Linear Algebra Ideas and Applications

Praise for the Third Edition

"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications."
—Electric Review

Learn foundational and advanced topics in linear algebra with this concise and approachable resource

A comprehensive introduction, Linear Algebra: Ideas and Applications, Fifth Edition provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique.

The book introduces each new concept in the context of explicit numerical examples, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs. Linear Algebra: Ideas and Applications, Fifth Edition also features:

• A new application section on section on Google’s Page Rank Algorithm.
• A new application section on pricing long term health insurance at a Continuing Care Retirement Community (CCRC).
• Many other illuminating applications of linear algebra with self-study questions for additional study.
• End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material
• Numerous computer exercises throughout using MATLAB code

Linear Algebra: Ideas and Applications, Fifth Edition is an excellent undergraduate-level textbook for one or two semester undergraduate courses in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.

RICHARD C. PENNEY, PHD is Emeritus Professor in the Department of Mathematics and former Director of the Mathematics/Statistics Actuarial Science Program at Purdue University. He has authored numerous journal articles, received several major teaching awards, and is an active researcher. He received his graduate education at MIT.

1 Systems of Linear Equations 1

1.1 The Vector Space of m × n Matrices 1

The Space R n 4

Linear Combinations and Linear Dependence 6

What Is a Vector Space? 10

Exercises 16

1.1.1 Computer Projects/Exercises/Exercises 21

Introduction to MATLAB 21

1.1.2 Applications to Graph Theory I 24

Exercises 26

1.2 Systems 27

Rank: The Maximum Number of Linearly Independent Equations 33

Exercises 37

1.2.1 Computer Projects/Exercises 39

The Translation Theorem 39

1.2.2 Applications to Circuit Theory 40

Exercises 44

1.3 Gaussian Elimination 45

Spanning in Polynomial Spaces 56

Computational Issues: Pivoting 59

Exercises 60

Computational Issues: Flops 65

1.3.1 Computer Projects/Exercises 66

Using tolerances in rref and rank 66

1.3.2 Applications to Traffic Flow 69

Exercises 70

1.4 Column Space and Nullspace 71

Subspaces 73

Exercises 82

1.4.1 Computer Projects/Exercises 90

The null Command 90

Chapter Summary 92

2 Linear Independence and Dimension 93

2.1 The Test for Linear Independence 93

Bases for the Column Space 99

Testing Functions for Independence 102

Exercises 104

2.1.1 Computer Projects/Exercises 108

Changing Pivot Columns 108

2.2 Dimension 109

Exercises 118

2.2.1 Computer Projects/Exercises 124

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 144

Random Matrices of a given Rank 144

Chapter Summary 145

3 Linear Transformations 147

3.1 The Linearity Properties 147

Exercises 154

3.1.1 Computer Projects/Exercises 159

2-D Computer Graphics 159

3.2 Matrix Multiplication (Composition) 161

Partitioned Matrices 168

Computational Issues: Parallel Computing 170

Exercises 171

3.2.1 Computer Projects/Exercises 176

3-D Computer Graphics 176

3.2.2 Applications to Graph Theory II 177

Exercises 179

3.2.3 Computer Projects/Exercises 179

Google’s Page Rank Algorithim 179

Exercises 182

3.3 Inverses 183

Computational Issues: Reduction versus Inverses 189

Exercises 191

3.3.1 Computer Projects/Exercises 196

Ill-Conditioned Systems 196

3.3.2 Applications to Economics: The Leontief open model 198

Exercises 203

3.4 The LU Factorization 204

Exercises 213

3.4.1 Computer Projects/Exercises 215

Row Exchanges in the LU Factorization 215

3.5 The Matrix of a Linear Transformation 216

Coordinates 216

Isomorphism 227

Invertible Linear Transformations 228

Exercises 230

3.5.1 Computer Projects/Exercises 234

Graphing in Skewed-Coordinates 234

3.5.2 Computer Projects/Exercises 236

Pricing Long Term Health Care Insurance 236

Exercises 240

Chapter Summary 241

4 Determinants 243

4.1 Definition of the Determinant 243

4.1.1 The Rest of the Proofs 251

Exercises 254

4.1.2 Computer Projects/Exercises 257

4.2 Reduction and Determinants 257

Uniqueness of the Determinant 262

Exercises 265

4.2.1 Volume 267

Exercises 270

4.3 A Formula for Inverses 270

Cramer’s Rule 272

Exercises 275

Chapter Summary 276

5 Eigenvectors and Eigenvalues 279

5.1 Eigenvectors 279

Exercises 288

5.1.1 Computer Projects/Exercises 291

Computing Roots of Polynomials 291

5.1.2 Application to Markov Chains 292

Application to the Auto Rental Business 292

Exercises 294

5.2 Diagonalization 296

Powers of Matrices 298

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

Complex Eigenvalues 314

Exercises 314

Chapter Summary 314

6 Orthogonality 317

6.1 The Scalar Product in R n 317

Orthogonal/Orthonormal Bases and Coordinates 321

Exercises 325

6.2 Projections: The Gram-Schmidt Process 327

The QR Decomposition 333

Uniqueness of the QR-factorization 336

Exercises 337

6.2.1 Computer Projects/Exercises 340

The Least Squares Solution 340

6.3 Fourier Series: Scalar Product Spaces 342

Exercises 348

6.3.1 Computer Projects/Exercises 352

Plotting Fourier Series 352

6.4 Orthogonal Matrices 353

Householder Matrices 359

Exercises 363

Discrete Wavelet Transform 366

6.4.1 Computer Projects/Exercises 367

6.5 Least Squares 369

Exercises 375

6.5.1 Computer Projects/Exercises 379

Finding the Orbit of an Asteroid 379

6.6 Quadratic Forms: Orthogonal Diagonalization 380

The Spectral Theorem 383

The Principal Axis Theorem 384

Exercises 390

6.6.1 Computer Projects/Exercises 393

The Principal Axis Theorem 393

6.7 The Singular Value Decomposition (SVD) 394

Application of the SVD to Least-Squares Problems 400

Exercises 402

Computing the SVD Using Householder Matrices 404

Diagonalizing Symmetric Matrices 406

6.8 Hermitian Symmetric and Unitary Matrices 408

Exercises 414

Chapter Summary 417

7 Generalized Eigenvectors 419

7.1 Generalized Eigenvectors 419

Exercises 427

7.2 Chain Bases 430

Jordan Form 436

Exercises 442

The Cayley-Hamilton Theorem 443

Chapter Summary 444

8 Numerical Techniques 445

8.1 Condition Number 445

Norms 445

Condition Number 447

Least Squares 450

Exercises 450

8.2 Computing Eigenvalues 451

Iteration 451

The QR Method 455

Exercises 461

Chapter Summary 462

Answers and Hints 464

Index 489

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