A first course in linear algebra by k kuttler

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A First Course in

LINEAR ALGEBRA

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CONTRIBUTIONS

Ilijas Farah, York University

Ken Kuttler, Brigham Young University

Lyryx Learning Team

Bruce Bauslaugh

Peter Chow

Nathan Friess

Stephanie Keyowski

Claude Laflamme

Martha Laflamme

Jennifer MacKenzie

Tamsyn Murnaghan

Bogdan Sava

Larissa Stone

Ryan Yee

Ehsun Zahedi

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A First Course in Linear Algebra an Open Text

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Some Important Proofs section

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Matrix Exponential based on original material by K. Kuttler.

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Contents

Contents iii

Preface 1

1 Systems of Equations 3

1.1 Systems of Equations, Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Systems Of Equations, Algebraic Procedures . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Uniqueness of the Reduced Row-Echelon Form . . . . . . . . . . . . . . . . . . 25

1.2.4 Rank and Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2.5 Balancing Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.2.6 Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.2.7 An Application to Resistor Networks . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Matrices 53

2.1 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.1 Addition of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.2 Scalar Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.1.3 Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1.4 The i jth Entry of a Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.1.5 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.1.6 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.1.7 The Identity and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.1.8 Finding the Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.9 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.1.10 More on Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.2 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2.1 Finding An LU Factorization By Inspection . . . . . . . . . . . . . . . . . . . . . 99

2.2.2 LU Factorization, Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2.3 Solving Systems using LU Factorization . . . . . . . . . . . . . . . . . . . . . . . 101

2.2.4 Justification for the Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 102

iii

iv CONTENTS

3 Determinants 107

3.1 Basic Techniques and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.1.1 Cofactors and 2×2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.1.2 The Determinant of a Triangular Matrix . . . . . . . . . . . . . . . . . . . . . . . 112

3.1.3 Properties of Determinants I: Examples . . . . . . . . . . . . . . . . . . . . . . . 114

3.1.4 Properties of Determinants II: Some Important Proofs . . . . . . . . . . . . . . . 118

3.1.5 Finding Determinants using Row Operations . . . . . . . . . . . . . . . . . . . . 123

3.2 Applications of the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.1 A Formula for the Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.2.3 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4 Rn 145

4.1 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2 Algebra in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2.1 Addition of Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.2.2 Scalar Multiplication of Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . 150

4.3 Geometric Meaning of Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.4 Length of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.5 Geometric Meaning of Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.6 Parametric Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.7 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.7.1 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.7.2 The Geometric Significance of the Dot Product . . . . . . . . . . . . . . . . . . . 170

4.7.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.8 Planes in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.9 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.9.1 The Box Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.10 Spanning, Linear Independence and Basis in Rn . . . . . . . . . . . . . . . . . . . . . . . 192

4.10.1 Spanning Set of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.10.2 Linearly Independent Set of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 194

4.10.3 A Short Application to Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.10.4 Subspaces and Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.10.5 Row Space, Column Space, and Null Space of a Matrix . . . . . . . . . . . . . . . 211

4.11 Orthogonality and the Gram Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . 232

4.11.1 Orthogonal and Orthonormal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 233

4.11.2 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

CONTENTS v

4.11.3 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

4.11.4 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

4.11.5 Least Squares Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4.12 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

4.12.1 Vectors and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

4.12.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

5 Linear Transformations 269

5.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

5.2 The Matrix of a Linear Transformation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

5.3 Properties of Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

5.4 Special Linear Transformations in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

5.5 One to One and Onto Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

5.6 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.7 The Kernel And Image Of A Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

5.8 The Matrix of a Linear Transformation II . . . . . . . . . . . . . . . . . . . . . . . . . . 315

5.9 The General Solution of a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

6 Complex Numbers 329

6.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

6.2 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

6.3 Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

6.4 The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

7 Spectral Theory 347

7.1 Eigenvalues and Eigenvectors of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 347

7.1.1 Definition of Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . 347

7.1.2 Finding Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 350

7.1.3 Eigenvalues and Eigenvectors for Special Types of Matrices . . . . . . . . . . . . 356

7.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

7.2.1 Similarity and Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

7.2.2 Diagonalizing a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

7.2.3 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

7.3 Applications of Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

7.3.1 Raising a Matrix to a High Power . . . . . . . . . . . . . . . . . . . . . . . . . . 373

7.3.2 Raising a Symmetric Matrix to a High Power . . . . . . . . . . . . . . . . . . . . 375

7.3.3 Markov Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

7.3.3.1 Eigenvalues of Markov Matrices . . . . . . . . . . . . . . . . . . . . . 384

vi CONTENTS

7.3.4 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

7.3.5 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

7.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

7.4.1 Orthogonal Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

7.4.2 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 409

7.4.3 Positive Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

7.4.3.1 The Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . . . 420

7.4.4 QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

7.4.4.1 The QR Factorization and Eigenvalues . . . . . . . . . . . . . . . . . . 424

7.4.4.2 Power Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

7.4.5 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

8 Some Curvilinear Coordinate Systems 439

8.1 Polar Coordinates and Polar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.2 Spherical and Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

9 Vector Spaces 455

9.1 Algebraic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

9.2 Spanning Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

9.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

9.4 Subspaces and Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

9.5 Sums and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

9.6 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

9.7 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

9.7.1 One to One and Onto Transformations . . . . . . . . . . . . . . . . . . . . . . . . 505

9.7.2 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

9.8 The Kernel And Image Of A Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

9.9 The Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

A Some Prerequisite Topics 537

A.1 Sets and Set Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

A.2 Well Ordering and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

B Selected Exercise Answers 543

Index 591

Preface

A First Course in Linear Algebra presents an introduction to the fascinating subject of linear algebra for

students who have a reasonable understanding of basic algebra. Major topics of linear algebra are pre-

sented in detail, with proofs of important theorems provided. Separate sections may be included in which

proofs are examined in further depth and in general these can be excluded without loss of contrinuity.

Where possible, applications of key concepts are explored. In an effort to assist those students who are

interested in continuing on in linear algebra connections to additional topics covered in advanced courses

are introduced.

Each chapter begins with a list of desired outcomes which a student should be able to achieve upon

completing the chapter. Throughout the text, examples and diagrams are given to reinforce ideas and

provide guidance on how to approach various problems. Students are encouraged to work through the

suggested exercises provided at the end of each section. Selected solutions to these exercises are given at

the end of the text.

As this is an open text, you are encouraged to interact with the textbook through annotating, revising,

and reusing to your advantage.

1

1. Systems of Equations

1.1 Systems of Equations, Geometry

Outcomes

A. Relate the types of solution sets of a system of two (three) variables to the intersections of

lines in a plane (the intersection of planes in three space)

As you may remember, linear equations like 2x+3y = 6 can be graphed as straight lines in the coordi- nate plane. We say that this equation is in two variables, in this case x and y. Suppose you have two such

equations, each of which can be graphed as a straight line, and consider the resulting graph of two lines.

What would it mean if there exists a point of intersection between the two lines? This point, which lies on

both graphs, gives x and y values for which both equations are true. In other words, this point gives the

ordered pair (x,y) that satisfy both equations. If the point (x,y) is a point of intersection, we say that (x,y) is a solution to the two equations. In linear algebra, we often are concerned with finding the solution(s)

to a system of equations, if such solutions exist. First, we consider graphical representations of solutions

and later we will consider the algebraic methods for finding solutions.

When looking for the intersection of two lines in a graph, several situations may arise. The follow-

ing picture demonstrates the possible situations when considering two equations (two lines in the graph)

involving two variables.

x

y

One Solution

x

y

No Solutions

x

y

Infinitely Many Solutions

In the first diagram, there is a unique point of intersection, which means that there is only one (unique)

solution to the two equations. In the second, there are no points of intersection and no solution. When no

solution exists, this means that the two lines are parallel and they never intersect. The third situation which

can occur, as demonstrated in diagram three, is that the two lines are really the same line. For example,

x+ y = 1 and 2x+ 2y = 2 are equations which when graphed yield the same line. In this case there are infinitely many points which are solutions of these two equations, as every ordered pair which is on the

graph of the line satisfies both equations. When considering linear systems of equations, there are always

three types of solutions possible; exactly one (unique) solution, infinitely many solutions, or no solution.

3

4 Systems of Equations

Example 1.1: A Graphical Solution

Use a graph to find the solution to the following system of equations

x+ y = 3 y− x = 5

Solution. Through graphing the above equations and identifying the point of intersection, we can find the

solution(s). Remember that we must have either one solution, infinitely many, or no solutions at all. The

following graph shows the two equations, as well as the intersection. Remember, the point of intersection

represents the solution of the two equations, or the (x,y) which satisfy both equations. In this case, there is one point of intersection at (−1,4) which means we have one unique solution, x =−1,y = 4.

−4 −3 −2 −1 1

2

4

6

x

y

(x,y) = (−1,4)

♠

In the above example, we investigated the intersection point of two equations in two variables, x and

y. Now we will consider the graphical solutions of three equations in two variables.

Consider a system of three equations in two variables. Again, these equations can be graphed as

straight lines in the plane, so that the resulting graph contains three straight lines. Recall the three possible

types of solutions; no solution, one solution, and infinitely many solutions. There are now more complex

ways of achieving these situations, due to the presence of the third line. For example, you can imagine

the case of three intersecting lines having no common point of intersection. Perhaps you can also imagine

three intersecting lines which do intersect at a single point. These two situations are illustrated below.

x

y

No Solution

x

y

One Solution

1.1. Systems of Equations, Geometry 5

Consider the first picture above. While all three lines intersect with one another, there is no common

point of intersection where all three lines meet at one point. Hence, there is no solution to the three

equations. Remember, a solution is a point (x,y) which satisfies all three equations. In the case of the second picture, the lines intersect at a common point. This means that there is one solution to the three

equations whose graphs are the given lines. You should take a moment now to draw the graph of a system

which results in three parallel lines. Next, try the graph of three identical lines. Which type of solution is

represented in each of these graphs?

We have now considered the graphical solutions of systems of two equations in two variables, as well

as three equations in two variables. However, there is no reason to limit our investigation to equations in

two variables. We will now consider equations in three variables.

You may recall that equations in three variables, such as 2x+ 4y− 5z = 8, form a plane. Above, we were looking for intersections of lines in order to identify any possible solutions. When graphically solving

systems of equations in three variables, we look for intersections of planes. These points of intersection

give the (x,y,z) that satisfy all the equations in the system. What types of solutions are possible when working with three variables? Consider the following picture involving two planes, which are given by

two equations in three variables.

Notice how these two planes intersect in a line. This means that the points (x,y,z) on this line satisfy both equations in the system. Since the line contains infinitely many points, this system has infinitely

many solutions.

It could also happen that the two planes fail to intersect. However, is it possible to have two planes

intersect at a single point? Take a moment to attempt drawing this situation, and convince yourself that it

is not possible! This means that when we have only two equations in three variables, there is no way to

have a unique solution! Hence, the types of solutions possible for two equations in three variables are no

solution or infinitely many solutions.

Now imagine adding a third plane. In other words, consider three equations in three variables. What

types of solutions are now possible? Consider the following diagram.

✠ New Plane

In this diagram, there is no point which lies in all three planes. There is no intersection between all

6 Systems of Equations

planes so there is no solution. The picture illustrates the situation in which the line of intersection of the

new plane with one of the original planes forms a line parallel to the line of intersection of the first two

planes. However, in three dimensions, it is possible for two lines to fail to intersect even though they are

not parallel. Such lines are called skew lines.

Recall that when working with two equations in three variables, it was not possible to have a unique

solution. Is it possible when considering three equations in three variables? In fact, it is possible, and we

demonstrate this situation in the following picture.

✠

New Plane

In this case, the three planes have a single point of intersection. Can you think of other types of

solutions possible? Another is that the three planes could intersect in a line, resulting in infinitely many

solutions, as in the following diagram.

We have now seen how three equations in three variables can have no solution, a unique solution, or

intersect in a line resulting in infinitely many solutions. It is also possible that the three equations graph

the same plane, which also leads to infinitely many solutions.

You can see that when working with equations in three variables, there are many more ways to achieve

the different types of solutions than when working with two variables. It may prove enlightening to spend

time imagining (and drawing) many possible scenarios, and you should take some time to try a few.

You should also take some time to imagine (and draw) graphs of systems in more than three variables.

Equations like x+y−2z+4w = 8 with more than three variables are often called hyper-planes. You may soon realize that it is tricky to draw the graphs of hyper-planes! Through the tools of linear algebra, we

can algebraically examine these types of systems which are difficult to graph. In the following section, we

will consider these algebraic tools.

1.2. Systems Of Equations, Algebraic Procedures 7

Exercises

Exercise 1.1.1 Graphically, find the point (x1,y1) which lies on both lines, x+ 3y = 1 and 4x− y = 3. That is, graph each line and see where they intersect.

Exercise 1.1.2 Graphically, find the point of intersection of the two lines 3x+ y = 3 and x+2y = 1. That is, graph each line and see where they intersect.

Exercise 1.1.3 You have a system of k equations in two variables, k ≥ 2. Explain the geometric signifi- cance of

(a) No solution.

(b) A unique solution.

(c) An infinite number of solutions.

1.2 Systems Of Equations, Algebraic Procedures

Outcomes

A. Use elementary operations to find the solution to a linear system of equations.

B. Find the row-echelon form and reduced row-echelon form of a matrix.

C. Determine whether a system of linear equations has no solution, a unique solution or an

infinite number of solutions from its row-echelon form.