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1 VECTOR SPACES 1

1.1 Introduction 1

1.2 Vector Spaces 6

1.3 Subspaces 15

1.4 Linear Combinations and Systems of Linear Equations 22

1.5 Linear Dependence and Linear Independence 33

1.6 Bases and Dimension 37

1.7 Maximal Linearly Independent Subsets 52

Index of Definitions for Chapter 1 55

2 LINEAR TRANSFORMATIONS AND MATRICES 57

2.1 Linear Transformations，Null Spaces，and Ranges 58

2.2 The Matrix Representation of a Linear Transformation 69

2.3 Composition of Linear Transformations and Matrix Multiplication 75

2.4 Invertibility and Isomorphisms 87

2.5 The Change of Coordinate Matrix 96

2.6 Dual Spaces 103

2.7 Homogeneous Linear Differential Equations with Constant Coefficients 110

Index of Definitions for Chapter 2 127

3 ELEMENTARY MATRIX OPERATIONS AND SYSTEMS OF LINEAR EQUATIONS 129

3.1 Elementary Matrix Operations and Elementary Matrices 130

3.2 The Rank of a Matrix and Matrix Inverses 135

3.3 Systems of Linear Equations—Theoretical Aspects 149

3.4 Systems of Linear Equations—Computational Aspects 161

Index of Definitions for Chapter 3 169

4 DETERMINANTS 171

4.1 Determinants of Order 2 172

4.2 Determinants of Order n 182

4.3 Properties of Determinants 190

4.4 The Classical Adjoint and Cramer’s Rule 203

4.5 Summary—Important Facts about Determinants 208

Index of Definitions for Chapter 4 215

5 DIAGONALIZATION 216

5.1 Eigenvalues and Eigenvectors 217

5.2 Diagonalizability 233

5.3 Matrix Limits and Markov Chains 252

5.4 Invariant Subspaces 280

5.5 The Cayley-Hamilton Theorem 287

5.6 The Minimal Polynomial 293

Index of Definitions for Chapter 5 300

6 CANONICAL FORMS 302

6.1 Generalized Eigenvectors 302

6.2 Jordan Canonical Form 319

6.3 Rational Canonical Form 339

Index of Definitions for Chapter 6 357

7 INNER PRODUCT SPACES 358

7.1 Inner Products and Norms 358

7.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements 367

7.3 The Adjoint of a Linear Operator 375

7.4 Einstein’s Special Theory of Relativity 380

7.5 Normal and Self-Adjoint Operators 393

7.6 Conditioning and the Rayleigh Quotient 400

7.7 Unitary and Orthogonal Operators and Their Matrices 408

7.8 The Geometry of Orthogonal Operators 420

7.9 Orthogonal Projections and the Spectral Theorem 429

7.10 Least Squares Approximation 436

7.11 Bilinear and Quadratic Forms 441

Index of Definitions for Chapter 7 466

APPENDICES 468

A Sets 468

B Functions 470

C Fields 472

D Complex Numbers 475

E Polynomials 479

ANSWERS TO SELECTED EXERCISES 488

LIST OF FREQUENTLY USED SYMBOLS 505

INDEX OF THEOREMS 506

INDEX 508