PREFACE XI

FEATURES OF THE TEXT XIII

ACKNOWLEDGMENTS XVII

ABOUT THE COMPANION WEBSITE XIX

**1 SYSTEMS OF LINEAR EQUATIONS 1**

1.1 The Vector Space of m × n Matrices 1

The Space Rn 4

Linear Combinations and Linear Dependence 6

What is a Vector Space? 11

Exercises 17

1.1.1 Computer Projects 22

1.1.2 Applications to Graph Theory I 25

Exercises 27

1.2 Systems 28

Rank: The Maximum Number of Linearly Independent Equations 35

Exercises 38

1.2.1 Computer Projects 41

1.2.2 Applications to Circuit Theory 41

Exercises 46

1.3 Gaussian Elimination 47

Spanning in Polynomial Spaces 58

Computational Issues: Pivoting 61

Exercises 63

Computational Issues: Counting Flops 68

1.3.1 Computer Projects 69

1.3.2 Applications to Traffic Flow 72

1.4 Column Space and Nullspace 74

Subspaces 77

Exercises 86

1.4.1 Computer Projects 94

Chapter Summary 95

**2 LINEAR INDEPENDENCE AND DIMENSION 97**

2.1 The Test for Linear Independence 97

Bases for the Column Space 104

Testing Functions for Independence 106

Exercises 108

2.1.1 Computer Projects 113

2.2 Dimension 114

Exercises 123

2.2.1 Computer Projects 127

2.2.2 Applications to Differential Equations 128

Exercises 131

2.3 Row Space and the rank-nullity theorem 132

Bases for the Row Space 134

Computational Issues: Computing Rank 142

Exercises 143

2.3.1 Computer Projects 146

Chapter Summary 147

**3 LINEAR TRANSFORMATIONS 149**

3.1 The Linearity Properties 149

Exercises 157

3.1.1 Computer Projects 162

3.2 Matrix Multiplication (Composition) 164

Partitioned Matrices 171

Computational Issues: Parallel Computing 172

Exercises 173

3.2.1 Computer Projects 178

3.2.2 Applications to Graph Theory II 180

Exercises 181

3.3 Inverses 182

Computational Issues: Reduction versus Inverses 188

Exercises 190

3.3.1 Computer Projects 195

3.3.2 Applications to Economics 197

Exercises 202

3.4 The LU Factorization 203

Exercises 212

3.4.1 Computer Projects 214

3.5 The Matrix of a Linear Transformation 215

Coordinates 215

Isomorphism 228

Invertible Linear Transformations 229

Exercises 230

3.5.1 Computer Projects 235

Chapter Summary 236

**4 DETERMINANTS 238**

4.1 Definition of the Determinant 238

4.1.1 The Rest of the Proofs 246

Exercises 249

4.1.2 Computer Projects 251

4.2 Reduction and Determinants 252

Uniqueness of the Determinant 256

Exercises 258

4.2.1 Volume 261

Exercises 263

4.3 A Formula for Inverses 264

Exercises 268

Chapter Summary 269

**5 EIGENVECTORS AND EIGENVALUES 271**

5.1 Eigenvectors 271

Exercises 279

5.1.1 Computer Projects 282

5.1.2 Application to Markov Processes 283

Exercises 285

5.2 Diagonalization 287

Powers of Matrices 288

Exercises 290

5.2.1 Computer Projects 292

5.2.2 Application to Systems of Differential Equations 293

Exercises 295

5.3 Complex Eigenvectors 296

Complex Vector Spaces 303

Exercises 304

5.3.1 Computer Projects 305

Chapter Summary 306

**6 ORTHOGONALITY 308**

6.1 The Scalar Product in RN 308

Orthogonal/Orthonormal Bases and Coordinates 312

Exercises 316

6.2 Projections: The Gram-Schmidt Process 318

The QR Decomposition 325

Uniqueness of the QR Factorization 327

Exercises 328

6.2.1 Computer Projects 331

6.3 Fourier Series: Scalar Product Spaces 333

Exercises 341

6.3.1 Application to Data Compression: Wavelets 344

Exercises 352

6.3.2 Computer Projects 353

6.4 Orthogonal Matrices 355

Householder Matrices 361

Exercises 364

Discrete Wavelet Transform 367

6.4.1 Computer Projects 369

6.5 Least Squares 370

Exercises 377

6.5.1 Computer Projects 380

6.6 Quadratic Forms: Orthogonal Diagonalization 381

The Spectral Theorem 385

The Principal Axis Theorem 386

Exercises 392

6.6.1 Computer Projects 395

6.7 The Singular Value Decomposition (SVD) 396

Application of the SVD to Least-Squares Problems 402

Exercises 404

Computing the SVD Using Householder Matrices 406

Diagonalizing Matrices Using Householder Matrices 408

6.8 Hermitian Symmetric and Unitary Matrices 410

Exercises 417

Chapter Summary 419

**7 GENERALIZED EIGENVECTORS 421**

7.1 Generalized Eigenvectors 421

Exercises 429

7.2 Chain Bases 431

Jordan Form 438

Exercises 443

The Cayley-Hamilton Theorem 445

Chapter Summary 445

**8 NUMERICAL TECHNIQUES 446**

8.1 Condition Number 446

Norms 446

Condition Number 448

Least Squares 451

Exercises 451

8.2 Computing Eigenvalues 452

Iteration 453

The QR Method 457

Exercises 462

Chapter Summary 464

ANSWERS AND HINTS 465

INDEX 487