A Course in Linear Algebra

by David B. Damiano, John B. Little

Overview

Suitable for advanced undergraduates and graduate students, this text offers a complete introduction to the basic concepts of linear algebra. Interesting and inspiring in its approach, it imparts an understanding of the subject's logical structure as well as the ways in which linear algebra provides solutions to problems in many branches of mathematics.
The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these concepts. This approach provides a ready-made context, motivation, and geometric interpretation for each new computational technique. Proofs and abstract problem-solving are introduced from the start, offering students an immediate opportunity to practice applying what they've learned. Each chapter contains an introduction, summary, and supplementary exercises. The text concludes with a pair of helpful appendixes and solutions to selected exercises.

Product Details

ISBN-13:	9780486469089
Publisher:	Dover Publications
Publication date:	09/14/2011
Series:	Dover Books on Mathematics
Pages:	464
Product dimensions:	6.10(w) x 9.10(h) x 1.00(d)

About the Author

The authors are Professors of Mathematics at College of the Holy Cross.

Table of Contents

Preface iii

Errata vii

A Guide to the Exercises xi

Chapter 1 Vector Spaces 1

Introduction 1

1.1 Vector Spaces 2

1.2 Subspaces 12

1.3 Linear Combinations 21

1.4 Linear Dependence and Linear Independence 26

1.5 Interlude on Solving Systems of Linear Equations 32

1.6 Bases and Dimension 47

Chapter Summary 58

Supplementary Exercises 59

Chapter 2 Linear Transformations 62

Introduction 62

2.1 Linear Transformations 63

2.2 Linear Transformations between Finite-Dimensional Spaces 73

2.3 Kernel and Image 84

2.4 Applications of the Dimension Theorem 95

2.5 Composition of Linear Transformations 106

2.6 The Inverse of a Linear Transformation 114

2.7 Change of Basis 122

Chapter Summary 129

Supplementary Exercises 130

Chapter 3 The Determinant Function 133

Introduction 133

3.1 The Determinant as Area 134

3.2 The Determinant of an n x n Matrix 140

3.3 Further Properties of the Determinant 153

Chapter Summary 160

Supplementary Exercises 160

Chapter 4 Eigenvalues, Eigenvectors, Diagonalization, and the Spectral Theorem in Rⁿ 162

Introduction 162

4.1 Eigenvalues and Eigenvectors 163

4.2 Diagonalizability 175

4.3 Geometry in Rn 184

4.4 Orthogonal Projections and the Gram-Schmidt Process 190

4.5 Symmetric Matrices 200

4.6 The Spectral Theorem 206

Chapter Summary 217

Supplementary Exercises 218

Chapter 5 Complex; Numbers and Complex Vector Spaces 224

Introduction 224

5.1 Complex Numbers 225

5.2 Vector Spaces Over a Field 234

5.3 Geometry in a Complex Vector Space 241

Chapter Summary 249

Supplementary Exercises 251

Chapter 6 Jordan Canonical Form 253

Introduction 253

6.1 Triangular Form 254

6.2 A Canonical Form for Nilpotent Mappings 263

6.3 Jordan Canonical Form 273

6.4 Computing Jordan Form 281

6.5 The Characteristic Polynomial and the Minimal Polynomial 287

Chapter Summary 294

Supplementary Exercises 295

Chapter 7 Differential Equations 299

Introduction 299

7.1 Two Motivating Examples 300

7.2 Constant Coefficient Linear Differential Equations The Diagonalizable Case 305

7.3 Constant (Coefficient Linear Differential Equations: The General Case 312

7.4 One Ordinary Differential Equation with Constant Coefficients 323

7.5 An Eigenvalue Problem 332

Chapter Summary 340

Supplementary Exercises 341

Appendix 1 Some Basic Logic and Set Theory 344

A1.1 Sets 344

A1.2 Statements and Logical Operators 345

A1.3 Statements with Quantifiers 348

A1.4 Further Notions from Set Theory 349

A1.5 Relations and Functions 351

A1.6 Injectivity, Surjectivity, and Bijectivity 354

A1.7 Composites and Inverse Mappings 354

A1.8 Some (Optional) Remarks on Mathematics and Logic 355

Appendix 2 Mathematical Induction 359

Solutions 367

Index 429