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1 SYSTEMS OF LINEAR EQUATIONS 1

Introductory Example：Linear Models in Economics and Engineering 1

1.1 Introduction to Systems of Linear Equations 2

1.2 Row Reduction and Echelon Forms 13

1.3 Applications of Linear Systems 25

Supplementary Exercises 34

2 VECTOR AND MATRIX EQUATIONS 37

Introductory Example：Nutrition Problems 37

2.1 Vectors in R” 38

2.2 The Equation Ax＝b 48

2.3 Solution Sets of Linear Systems 56

2.4 Linear Independence 63

2.5 Introduction to Linear Transformations 71

2.6 The Matrix of a Linear Transformation 79

2.7 Applications to Nutrition and Population Movement 85

Supplementary Exercises 92

3 MATRIX ALGEBRA 95

Introductory Example：Computer Graphics in Automotive Design 95

3.1 Matrix Operations 96

3.2 The Inverse of a Matrix 107

3.3 Characterizations of Invertible Matrices 116

3.4 Partitioned Matrices 121

3.5 Matrix Factorizations 128

3.6 Iterative Solutions of Linear Systems 137

3.7 The Leontief Input-Output Model 142

3.8 Applications to Computer Graphics 148

Supplementary Exercises 158

4 DETERMINANTS 161

Introductory Example：Determinants in Analytic Geometry 161

4.1 Introduction to Determinants 162

4.2 Properties of Determinants 168

4.3 Cramer’s Rule，Volume，and Linear Transformations 176

Supplementary Exercises 186

5 VECTOR SPACES 189

Introductory Example：Space Flight and Control Systems 189

5.1 Vector Spaces and Subspaces 190

5.2 Null Spaces，Column Spaces，and Linear Transformations 200

5.3 Linearly Independent Sets； Bases 211

5.4 Coordinate Systems 219

5.5 The Dimension of a Vector Space 229

5.6 Rank 235

5.7 Change of Basis 243

5.8 Applications to Difference Equations 248

5.9 Applications to Markov Chains 259

Supplementary Exercises 269

6 EIGENVALUES AND EIGENVECTORS 271

Introductory Example：Dynamical Systems and Spotted Owls 271

6.1 Eigenvectors and Eigenvalues 273

6.2 The Characteristic Equation 280

6.3 Diagonalization 288

6.4 Eigenvectors and Linear Transformations 296

6.5 Complex Eigenvalues 303

6.6 Applications to Dynamical Systems 310

6.7 Iterative Estimates for Eigenvalues 321

Supplementary Exercises 329

7 ORTHOGONALITY AND LEAST-SQUARES 331

Introductory Example：Readjusting the North American Datum 331

7.1 Inner Product，Length，and Orthogonality 333

7.2 Orthogonal Sets 342

7.3 Orthogonal Projections 352

7.4 The Gram-Schmidt Process 359

7.5 Least-Squares Problems 366

7.6 Applications to Linear Models 376

7.7 Inner Product Spaces 384

7.8 Applications of Inner Product Spaces 393

Supplementary Exercises 401

8 SYMMETRIC MATRICES AND QUADRATIC FORMS 403

Introductory Example：Multichannel Image Processing 403

8.1 Diagonalization of Symmetric Matrices 405

8.2 Quadratic Forms 411

8.3 Constrained Optimization 419

8.4 The Singular Value Decomposition 426

8.5 Applications to Image Processing and Statistics 435

Supplementary Exercises 444

APPENDICES 447

A Uniqueness of the Reduced Echelon Form 447

B Complex Numbers 449

GLOSSARY 455

ANSWERS TO ODD-NUMBERED EXERCISES 467