Contents

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Preface 8

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A Note to Students 15

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Chapter 1 Linear Equations in Linear Algebra 17

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INTRODUCTORY EXAMPLE: Linear Models in Economics and Engineering 17

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1.1 Systems of Linear Equations 18

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1.2 Row Reduction and Echelon Forms 28

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1.3 Vector Equations 40

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1.4 The Matrix Equation Ax = b 51

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1.5 Solution Sets of Linear Systems 59

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1.6 Applications of Linear Systems 66

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1.7 Linear Independence 72

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1.8 Introduction to Linear Transformations 79

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1.9 The Matrix of a Linear Transformation 87

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1.10 Linear Models in Business, Science, and Engineering 97

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Supplementary Exercises 105

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Chapter 2 Matrix Algebra 109

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INTRODUCTORY EXAMPLE: Computer Models in Aircraft Design 109

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2.1 Matrix Operations 110

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2.2 The Inverse of a Matrix 120

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2.3 Characterizations of Invertible Matrices 129

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2.4 Partitioned Matrices 135

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2.5 Matrix Factorizations 141

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2.6 The Leontief Input�COutput Model 150

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2.7 Applications to Computer Graphics 156

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2.8 Subspaces of 164

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2.9 Dimension and Rank 171

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Supplementary Exercises 178

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Chapter 3 Determinants 181

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INTRODUCTORY EXAMPLE: Random Paths and Distortion 181

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3.1 Introduction to Determinants 182

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3.2 Properties of Determinants 187

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3.3 Cramer��s Rule, Volume, and Linear Transformations 195

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Supplementary Exercises 204

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Chapter 4 Vector Spaces 207

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INTRODUCTORY EXAMPLE: Space Flight and Control Systems 207

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4.1 Vector Spaces and Subspaces 208

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4.2 Null Spaces, Column Spaces, and Linear Transformations 216

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4.3 Linearly Independent Sets; Bases 226

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4.4 Coordinate Systems 234

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4.5 The Dimension of a Vector Space 243

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4.6 Rank 248

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4.7 Change of Basis 257

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4.8 Applications to Difference Equations 262

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4.9 Applications to Markov Chains 271

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Supplementary Exercises 280

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Chapter 5 Eigenvalues and Eigenvectors 283

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INTRODUCTORY EXAMPLE: Dynamical Systems and Spotted Owls 283

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5.1 Eigenvectors and Eigenvalues 284

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5.2 The Characteristic Equation 292

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5.3 Diagonalization 299

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5.4 Eigenvectors and Linear Transformations 306

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5.5 Complex Eigenvalues 313

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5.6 Discrete Dynamical Systems 319

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5.7 Applications to Differential Equations 329

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5.8 Iterative Estimates for Eigenvalues 337

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Supplementary Exercises 344

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Chapter 6 Orthogonality and Least Squares 347

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INTRODUCTORY EXAMPLE: The North American Datum and GPS Navigation 347

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6.1 Inner Product, Length, and Orthogonality 348�ȷe���L���c������

6.2 Orthogonal Sets 356

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6.3 Orthogonal Projections 365

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6.4 The Gram�CSchmidt Process 372

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6.5 Least-Squares Problems 378

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6.6 Applications to Linear Models 386

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6.7 Inner Product Spaces 394

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6.8 Applications of Inner Product Spaces 401

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Supplementary Exercises 408

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Chapter 7 Symmetric Matrices and Quadratic Forms 411

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INTRODUCTORY EXAMPLE: Multichannel Image Processing 411

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7.1 Diagonalization of Symmetric Matrices 413

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7.2 Quadratic Forms 419

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7.3 Constrained Optimization 426

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7.4 The Singular Value Decomposition 432

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7.5 Applications to Image Processing and Statistics 442

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Supplementary Exercises 450

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Chapter 8 The Geometry of Vector Spaces 453

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INTRODUCTORY EXAMPLE: The Platonic Solids 453

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8.1 Af?ne Combinations 454

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8.2 Af?ne Independence 462

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8.3 Convex Combinations 472

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8.4 Hyperplanes 479

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8.5 Polytopes 487

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8.6 Curves and Surfaces 499

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Chapter 9 Optimization (Online)

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INTRODUCTORY EXAMPLE: The Berlin Airlift

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