CHAPTER 1 Systems of Linear Equations and Matrices 1

1.1 Introduction to Systems of Linear Equations 2

1.2 Gaussian Elimination 11

1.3 Matrices and Matrix Operations 25

1.4 Inverses; Algebraic Properties of Matrices 39

1.5 Elementary Matrices and a Method for Finding A^-1 52

1.6 More on Linear Systems and Invertible Matrices 61

1.7 Diagonal, Triangular, and Symmetric Matrices 67

1.8 Matrix Transformations 75

1.9 Applications of Linear Systems 84

Network Analysis (Traffic Flow) 84

Electrical Circuits 86

Balancing Chemical Equations 88

Polynomial Interpolation 91

1.10 Application:Leontief Input-Output Models 96

CHAPTER 2 Determinants 105

2.1 Determinants by Cofactor Expansion 105

2.2 Evaluating Determinants by Row Reduction 113

2.3 Properties of Determinants; Cramer’s Rule 118

CHAPTER 3 Euclidean Vector Spaces 131

3.1 Vectors in 2-Space, 3-Space, and n-Space 131

3.2 Norm, Dot Product, and Distance in Rⁿ 142

3.3 Orthogonality 155       

3.4The Geometry of Linear Systems 164

3.5 Cross Product 172

CHAPTER 4 General Vector Spaces 83

4.1 Real Vector Spaces 183

4.2 Subspaces 191

4.3 Linear Independence 202

4.4 Coordinates and Basis 212

4.5 Dimension 221

4.6 Change of Basis 229

4.7 Row Space, Column Space, and Null Space 237

4.8 Rank, Nullity, and the Fundamental Matrix Spaces 248

4.9 Basic Matrix Transformations in R² and R³ 259

4.10 Properties of Matrix Transformations 270

4.11Application: Geometry of Matrix Operators on R² 280

CHAPTER 5 Eigenvalues and Eigenvectors 291

5.1 Eigenvalues and Eigenvectors 291

5.2 Diagonalization 302

5.3 Complex Vector Spaces 313

5.4 Application: Differential Equations 326

5.5 Application: Dynamical Systems and Markov Chains 332

CHAPTER 6 Inner Product Spaces 345

6.1 Inner Products 345

6.2 Angle and Orthogonality in Inner Product Spaces 355

6.3 Gram-Schmidt Process; QR-Decomposition 364

6.4 Best Approximation; Least Squares 378

6.5 Application: Mathematical Modeling Using Least Squares 387

6.6 Application: Function Approximation; Fourier Series 394

CHAPTER 7 Diagonalization and Quadratic Forms 401

7.1 Orthogonal Matrices 401

7.2 Orthogonal Diagonalization 409

7.3 Quadratic Forms 417

7.4 Optimization Using Quadratic Forms 429

7.5 Hermitian, Unitary, and Normal Matrices 437

CHAPTER 8 General Linear Transformations 447

8.1 General Linear Transformations 447

8.2 Compositions and Inverse Transformations 458

8.3 Isomorphism 466

8.4 Matrices for General Linear Transformations 472

8.5 Similarity 481

CHAPTER 9 Numerical Methods 491

9.1 LU-Decompositions 491

9.2 The Power Method 501

9.3 Comparison of Procedures for Solving Linear Systems 509

9.4 Singular Value Decomposition 514

9.5 Application: Data Compression Using Singular Value Decomposition 521

CHAPTER 10 Applications of Linear Algebra 527

10.1 Constructing Curves and Surfaces Through Specified Points 528

10.2 The Earliest Applications of Linear Algebra 533

10.3 Cubic Spline Interpolation 540

10.4 Markov Chains 551

10.5 Graph Theory 561

10.6 Games of Strategy 570

10.7 Leontief Economic Models 579

10.8 Forest Management 588

10.9 Computer Graphics 595

10.10 Equilibrium Temperature Distributions 603

10.11 Computed Tomography 613

10.12 Fractals 624

10.13 Chaos 639

10.14 Cryptography 652

10.15 Genetics 663

10.16 Age-Specific Population Growth 673

10.17 Harvesting of Animal Populations 683

10.18 A Least Squares Model for Human Hearing 691

10.19 Warps and Morphs 697

10.20 Internet Search Engines 706

APPENDIX A Working with Proofs A1

APPENDIX B Complex Numbers A5

Answers to exercises A13

Index I1

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