Linear Algebra / Edition 3

by Stephen H. Friedberg, Lawrence E. Spence, Arnold J. Insel

Overview

For courses in Advanced Linear Algebra.

Illustrates the power of linear algebra through practical applications

This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry, and physics appear throughout, and can be included at the instructor’s discretion.

0134860241 / 9780134860244 Linear Algebra, 5/e

Product Details

ISBN-13:	9780132338592
Publisher:	Prentice Hall Professional Technical Reference
Publication date:	08/01/1996
Edition description:	Older Edition
Pages:	557
Product dimensions:	6.24(w) x 9.32(h) x 1.05(d)

About the Author

Stephen H. Friedberg holds a BA in mathematics from Boston University and MS and PhD degrees in mathematics from Northwestern University, and was awarded a Moore Postdoctoral Instructorship at MIT. He served as a director for CUPM, the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics. He was a faculty member at Illinois State University for 32 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1990. He has also taught at the University of London, the University of Missouri, and at Illinois Wesleyan University. He has authored or coauthored articles and books in analysis and linear algebra.

Arnold J. Insel received BA and MA degrees in mathematics from the University of Florida and a PhD from the University of California at Berkeley. He served as a faculty member at Illinois State University for 31 years and at Illinois Wesleyan University for two years. In addition to authoring and co-authoring articles and books in linear algebra, he has written articles in lattice theory, topology, and topological groups.

Lawrence E. Spence holds a BA from Towson State College and MS and PhD degrees in mathematics from Michigan State University. He served as a faculty member at Illinois State University for 34 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1987. He is an author or co-author of nine college mathematics textbooks, as well as articles in mathematics journals in the areas of discrete mathematics and linear algebra.

Table of Contents

* Sections denoted by an asterisk are optional.

1. Vector Spaces

1.1 Introduction

1.2 Vector Spaces

1.3 Subspaces

1.4 Linear Combinations and Systems of Linear Equations

1.5 Linear Dependence and Linear Independence

1.6 Bases and Dimension

1.7* Maximal Linearly Independent Subsets

Index of Definitions

2. Linear Transformations and Matrices

2.1 Linear Transformations, Null Spaces, and Ranges

2.2 The Matrix Representation of a Linear Transformation

2.3 Composition of Linear Transformations and Matrix Multiplication

2.4 Invertibility and Isomorphisms

2.5 The Change of Coordinate Matrix

2.6* Dual Spaces

2.7* Homogeneous Linear Differential Equations with Constant Coefficients

Index of Definitions

3. Elementary Matrix Operations and Systems of Linear Equations

3.1 Elementary Matrix Operations and Elementary Matrices

3.2 The Rank of a Matrix and Matrix Inverses

3.3 Systems of Linear Equations – Theoretical Aspects

3.4 Systems of Linear Equations – Computational Aspects

Index of Definitions

4. Determinants

4.1 Determinants of Order 2

4.2 Determinants of Order n

4.3 Properties of Determinants

4.4 Summary|Important Facts about Determinants

4.5* A Characterization of the Determinant

Index of Definitions

5. Diagonalization

5.1 Eigenvalues and Eigenvectors

5.2 Diagonalizability

5.3* Matrix Limits and Markov Chains

5.4 Invariant Subspaces and the Cayley–Hamilton Theorem

Index of Definitions

6. Inner Product Spaces

6.1 Inner Products and Norms

6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements

6.3 The Adjoint of a Linear Operator

6.4 Normal and Self-Adjoint Operators

6.5 Unitary and Orthogonal Operators and Their Matrices

6.6 Orthogonal Projections and the Spectral Theorem

6.7* The Singular Value Decomposition and the Pseudoinverse

6.8* Bilinear and Quadratic Forms

6.9* Einstein's Special Theory of Relativity

6.10* Conditioning and the Rayleigh Quotient

6.11* The Geometry of Orthogonal Operators

Index of Definitions

7. Canonical Forms

7.1 The Jordan Canonical Form I

7.2 The Jordan Canonical Form II

7.3 The Minimal Polynomial

7.4* The Rational Canonical Form

Index of Definitions

Appendices

A. Sets

B. Functions

C. Fields

D. Complex Numbers

E. Polynomials

Answers to Selected Exercises

Index

Preface

The language and concepts of matrix theory and, more generally, of linear algebra have come into widespread usage in the social and natural sciences, computer science, and statistics. In addition, linear algebra continues to be of great importance in modern treatments of geometry and analysis.

The primary purpose of this fourth edition of Linear Algebra is to present a careful treatment of the principal topics of linear algebra and to illustrate the power of the subject through a variety of applications. Our major thrust emphasizes the symbiotic relationship between linear transformations and matrices. However, where appropriate, theorems are stated in the more general infinite-dimensional case. For example, this theory is applied to finding solutions to a homogeneous linear differential equation and the best approximation by a trigonometric polynomial to a continuous function.

Although the only formal prerequisite for this book is a one-year course in calculus, it requires the mathematical sophistication of typical junior and senior mathematics majors. This book is especially suited for a second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis.

The book is organized to permit a number of different courses (ranging from three to eight semester hours in length) to be taught from it. The core material (vector spaces, linear transformations and matrices, systems of linear equations, determinants, diagonalization, and inner product spaces) is found in Chapters 1 through 5 and Sections 6.1 through 6.5. Chapters 6 and 7, on inner product spaces and canonical forms, are completely independent and may be studied in either order. In addition, throughout the book are applications to such areas as differential equations, economics, geometry, and physics. These applications are not central to the mathematical development, however, and may be excluded at the discretion of the instructor.

We have attempted to make it possible for many of the important topics of linear algebra to be covered in a one-semester course. This goal has led us to develop the major topics with fewer preliminaries than in a traditional approach. (Our treatment of the Jordan canonical form, for instance, does not require any theory of polynomials.) The resulting economy permits us to cover the core material of the book (omitting many of the optional sections and a detailed discussion of determinants) in a one-semester four-hour course for students who have had some prior exposure to linear algebra.

Chapter 1 of the book presents the basic theory of vector spaces: subspaces, linear combinations, linear dependence and independence, bases, and dimension. The chapter concludes with an optional section in which eve prove that every infinite-dimensional vector space has a basis.

Linear transformations and their relationship to matrices are the subject of Chapter 2. We discuss the null space and range of a linear transformation, matrix representations of a linear transformation, isomorphisms, and change of coordinates. Optional sections on dual spaces and homogeneous linear differential equations end the chapter.

The application of vector space theory and linear transformations to systems of linear equations is found in Chapter 3. We have chosen to defer this important subject so that it can be presented as a consequence of the preceding material. This approach allows the familiar topic of linear systems to illuminate the abstract theory and permits us to avoid messy matrix computations in the presentation of Chapters 1 and 2. There are occasional examples in these chapters, however, where we solve systems of linear equations. (Of course, these examples are not a part of the theoretical development.) The necessary background is contained in Section 1.4.

Determinants, the subject of Chapter 4, are of much less importance than they once were. In a short course (less than one year), we prefer to treat determinants lightly so that more time may be devoted to the material in Chapters 5 through 7. Consequently we have presented two alternatives in Chapter 4—a complete development of the theory (Sections 4.1 through 4.3) and a summary of important facts that are needed for the remaining chapters (Section 4.4). Optional Section 4.5 presents an axiomatic development of the determinant.

Chapter 5 discusses eigenvalues, eigenvectors, and diagonalization. One of the most important applications of this material occurs in computing matrix limits. We have therefore included an optional section on matrix limits and Markov chains in this chapter even though the most general statement of some of the results requires a knowledge of the Jordan canonical form. Section 5.4 contains material on invariant subspaces and the Cayley-Hamilton theorem.

Inner product spaces are the subject of Chapter 6. The basic mathematical theory (inner products; the Gram-Schmidt process; orthogonal complements; the adjoint of an operator; normal, self-adjoint, orthogonal and unitary operators; orthogonal projections; and the spectral theorem) is contained in Sections 6.1 through 6.6. Sections 6.7 through 6.11 contain diverse applications of the rich inner product space structure.

Canonical forms are treated in Chapter 7. Sections 7.1 and 7.2 develop the Jordan canonical form, Section 7.3 presents the minimal polynomial, and Section 7.4 discusses the rational canonical form.

There are five appendices. The first four, which discuss sets, functions, fields, and complex numbers, respectively, are intended to review basic ideas used throughout the book. Appendix E on polynomials is used primarily in Chapters 5 and 7, especially in Section 7.4. We prefer to cite particular results from the appendices as needed rather than to discuss the appendices independently.

DIFFERENCES BETWEEN THE THIRD AND FOURTH EDITIONS

The principal content change of this fourth edition is the inclusion of a new section (Section 6.7) discussing the singular value decomposition and the pseudoinverse of a matrix or a linear transformation between finite-dimensional inner product spaces. Our approach is to treat this material as a generalization of our characterization of normal and self-adjoint operators.

The organization of the text is essentially the same as in the third edition. Nevertheless, this edition contains many significant local changes that improve the book. Section 5.1 (Eigenvalues and Eigenvectors) has been streamlined, and some material previously in Section 5.1 has been moved to Section 2.5 (The Change of Coordinate Matrix). Further improvements include revised proofs of some theorems, additional examples, new exercises, and literally hundreds of minor editorial changes.

We are especially indebted to Jane M. Day (San Jose State University) for her extensive and detailed comments on the fourth edition manuscript. Additional comments were provided by the following reviewers of the fourth edition manuscript: Thomas Banchoff (Brown University), Christopher Heil (Georgia Institute of Technology), and Thomas Shemanske (Dartmouth College).