【实例简介】linear algebra with application

【实例截图】

【核心代码】

Contents

Preface ix

1 Matrices and Systems of Equations 1

1.1 Systems of Linear Equations 1

1.2 Row Echelon Form 11

1.3 Matrix Arithmetic 27

1.4 Matrix Algebra 46

1.5 Elementary Matrices 60

1.6 Partitioned Matrices 70

MATLAB Exercises 80

Chapter Test A—True or False 84

Chapter Test B 85

2 Determinants 87

2.1 The Determinant of a Matrix 87

2.2 Properties of Determinants 94

2.3 Additional Topics and Applications 101

MATLAB Exercises 109

Chapter Test A—True or False 111

Chapter Test B 111

3 Vector Spaces 112

3.1 Definition and Examples 112

3.2 Subspaces 119

3.3 Linear Independence 130

3.4 Basis and Dimension 141

3.5 Change of Basis 147

3.6 Row Space and Column Space 157

MATLAB Exercises 165

Chapter Test A—True or False 166

Chapter Test B 167

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vi Contents

4 Linear Transformations 169

4.1 Definition and Examples 169

4.2 Matrix Representations of Linear Transformations 178

4.3 Similarity 192

MATLAB Exercises 198

Chapter Test A—True or False 199

Chapter Test B 200

5 Orthogonality 201

5.1 The Scalar Product in Rn 202

5.2 Orthogonal Subspaces 217

5.3 Least Squares Problems 225

5.4 Inner Product Spaces 238

5.5 Orthonormal Sets 247

5.6 The Gram–Schmidt Orthogonalization Process 266

5.7 Orthogonal Polynomials 275

MATLAB Exercises 283

Chapter Test A—True or False 285

Chapter Test B 285

6 Eigenvalues 287

6.1 Eigenvalues and Eigenvectors 288

6.2 Systems of Linear Differential Equations 301

6.3 Diagonalization 312

6.4 Hermitian Matrices 330

6.5 The Singular Value Decomposition 342

6.6 Quadratic Forms 356

6.7 Positive Definite Matrices 370

6.8 Nonnegative Matrices 377

MATLAB Exercises 387

Chapter Test A—True or False 393

Chapter Test B 393

7 Numerical Linear Algebra 395

7.1 Floating-Point Numbers 396

7.2 Gaussian Elimination 404

Contents vii

7.3 Pivoting Strategies 409

7.4 Matrix Norms and Condition Numbers 415

7.5 Orthogonal Transformations 429

7.6 The Eigenvalue Problem 440

7.7 Least Squares Problems 451

MATLAB Exercises 463

Chapter Test A—True or False 468

Chapter Test B 468

8 Iterative Methods Online∗

8.1 Basic Iterative Methods

9 Canonical Forms Online∗

9.1 Nilpotent Operators

9.2 The Jordan Canonical Form

Appendix: MATLAB 471

Bibliography 483

Answers to Selected Exercises 486

Index 499