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
