Linear Algebra A Modern Introduction

Linear Algebra
 
Author:
David Poole
Publisher: Cengage Learning
ISBN No: 978-1-285-46324-7
Release at: 2015
Pages: 722
Edition:
First Edition
File Size: 68 MB
File Type: pdf
Language: English



Content of Linear Algebra A Modern Introduction



Chapter 1 Vectors 1


1.0 Introduction: The Racetrack Game 1

1.1 The Geometry and Algebra of Vectors 3

1.2 Length and Angle: The Dot Product 18

1.3 Exploration: Vectors and Geometry 32

Lines and Planes 34

Exploration: The Cross Product 48

Writing Project: The Origins of the Dot Product and Cross Product 49

1.4 Applications 50

Force Vectors 50

Chapter Review 55 


Chapter 2  Systems of Linear Equations 57


2.0 Introduction: Triviality 57

2.1 Introduction to Systems of Linear Equations 58

2.2 Direct Methods for Solving Linear Systems 64

Writing Project: A History of Gaussian Elimination 82

Explorations: Lies My Computer Told Me 83

Partial Pivoting 84

Counting Operations: An Introduction to the

Analysis of Algorithms 85

2.3 Spanning Sets and Linear Independence 88

2.4 Applications 99

Allocation of Resources 99

Balancing Chemical Equations 101

Network Analysis 102

Electrical Networks 104

Linear Economic Models 107

Finite Linear Games 109

Vignette: The Global Positioning System 121

2.5 Iterative Methods for Solving Linear Systems 124

Chapter Review 134


Chapter 3 Matrices 136


3.0 Introduction: Matrices in Action 136

3. 1 Matrix Operations 138

3.2 Matrix Algebra 154

3.3 The Inverse of a Matrix 163

3.4 The LU Factorization 180

3.5 Subspaces, Basis, Dimension, and Rank

3.6 Introduction to Linear Transformations

Vignette: Robotics 226

3.7 Applications 230

Markov Chains 230

Linear Economic Models 235

Population Growth 239

Graphs and Digraphs 241

Chapter Review 251


Chapter 4  Eigenvalues and Eigenvectors 253


4.0 Introduction: A Dynamical System on Graphs 253

4.1 Introduction to Eigenvalues and Eigenvectors 254

4.2 Determinants 263

Writing Project: Which Came First: The Matrix or the Determinant?

Vignette: Lewis Carroll's Condensation Method 284

Exploration: Geometric Applications of Determinants 286

4.3 Eigenvalues and Eigenvectors of n X n Matrices 292

Writing Project: The History of Eigenvalues 301

3.4 Similarity and Diagonalization 301

3.5 Iterative Methods for Computing Eigenvalues 31 1

3.6 Applications and the Perron-Frobenius Theorem 325

Markov Chains 325

Population Growth 330

The Perron-Frobenius Theorem 332

Linear Recurrence Relations 335

Systems of Linear Differential Equations 340

Discrete Linear Dynamical Systems 348

Vignette: Ranking Sports Teams and Searching the Internet 356

Chapter Review 364


Chapter 5 Orthogonality 366


5.0 Introduction: Shadows on a Wall 366

5.1 Orthogonality in IR" 368

5.2 Orthogonal Complements and Orthogonal Projections 378

5.3 The Gram-Schmidt Process and the QR Factorization 388

Explorations: The Modified QR Factorization 396

Approximating Eigenvalues with the QR Algorithm 398

5.4 Orthogonal Diagonalization of Symmetric Matrices 400

5.5 Applications 408

Quadratic Forms 408

Graphing Quadratic Equations 415

Chapter Review 425


Chapter 6 Vector Spaces 427


6.0 Introduction: Fibonacci in (Vector) Space 427

6. 1 Vector Spaces and Subspaces 429

6.2 Writing Project: The Rise of Vector Spaces 443

6.3 Linear Independence, Basis, and Dimension 443

6.4 Exploration: Magic Squares 460

6.5 Change of Basis 463

6.6 Linear Transformations 472

The Kernel and Range of a Linear Transformation 481

The Matrix of a Linear Transformation 497

Exploration: Tilings, Lattices, and the Crystallographic Restriction

6.7 Applications 518

Homogeneous Linear Differential Equations 518

Chapter Review 527


Chapter 7 Distance and Approximation 529

7.0 Introduction: Taxicab Geometry 529

7. 1 Inner Product Spaces 531

Explorations: Vectors and Matrices with Complex Entries 543

Geometric Inequalities and Optimization Problems 547

7.2 Norms and Distance Functions 552

7.3 Least Squares Approximation 568

7.4 The Singular Value Decomposition 590

Vignette: Digital Image Compression 607

7.5 Applications 610

Approximation of Functions 610

Chapter Review 618


Chapter 8 Codes Online only 620


8. 1 Code Vectors 620

Vignette: The Codabar System 626

8.2 Error-Correcting Codes 627

8.3 Dual Codes 632

8.4 Linear Codes 639

8.5 The Minimum Distance of a Code 644


APPENDIX A Mathematical Notation and Methods of Proof Al

APPENDIX B Mathematical Induction B 1

APPENDIX C Complex Numbers Cl

APPENDIX D Polynomials D 1

APPENDIX E Technology Bytes Online only

Answers to Selected Odd-Numbered Exercises ANSI

Index II

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