9780130084514

This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms.For statisticians and engineers.

Preface

ix

Vector Spaces

1

(63)

Introduction

1

(5)

Vector Spaces

6

(10)

Subspaces

16

(8)

Linear Combinations and Systems of Linear Equations

24

(11)

Linear Dependence and Linear Independence

35

(7)

Bases and Dimension

42

(16)

Maximal Linearly Independent Subsets

58

(6)

Index of Definitions

62

(2)

Linear Transformations and Matrices

64

(83)

Linear Transformations, Null Spaces, and Ranges

64

(15)

The Matrix Representation of a Linear Transformation

79

(7)

Composition of Linear Transformations and Matrix Multiplication

86

(13)

Invertibility and Isomorphisms

99

(11)

The Change of Coordinate Matrix

110

(9)

Dual Spaces

119

(8)

Homogeneous Linear Differential Equations with Constant Coefficients

127

(20)

Index of Definitions

145

(2)

Elementary Matrix Operations and Systems of Linear Equations

147

(52)

Elementary Matrix Operations and Elementary Matrices

147

(5)

The Rank of a Matrix and Matrix Inverses

152

(16)

Systems of Linear Equations---Theoretical Aspects

168

(14)

Systems of Linear Equations---Computational Aspects

182

(17)

Index of Definitions

198

(1)

Determinants

199

(46)

Determinants of Order 2

199

(10)

Determinants of Order n

209

(13)

Properties of Determinants

222

(10)

Summary---Important Facts about Determinants

232

(6)

A Characterization of the Determinant

238

(7)

Index of Definitions

244

(1)

Diagonalization

245

(84)

Eigenvalues and Eigenvectors

245

(16)

Diagonalizability

261

(22)

Matrix Limits and Markov Chains

283

(30)

Invariant Subspaces and the Cayley--Hamilton Theorem

313

(16)

Index of Definitions

328

(1)

Inner Product Spaces

329

(153)

Inner Products and Norms

329

(12)

The Gram--Schmidt Orthogonalization Process and Orthogonal Complements

341

(16)

The Adjoint of a Linear Operator

357

(12)

Normal and Self-Adjoint Operators

369

(10)

Unitary and Orthogonal Operators and Their Matrices

379

(19)

Orthogonal Projections and the Spectral Theorem

398

(7)

The Singular Value Decomposition and the Pseudoinverse

405

(17)

Bilinear and Quadratic Forms

422

(29)

Einstein's Special Theory of Relativity

451

(13)

Conditioning and the Rayleigh Quotient

464

(8)

The Geometry of Orthogonal Operators

472

(10)

Index of Definitions

480

(2)

Canonical Forms

482

(66)

The Jordan Canonical Form I

482

(15)

The Jordan Canonical Form II

497

(19)

The Minimal Polynomial

516

(8)

The Rational Canonical Form

524

(24)

Index of Definitions

548

(1)

Appendices

549

(22)

A Sets

549

(2)

B Functions

551

(1)

C Fields

552

(3)

D Complex Numbers

555

(6)

E Polynomials

561

(10)

Answers to Selected Exercises

571

(18)

Index

589

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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 ofLinear Algebrais 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

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Linear Algebra

Linear Algebra

0130084514

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