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Chapter 1 MATRICES AND GAUSSIAN ELIMINATION 1

1.1 Introduction 1

1.2 The Geometry of Linear Equations 3

1.3 An Example of Gaussian Elimination 11

1.4 Matrix Notation and Matrix Multiplication 19

1.5 Triangular Factors and Row Exchanges 32

1.6 Inverses and Transposes 45

1.7 Special Matrices and Applications 58

Review Exercises: Chapter 1 65

Chapter 2 VECTOR SPACES 69

2.1 Vector Spaces and Subspaces 69

2.2 Solving Ax = 0 and Ax = b 77

2.3 Linear Independence, Basis, and Dimension 92

2.4 The Four Fundamental Subspaces 102

2.5 Graphs and Networks 114

2.6 Linear Transformations 125

Review Exercises: Chapter 2 137

Chapter 3 ORTHOGONALITY 141

3.1 Orthogonal Vectors and Subspaces 141

3.2 Cosines and Projections onto Lines 152

3.3 Projections and Least Squares 160

3.4 Orthogonal Bases and Gram-Schmidt 174

3.5 The Fast Fourier Transform 188

Review Exercises: Chapter 3 198

Chapter 4 DETERMINANTS 201

4.1 Introduction 201

4.2 Properties of the Determinant 203

4.3 Formulas for the Determinant 210

4.4 Applications of Determinants 220

Review Exercises: Chapter 4 230

Chapter 5 EIGENVALUES AND EIGENVECTORS 233

5.1 Introduction 233

5.2 Diagonalization of a Matrix 245

5.3 Difference Equations and Powers Ak 254

5.4 Differential Equations and eAt 266

5.5 Complex Matrices 280

5.6 Similarity Transformations 293

Review Exercises: Chapter 5 307

Chapter 6 POSITIVE DEFINITE MATRICES 311

6.1 Minima, Maxima, and Saddle Points 311

6.2 Tests for Positive Definiteness 318

6.3 Singular Value Decomposition 331

6.4 Minimum Principles 339

6.5 The Finite Element Method 346

Chapter 7 COMPUTATIONS WITH MATRICES 351

7.1 Introduction 351

7.2 Matrix Norm and Condition Number 352

7.3 Computation of Eigenvalues 359

7.4 Iterative Methods for Ax = b 367

Chapter 8 LINEAR PROGRAMMING AND GAME THEORY 377

8.1 Linear Inequalities 377

8.2 The Simplex Method 382

8.3 The Dual Problem 392

8.4 Network Models 401

8.5 Game Theory 408

Appendix A INTERSECTION, SUM, AND PRODUCT OF SPACES 415

Appendix B THE JORDAN FORM 422

Solutions to Selected Exercises 428

Matrix Factorizations 474

Glossary 476

MATLAB Teaching Codes 481

Index 482

Linear Algebra in a Nutshell 488