You should use wxMaxima to explore the determinants of matrices. Refer to the
getting started section in Activity 3 of Unit 1 in this module for further information.
Web References:
The Linear Algebra Toolkit (visited 07.11.06)
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=det
 Explore determinants with this device in the toolkit.
Wolfram MathWorld (visited 07.11.06)
http://mathworld.wolfram.com/Determinant.html
 Read this entry for Determinants.
 Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Determinant
 Read this entry for Determinants.
 Follow links to explain specific concepts as you need to.
MacTutor History of Mathematics (visited 07.11.06)
http://www-history.mcs.standrews.ac.uk/HistTopics/Matrices_and_determinants.html
 Explore the history of matrices and determinants
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1.4
The Determinant

Definition 1 p288 Determinant

Reading p288-290
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
Exploration 1.1p288-290. You need to read this section very carefully as it
gives you an intuitive treatment of the properties of determinants.
DO THIS.

Exercise 1.1, 1.3, 1.5, 1.6, 1.7, 1.8, 1.9.

Do 1.3- 1.18 with a colleague
ANSWERS: P163-165
1.4.1 Properties of determinants
Important Note:
You need to compare these properties with the properties of matrices under
row operations which you did during the exercise on systems of linear
equations and to recap your knowledge of Gaussian elimination

Reading and examples p293-295

Lemma 2.6 p295
Module Development Template
Reading and Activity from:
Linear Algebra by Jim
Hefferon
72
DO THIS

Exercise 2.7, 2.9, 2.10, 2.13. 295- 297

Do 2.16-2.22 with a colleague
ANSWERS: P165-168
1.4.2 The Permutation Expansion

Reading and examples p297- 298

Definition 3.2 p298 multilinear map

Lemma 3.3 p299

Definition 3.7 p301 n- permutation

Definition 3.9 p302 permutation expansion

Theorems 3.11 and 3.12 p303

Corollary 3.13 p303
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercise 3.16, 3.18, 3.20, 3.30, 3.33, 3.34
ANSWERS: P168-170
1.4.3 Determinants Exist

Reading p306-311

Definition 4.1 p307 inversion

Lemma 4.3 p307

Definition 4.4 p308 signum

Lemma 4.7 p309

Theorem 4.9 p311 T  T trans
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises 4.10 -4.17 p311-312 with a colleague
ANSWERS: P171-172
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1.5
Geometry of Determinants
1.5.1 Determinants as size functions

Reading and examples p313- 317

Definition 1.3 p315 box or parallelepiped

Theorem 1.5 p315

Corollary 1.7 T 1  1/ T
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises 1.8- 1.12

Do 1.15, 1.19, 1.24, 1.26, with a colleague
ANSWERS: P172-175
1.6
Other Formulas
1.6.1 Laplace’s Expansion
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and exercises p320-323

Definition 1.2 p321. Note the introduction of the concepts of minor and
cofactor

Theorem 1.5 p231 Laplace Expansion of determinants

Definition 1.8 p322 adjoint

Theorem 1.9 p 322

Corollary 1.11 p323
DO THIS

Exercises 1.13- 1.18

Do 1.23-1.28 with a colleague.
ANSWERS: P176-179
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1.6.2 Crammer’s Rule

Reading and examples p325-326
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Discuss the exercises with a colleague. p326-327
ANSWERS: P178-179
1.7
Synthesis
This activity extended your knowledge of matrices introduced you to the theory of
determinants a key concept of the theory of matrices. You also familiarized
yourself with an important application of determinants as you used them to solve
systems of linear equations
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Activity 2: Similarity
2.0
Objectives
By the end of this unit you should be able to:

State what is meant by a complex vector space

State and explain what is meant by similar matrices

Define a diagonizable transformation

Explain the relationship between a diagonizable matrix and a diagonal
matrix

State and prove the conditions that characterize a diagonizable matrix

Find the characteristic polynomial of an nxn matrix or a linear
transformation

Find by calculation a basis of the eigenspace corresponding to a given
eigenvalue of a matrix

Prove important properties of eigenvalues and eigenvectors

State and prove the Cayley –Hamilton theorem.
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2.1
Glossary
Use Wikipedia for comprehensive definitions of these terms. Go to:
http://en.wikipedia.org/wiki/Main_Page and type the term into the search box.

Complex vector spaces

Division theorem for polynomials

Similar matrices

Diagonizability

Diagonizable matrix

Eigenvalue

Eigenvector

Characteristic polynomial

Characteristic equation

Eigenspace

Nilpotence

Generalized rangespace

Generalized nullspace

Strings

T-string basis

Jordan form

Minimal polynomial

Cayley-hamilton theorem

Jordan canonical form
2.2
Introduction
In this activity we look at those mappings that transform a vector space into a
scalar multiple of itself. This is a consequence of many problems in mathematics
where it is important to determine those scalars  for which the equation
t   has non zero solutions for a given linear mapping t: V  V. The scalars
for which the equation holds are called eigenvalues and the corresponding
vectors  are called eigenvectors. In this activity we will explore this equation
and discuss some of its applications.
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Note: You will need your copy of Linear Algebra by Jim Hefferon throughout
activity 2.
Module Development Template
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2.3
Internet and Software Resources
Software
You should use wxMaxima to further explore properties of matrices. Refer to the
getting started section in Activity 3 of Unit 1 in this module for further information.
Notably you will want to use these functions:

Diagmatrix
4 0 0
Example: type A:diagmatrix(3,4) to produce a matrix A   0 4 0 
0 0 4



Eigenvalues
Example: eigenvalues(A) shows the eigenvalues of the matrix A

Eigenvectors
Example: eigenvectors(A) shows the eigenvectors of the matrix A
Web Reference:
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Diagonalizable_matrix
 Read this entry for diagonalizable matrices.
 Follow links to explain specific concepts as you need to.
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2.4
Complex Vector Spaces
2.4.1 Some Number Theory
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
For background reading on this section and some of the proofs for theorems
you need to revisit your work in the module on Number theory.

Reading and examples p343- 346

Theorem 1.1 Division theorem for polynomials p 344

Corollary 1.3, 1.4 p344

Theorem 1.5 p345 Irriducibility

Corollary 1.6, 1.10 p345 (Fundamental Theorem of Algebra)
2.4.2 Similarity

Reading and examples p347- 348

Definition 1.1 Similar matrices
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercise 1.4, 1.5, 1.6, 1.7, p349

Do 1.12, 1.17, 1.19, 1.21 with a colleague
(ANSWERS: P183-186)
2.4.3 Diagonalizability

Reading and examples p349-352

Definition 2.1 p350 Diagonalizable transformation, diagonalizable
matrix

Corollary 2.4., p350
DO THIS

Exercise p352-353: 2.6-2.12, 2.14- 2.18
(ANSWERS: P186-190)
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2.4.4 Eigenvalues and eigenvectors

Reading and examples p353-358

Definition 3.1 p353 eigenvalue and eigenvector
Reading and Activity from:
Linear Algebra by Jim
Hefferon
Important Note:
You need to familiarize yourself with remark 3.5 and its contents.

Definition 3.5 p354 eigenvalue and eigenvector of a square matrix

Definition 3.9 p356 characteristic polynomial, characteristic equation;

Definition 3.11 p 356 eigenspace

Lemma 3.12 p356

Theorem 3.17 p357

Lemma 3.19 p358
DO THIS

Exercises p358-360: 3.20-3.24, 3.28, 3.36, 3.40
(ANSWERS: P190-195)
2.4.5 Nilpotence
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p361-374

Self composition p361-363

Lemma 1.3 p362 Descending chain

Definition 1.7 p363 generalized range space and generalized nullspace
DO THIS

Exercises p364 1.8-1.11.

Do 1.12, 1.13, 1.15 with a colleague
(ANSWERS: P195-196)
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2.4.6 Strings
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p364-372

Lemma 2.1 p364

Definition 2.6 p366 nilpotent matrix, nilpotent transformation, index of
nilpotency.

Definition 2.10 p367 t-string, t- string basis

Theorem 2.13 p369

Corollary 2.14 p370
DO THIS

Exercises p372-374: 2.17-2.20

Do 2.28, 2.35 with a colleague
ANSWERS: P196-201
2.4.7 Jordan form
Reading and Activity from:
Linear Algebra by Jim
Hefferon

Reading and examples p375-392 Polynomials of maps and matrices

Reading and examples p375-380

Definition 1.3, 1.5 p376 minimal polynomial

Lemma1.7 p377

Theorem 1.8 p378 The Cayley Hamilton theorem
Important Note:
Lemma 1.9, 1.10, 1.11 p378-379 prove Cayley Hamilton’s theorem.
Other sources have stated the theorem in the following way:

let A be any nxn matrix with the characteristic polynomial
Pn ( )  (1) n  n  an 1 n 1  ...  a1  a0
then
Pn ( A)  (1) n An  an 1 An 1  ...  a1 A  a0 I  0
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DO THIS

Exercises p380-382 1.13-1.17, 1.22, 1.29, 1.32, 1.31
(ANSWERS: P201-207)
2.4.8 Jordan Canonical Form

Reading and examples p382- 392

Lemma 2.2 p382

Definition 2.6 p385 t-invarience

Lemma 2.7 p 385

Lemma 2.8, 2.9, p386

Lemma 2.11p387

Theorem 2.12 pg 388
Reading and Activity from:
Linear Algebra by Jim
Hefferon
DO THIS

Exercises 2.18-2.21, 2.24, 2.28, 2.34 P392-394
(ANSWERS: P207-214)
2.5
Synthesis
In this unit you learnt about finding the characteristic polynomial of a square
matrix or a linear transformation and how to use it to find the characteristic
eigenvalues and eigenvectors. You also learnt how to compute a basis for an
eigenspace corresponding to a given eigenvalue of a matrix. Finally you learnt
the conditions that characterize a diagonaizable matrix and the computation of a
diagonal matrix similar to a diagonalizable matrix.
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15. Synthesis of the Module
In Unit1 we introduced the basic concepts of linear equations and systtems of
linear equationsand progressed to the discusision of the various ways of solving
these. We also introduced you to ways of mathematizing problem situations- how
to look at real life situations and transform them into mathematical models. This
is a very imporatnt component of thinking in the teaching and learning of
mathematics.
We went on to discuss the concepts of matrices and vectors, leading to a critical
and fundamental theme of linear algebra- that of vector spaces, their pro[perties
and ather related concepts such as suspaces, basis and dimension. In the Unit,
you will have noticed that we endeavoured to relate the content to real life
situations. You will also have noticed that this was done quite sparingly in Unit2
as the content and the concepts therein became more theoretical and abstract.
The concepts of vactor spaces and linear maps were then extended to the
concepts of eigenvalues and eigenvectors and their related applications.
Finally, we tried to make you recognize that mathematical knowledge is
organised and centralized around definitions, lemmas, propositions, theorems,
example, non-examples, exercises and so on. I hope you were able to test your
understanding of these by carefully and diligently going through the numerous
worked examples in your compulsory readings and doing more than the
exercises stipulated in the module.
Real mathematical understanding is brought about by doing exercises, both
individually and collectively with colleagues and other mathematically informed
parties. We hope that you got into habit of working through as many exercises as
possible!
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16. Summative Evaluation
Module 4: Linear Algebra
Assessments and Solutions
Summative Test Questions
Time 4 hours
Answer all questions
1
Solve the following system of equations using the Gauss-Jordan Method.
x1  3x2  2 x3  2 x5  0
2 x1  6 x2  5 x3  2 x4  4 x5  3x6  1
5 x3  10 x4  15 x6  5
2 x1  6 x2  8 x4  4 x5  18 x6  6
2 (a) Define a vector space U stating all the axioms.
(b)
Prove that the set
 x 

 

 y

L     4 x  y  z  w  0 


 z

 w 

is a vector space under the operations inherited from
(c)
4
.
Prove that the set
 x 

 

3
L   y  
x  y  z  1
 z 

 

is not a vector space under the operations inherited from
3
.
3(a) Define a subspace W  V .
(b)
Let A be a fixed m n matrix with real entries. Let
N  x x  n and Ax  0 .

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
88
Prove that N is a subspace of
4
n
.
Let
0 1 1 
Q  1 0 1 
1 1 0 
by using row operations show that the inverse, Q 1 , is given by
 1 1 1 
1
Q   1 1 1 
2
 1 1 1
1
5(a)
(b)
Define the two properties of a linear transformation T : U V .
Let M mn denote the space of m n matrices with real entries and M nm the
space of n  m matrices with real entries. Consider T : M mn  M nm defined
by T ( A)  AT , where a is an m n matrix. Prove that T is a Linear
Transformation.
6
Let U and V be vector spaces, T : U V a linear transformation. Prove
each of the following:
T (0)  0
(i)
T ( x   y )   T ( x)   T ( y )
(ii)
(iii)
 n
 n
T    i xi     iT ( xi )
 i 1
 i 1
3
7
Verify that the following is a basis for
 1   3   0  
      
 2  ,  2  ,  0  
 3   1   1  
      
8
Find an orthonormal basis for the subspace of
 1
 0
 0
 
 
 
x1  1 , x2   1  , x3   0 
 1
1
1
 
 
 
Hint: Use Gram-Schmidt process.
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spanned by x1, x2, x3 if
89
9
 3 2 0
Let A   2 3 0 


 0 0 5 
Find the eigenvalues and corresponding eigenvectors of the matrix A.
END OF EXAMINATION PAPER
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Summative Test Solutions
1
The augmented matrix for the system is
 1 3 2 0 2 0 0 


 2 6 5 2 4 3 1 r  2r  r , r  2r  r
1
2
4
1
4
 0 0 5 10 0 15 5  2


 2 6 0 8 4 18 6 
1

0
0

0
3 2
0

0 1 2 0 3 1
r2   r2 , r3  r3  r2 , r4  r4  4r2
0 5 10 0 15 5 

0 4 8 0 18 6 
1

0
0

0
3 2 0 2 0 0 

0 1 2 0 3 1
r3  r4 , r3  16 r3
0 0 0 0 0 0

0 0 0 0 6 2 
1

0
0

0
3 2 0 2 0 0 

0 1 2 0 3 1
r2  r2  3r3
0 0 0 0 1 13 

0 0 0 0 0 0 
1

0
0

 0
3 0 4 2 0 0

0 1 2 0 0 0
0 0 0 0 1 13 

0 0 0 0 0 0 
0
2
0
Thus x6  13 , x3  2 x4
x1  3x2  4 x4  2 x5  0
Let x4  s , x3  2s , x5  t , x2  r , x1  3r  4s  2t .
So we have infinitely many solutions.
2
A vector space (over ) consists of a set U along with two operations “+”
and “  ” subject to the following axioms:
(A1) Given v , w U v  w  U
(A2) v  w  w  v
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( v  w)  u  v  ( w  u ) for v , w, u U
0  U : v  0  v ,  v  U
 v U w U such that v  w  0 (additive inverse)
For each scalar s 
sv  U ,  v  U
(A7) For the scalars s, t 
(s  t )  v  s  v  t  v
(A8) t  ( v  w)  t  v  t  w, t 
(A9) (ts )  v  t  ( s  v ), s, t 
(A10) 1  v  v
(A3)
(A4)
(A5)
(A6)
(b)
Note that
 x1   x2   x1  x2 
    

 y1    y2    y1  y2  is in L because
 z1   z2   z1  z2 
    

 w1   w2   w1  w2 
( x1  x2 )  ( y1  y2 )  ( z1  z2 )  ( w1  w2 )
 ( x1  y1  z1  w1 )  ( x2  y2  z2  w2 )
=0+0
=0
Hence L is a vector space.
Note that
 x1   x2   x1  x2 
    

 y1    y2    y1  y2 
z  z  z z 
 1  2  1 2 
Therefore
( x1  x2 )  ( y1  y2 )  ( z1  z2 )
= ( x1  y1  z1 )  ( x2  y2  z2 )
=1+1
=2
Hence the set is not a vector space.
(c)
3(a) For any vector space V, a subspace W is a subset that is itself a vector
space, under the inherited operations.
(b)
First, suppose that x1  N and x2  N , then
A( x1  x2 )  Ax1  Ax2
=0+0
=0
Thus x1  x2  N . Next if x  N and  is a scalar
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A( x)   A( x)
=  0
=0
Thus  x  N .
Therefore we see that N is a subspace of
4
n
.
There augmented matrix is
0 1 1 1 0 0 


1 0 1 0 1 0  r1  r3
1 1 0 0 0 1 
1 1 0 0 0 1 


1 0 1 0 1 0  r2  r2  r1
0 1 1 1 0 0 
1 1 0 0 0 1 


0 1 1 0 1 1 r3  r3  r2
0 1 1 1 0 0 
1 1

0 1
0 0
1 1

0 1
0 0

0 0 0
1 0 1
2 1 1
0 0 0
1 0 1
1
1
2
1
2
1

1 r3  12 r3
1
1

1  r2  r2  r3
 12 
1 1 0 0 0 1 

1
1
1
0 1 0  2 2  2  r1  r1  r2 ; r2   r2
0 0 1 12 12  12 


1
1 0 0  12 12

2

1
1
1 
0 1 0 2  2 2 
1
0 0 1 12
 12 
2

1
  12 12
2 

1
1
1
1 
Thus Q   2  2 2 
1
 12
 12 
2
5(a)
Let V and W be vector spaces and T be a function T : V  W . For T to be
linear:
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(i)
(ii)
(b)
6
T ( x  y )  T ( x )  T ( y )  x, y  V
T ( x)   T ( x) x V and  
Since the transpose of an m n matrix is an n  m matrix, T is a welldefined function from M mn to M nm . If A and B are m n matrices and  is
a scalar.
T ( A  B)  ( A  B)T by definition of T
 AT  BT by distribution law
 T ( A)  T ( B) by definition of T
and
T ( A)  ( A)T by definition of T
  AT
  T ( A) by definition of T
(i)
T (0)  T (0  x)  0  T ( x)  0
(ii)
T ( x   y )  T ( x)  T (  y )
  T ( x)   T ( y )
Proof by mathematical induction
If n=1 then
T (1 x1 )  1T ( x1 ) true
If n  1
 n

 n1

T    i xi   T    i xi   n xn 
 i 1

 i 1

n 1


 T    i xi   T  n xn 
 i 1

By induction hypothesis
 n 1
 n 1
T    i xi     iT ( xi )
 i 1
 i 1
(iii)
n 1
Thus we have
  T ( x )  T  x 
i 1
i
i
n n
n 1
   iT ( xi )   n T  xn 
i 1
n
   iT ( xi )
i 1
7
1
 3
0




Let x1   2  ; x2   2  ; x3   0 
 3
1
1
 
 
 
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To show that this set of vectors spans 3 , we must show that there exist
scalars 1 ,  2 ,  3 such that x  1 x1   2 x2  3 x3 .
3 0 1 
2 0  2 
1 1   3 
3 0
1 3
2 0
 2  6  4
2 2
3 1 1
We see that the system of equations is solvable.
Hence the system of equations spans 3 .
To prove linear independence we have 1 x1   2 x2  3 x3  0
 x  1
 y   2
  
 z   3
1
Now 2
1 3 0  x1  0
 2 2 0   x   0

 2  
 3 1 1   x3  0
Solving this matrix we have the following augmented matrix
1 3 0 0 


 2 2 0 0  r1  2r1
 3 1 1 0 
2 6 0 0 


1
 2 2 0 0  r1  r1  r2 ; r2  2 r2 ; r3  r3  3r2
 3 1 1 0 
0 4 0 0 


1 1 0 0  r3  2r3  r1
0 2 1 0 
0 4 0 0 


1
1
1 1 0 0  r1   4 r1 ; r3  2 r3
0 0 2 0 
0 1 0 0 


1 1 0 0  r2  r2  r1
0 0 1 0 
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0 1 0 0 


1 0 0 0 
0 0 1 0 
Thus x1  0, x2  0, x3  0
Hence linear independence and so the set of vectors is a basis for
3
.
 1
 0
 0




8
Let x1  1 ; x2   1  ; x3   0 
 1
1
1
 
 
 
Using Gram-Schmidt
v1 
x1
(1,1,1)  1 1 1 


,
,

x1
3
 3 3 3
x2  x2 , v1 v1  (0,1,1) 
v2 
x2  x2 , v1 v1
x2  x2 , v1 v1

2  1 1 1   2 1 1
,
,

   , , 
3  3 3 3   3 3 3
3  2 1 1  2 1 1 
,
,
 , ,   

6  3 3 3 
6 6 6
x3  x3 , v1 v1  x3 , v2 v2  (0, 0,1) 
1  1 1 1  1  2 1 1 
,
,
,
,




3 3 3 3
6
6 6 6
  0,  12 , 12 
v3 
x3  x3 , v1 v1  x3 , v2 v2
x3  x3 , v1 v1  x3 , v2 v2

 0,  12 , 12 
 0,  12 , 12 
 2  0,  12 , 12 
 1 1 1 
 2 1 1 
Thus v1  
,
,
,
,
 ; v2   
 ; v3  (0, 
6 6 6
 3 3 3

orthonormal basis for
9
3
1
2
,
1
2
) form an
.
The characteristic equation of A is   I  A   0
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  0 0   3 2 0 

 

 0  0    2 3 0   0
 0 0    0 0 5

 

 3
2
0
2
 3
0
0
0
 5
 (  3)
0
 3
0
0
 5
2
2
0
0  5
 (  3)(  3)(  5)  4(  5)
 (  5)  2  6  9  4 
 (  5)  2  6  5
 (  5)2 (  1)  0
Thus   5 twice or   1
When   5
2
0   x1 
  3
 2
 3
0   x2   0

 0
0
  5  x3 
2 2 0 0


 2 2 0 0  r2  r2  r1
 0 0 0 0 
2 2 0 0


0 0 0 0
 0 0 0 0 
Let x2  s, x1   s, x3  t
s 
 1 0


thus x  s  s  1   t 0
 
   
 t 
 0  1 
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 1
0 


the eigenvectors are  1  and  0  .
 1 
 0 
Now when   1
2 2 0 0 


1
 2 2 0 0  r2  r2  r1 ; r3   4 r3
 0 0 4 0 
2 2

0 0
 0 0
Setting
0 0

0 0
1 0 
x1  s, x2  s, x3  0 . The eigenvector corresponding to   1 is
1 
x  1  .
 0 
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17. References
Jim Hefferon, Linear Algebra, Saint Michael’s College, Colchester, Vermont USA
05439, 2006. Web reference: http://joshua.smcvt.edu
18. Main Author of the Module
Module Developer Writing Tip. Module Developers should provide a brief biography (50-75 words), a picture, title and
contact information (email).
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