《LINEAR ALGEBRA》

1VECTOR SPACES1

1.1 Introduction1

1.2 Vector Spaces6

1.3 Subspaces15

1.4 Linear Combinations and Systems of Linear Equations22

1.5 Linear Dependence and Linear Independence33

1.6 Bases and Dimension37

1.7 Maximal Linearly Independent Subsets52

Index of Definitions for Chapter 155

2LINEAR TRANSFORMATIONS AND MATRICES57

2.1 Linear Transformations，Null Spaces，and Ranges58

2.2 The Matrix Representation of a Linear Transformation69

2.3 Composition of Linear Transformations and Matrix Multiplication75

2.4 Invertibility and Isomorphisms87

2.5 The Change of Coordinate Matrix96

2.6 Dual Spaces103

2.7 Homogeneous Linear Differential Equations with Constant Coefficients110

Index of Definitions for Chapter 2127

3ELEMENTARY MATRIX OPERATIONS AND SYSTEMS OF LINEAR EQUATIONS129

3.1 Elementary Matrix Operations and Elementary Matrices130

3.2 The Rank of a Matrix and Matrix Inverses135

3.3 Systems of Linear Equations—Theoretical Aspects149

3.4 Systems of Linear Equations—Computational Aspects161

Index of Definitions for Chapter 3169

4DETERMINANTS171

4.1 Determinants of Order 2172

4.2 Determinants of Order n182

4.3 Properties of Determinants190

4.4 The Classical Adjoint and Cramer’s Rule203

4.5 Summary—Important Facts about Determinants208

Index of Definitions for Chapter 4215

5DIAGONALIZATION216

5.1 Eigenvalues and Eigenvectors217

5.2 Diagonalizability233

5.3 Matrix Limits and Markov Chains252

5.4 Invariant Subspaces280

5.5 The Cayley-Hamilton Theorem287

5.6 The Minimal Polynomial293

Index of Definitions for Chapter 5300

6CANONICAL FORMS302

6.1 Generalized Eigenvectors302

6.2 Jordan Canonical Form319

6.3 Rational Canonical Form339

Index of Definitions for Chapter 6357

7INNER PRODUCT SPACES358

7.1 Inner Products and Norms358

7.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements367

7.3 The Adjoint of a Linear Operator375

7.4 Einstein’s Special Theory of Relativity380

7.5 Normal and Self-Adjoint Operators393

7.6 Conditioning and the Rayleigh Quotient400

7.7 Unitary and Orthogonal Operators and Their Matrices408

7.8 The Geometry of Orthogonal Operators420

7.9 Orthogonal Projections and the Spectral Theorem429

7.10 Least Squares Approximation436

7.11 Bilinear and Quadratic Forms441

Index of Definitions for Chapter 7466

APPENDICES468

ASets468

B Functions470

C Fields472

D Complex Numbers475

E Polynomials479

ANSWERS TO SELECTED EXERCISES488

LIST OF FREQUENTLY USED SYMBOLS505

INDEX OF THEOREMS506

INDEX508