Linear Algebra

Preface xi

Features of the Text xiii

Acknowledgments xvii

About the Companion Website xviii

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 7

What Is a Vector Space? 11

Why Prove Anything? 15

Exercises 16

1.1.1 Computer Projects/Exercises/Exercises 22

Exercises 24

1.1.2 Applications to Graph Theory I 25

Exercises 27

1.2 Systems 27

Rank: The Maximum Number of Linearly Independent Equations 34

Exercises 37

1.2.1 Computer Projects/Exercises 39

Exercises 39

1.2.2 Applications to Circuit Theory 40

Exercises 44

1.3 Gaussian Elimination 46

Spanning in Polynomial Spaces 56

Computational Issues: Pivoting 59

Exercises 60

1.3.1 Using tolerances in MATLAB's rref and rank 66

Using Tolerances in rref and Rank 66

Exercises 67

1.3.2 Applications to Traffic Flow 68

Exercises 70

1.4 Column Space and Nullspace 71

Subspaces 74

Exercises 82

1.4.1 Computer Projects/Exercises 89

Exercises 90

Chapter Summary 91

2 Linear Independence and Dimension 93

2.1 The Test for Linear Independence 93

Bases for the Column Space 100

Testing Functions for Independence 102

Exercises 104

2.1.1 Computer Projects/Exercises 108

Exercises 108

2.2 Dimension 109

Exercises 118

2.2.1 Computer Projects/Exercises 123

Exercises 123

2.2.2 Applications to Differential Equations 125

Exercises 128

2.3 Row Space and the Rank-Nullity Theorem 128

Bases for the Row Space 130

Computational Issues: Computing Rank 138

Exercises 140

2.3.1 Computer Projects/Exercises 143

Exercises 143

Chapter Summary 144

3 Linear Transformations 147

3.1 The Linearity Properties 147

Exercises 155

3.1.1 Computer Projects/Exercises 160

Exercises 161

3.2 Matrix Multiplication (Composition) 162

Partitioned Matrices 169

Computational Issues: Parallel Computing 171

Exercises 171

3.2.1 Computer Projects/Exercises 177

3-D Computer Graphics 177

Exercises 177

3.2.2 Applications to Graph Theory II 178

Exercises 180

3.2.3 Computer Projects/Exercises 180

Google's Page Rank Algorithm 180

Exercises 183

3.3 Inverses 184

Computational Issues: Reduction versus Inverses 190

Exercises 192

3.3.1 Computer Projects/Exercises 197

Ill-Conditioned Systems 197

Exercises 197

3.3.2 Applications to Economics: The Leontief Open Model 199

Exercises 204

3.4 The LU Factorization 205

Exercises 213

3.4.1 Computer Projects/Exercises 216

Exercises 216

3.5 The Matrix of a Linear Transformation 217

Coordinates 217

Application to Differential Equations 225

Isomorphism 228

Invertible Linear Transformations 229

Exercises 231

3.5.1 Computer Projects/Exercises 236

Graphing in Skewed-Coordinates 236

Exercises 236

3.5.2 Computer Projects/Exercises 237

Pricing Long Term Health Care Insurance 237

Exercises 242

Chapter Summary 242

4 Determinants 245

4.1 Definition of the Determinant 245

4.1.1 The Rest of the Proofs 252

Exercises 256

4.1.2 Computer Projects/Exercises 258

4.2 Reduction and Determinants 259

Exercises 266

4.2.1 Volume 268

Exercises 271

4.3 A Formula for Inverses 271

Exercises 275

Chapter Summary 276

5 Eigenvectors and Eigenvalues 279

5.1 Eigenvectors 279

Exercises 288

5.1.1 Computer Projects/Exercises 291

Exercises 291

5.1.2 Application to Markov Chains 291

Exercises 294

5.2 Diagonalization 295

Powers of Matrices 297

Exercises 299

5.2.1 Application to Systems of Differential Equations 301

Exercises 304

5.3 Complex Eigenvectors 304

Complex Vector Spaces 311

Exercises 312

5.3.1 Computer Projects/Exercises 314

Exercises 314

Chapter Summary 314

6 Orthogonality 317

6.1 The Scalar Product in Rⁿ 317

Orthogonal/Orthonormal Bases and Coordinates 321

Exercises 326

6.2 Projections: The Gram-Schmidt Process 328

The QR Decomposition 334

Uniqueness of the QR Factorization 337

Exercises 338

6.2.1 Computer Projects/Exercises 341

Exercises 342

6.3 Fourier Series: Scalar Product Spaces 342

Exercises 350

6.3.1 Computer Projects/Exercises 353

Exercises 354

6.4 Orthogonal Matrices 355

Householder Matrices 360

Exercises 364

6.4.1 Computer Projects/Exercises 369

Exercises 369

6.5 Least Squares 370

Exercises 377

6.5.1 Computer Projects/Exercises 380

Exercises 380

6.6 Quadratic Forms: Orthogonal Diagonalization 381

The Spectral Theorem 384

The Principal Axis Theorem 385

Exercises 392

6.6.1 Computer Projects/Exercises 394

Exercises 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 409

Exercises 416

Chapter Summary 418

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 444

Chapter Summary 445

8 Numerical Techniques 447

8.1 Condition Number 447

Condition Number 449

Least Squares 452

Exercises 453

8.2 Computing Eigenvalues 454

Iteration 454

The QR Method 458

Exercises 464

Chapter Summary 465

Answers and Hints 467

Index 491