CONTENTS
Preface vii
Foreword to the Instructor xiii
Foreword to the Student xvii
Chapter 1 Vectors and Matrices 1
1. Vectors 1
2. Dot Product 18
3. Hyperplanes in Rn 28
4. Systems of Linear Equations and Gaussian Elimination 36
5. The Theory of Linear Systems 53
6. Some Applications 64
Chapter 2 Matrix Algebra 81
1. Matrix Operations 81
2. Linear Transformations: An Introduction 91
3. Inverse Matrices 102
4. Elementary Matrices: Rows Get Equal Time 110
5. The Transpose 119
Chapter 3 Vector Spaces 127
1. Subspaces of Rn 127
2. The Four Fundamental Subspaces 136
3. Linear Independence and Basis 143
4. Dimension and Its Consequences 157
5. A Graphic Example 170
6. Abstract Vector Spaces 176
Chapter 4 Projections and Linear Transformations 191
1. Inconsistent Systems and Projection 191
2. Orthogonal Bases 200
3. The Matrix of a Linear Transformation and the
Change-of-Basis Formula 208
4. Linear Transformations on Abstract Vector Spaces 224
Chapter 5 Determinants 239
1. Properties of Determinants 239
2. Cofactors and Cramer’s Rule 245
3. Signed Area in R2 and Signed Volume in R3 255
Chapter 6 Eigenvalues and Eigenvectors 261
1. The Characteristic Polynomial 261
2. Diagonalizability 270
3. Applications 277
4. The Spectral Theorem 286
Chapter 7 Further Topics 299
1. Complex Eigenvalues and Jordan Canonical Form 299
2. Computer Graphics and Geometry 314
3. Matrix Exponentials and Differential Equations 331
For Further Reading 349
Answers to Selected Exercises 351
List of Blue Boxes 367
Index 369
PREFACE
One of the most enticing aspects of mathematics, we have found, is the interplay of
ideas from seemingly disparate disciplines of the subject. Linear algebra provides
a beautiful illustration of this, in that it is by nature both algebraic and geometric.
Our intuition concerning lines and planes in space acquires an algebraic interpretation that
then makes sense more generally in higher dimensions. What’s more, in our discussion of
the vector space concept, we will see that questions from analysis and differential equations
can be approached through linear algebra. Indeed, it is fair to say that linear algebra lies
at the foundation of modern mathematics, physics, statistics, and many other disciplines.
Linear problems appear in geometry, analysis, and many applied areas. It is this multifaceted
aspect of linear algebra that we hope both the instructor and the students will find appealing
as they work through this book.
From a pedagogical point of view, linear algebra is an ideal subject for students to learn
to think about mathematical concepts and to write rigorous mathematical arguments. One
of our goals in writing this text—aside from presenting the standard computational aspects
and some interesting applications—is to guide the student in this endeavor. We hope this
book will be a thought-provoking introduction to the subject and its myriad applications,
one that will be interesting to the science or engineering student but will also help the
mathematics student make the transition to more abstract advanced courses.
We have tried to keep the prerequisites for this book to a minimum. Although many
of our students will have had a course in multivariable calculus, we do not presuppose any
exposure to vectors or vector algebra. We assume only a passing acquaintance with the
derivative and integral in Section 6 of Chapter 3 and Section 4 of Chapter 4. Of course,
in the discussion of differential equations in Section 3 of Chapter 7, we expect a bit more,
including some familiarity with power series, in order for students to understand the matrix
exponential.
In the second edition, we have added approximately 20% more examples (a number of
which are sample proofs) and exercises—most computational, so that there are now over
210 examples and 545 exercises (many with multiple parts). We have also added solutions
to many more exercises at the back of the book, hoping that this will help some of the
students; in the case of exercises requiring proofs, these will provide additional worked
examples that many students have requested. We continue to believe that good exercises
are ultimately what makes a superior mathematics text.
In brief, here are some of the distinctive features of our approach:
• We introduce geometry from the start, using vector algebra to do a bit of analytic
geometry in the first section and the dot product in the second.
• We emphasize concepts and understanding why, doing proofs in the text and asking
the student to do plenty in the exercises. To help the student adjust to a higher level
of mathematical rigor, throughout the early portion of the text we provide “blue
boxes” discussing matters of logic and proof technique or advice on formulating
problem-solving strategies. Acomplete list of the blue boxes is included at the end
of the book for the instructor’s and the students’ reference.
• We use rotations, reflections, and projections in R2 as a first brush with the notion of
a linear transformation when we introduce matrix multiplication; we then treat linear
transformations generally in concert with the discussion of projections. Thus, we
motivate the change-of-basis formula by starting with a coordinate system in which
a geometrically defined linear transformation is clearly understood and asking for
its standard matrix.
• We emphasize orthogonal complements and their role in finding a homogeneous
system of linear equations that defines a given subspace of Rn.
• In the last chapter we include topics for the advanced student, such as Jordan
canonical form, a classification of the motions of R2 and R3, and a discussion of
how Mathematica draws two-dimensional images of three-dimensional shapes.
The historical notes at the end of each chapter, prepared with the generous assistance of
Paul Lorczak for the first edition, have been left as is. We hope that they give readers an
idea how the subject developed and who the key players were.
A few words on miscellaneous symbols that appear in the text: We have marked with
an asterisk (∗) the problems for which there are answers or solutions at the back of the text.
As a guide for the new teacher, we have also marked with a sharp () those “theoretical”
exercises that are important and to which reference is made later. We indicate the end of a
proof by the symbol .
