Elementary Linear Algebra

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 40

1.5 Elementary Matrices and a Method for Finding A⁻¹ 53

1.6 More on Linear Systems and Invertible Matrices 62

1.7 Diagonal, Triangular, and Symmetric Matrices 69

1.8 Introduction to Linear Transformations 76

1.9 Compositions of Matrix Transformations 90

1.10 Applications of Linear Systems 98

• Network Analysis 98

• Electrical Circuits 100

• Balancing Chemical Equations 103

• Polynomial Interpolation 105

1.11 Leontief Input-Output Models 110

2 Determinants 118

2.1 Determinants by Cofactor Expansion 118

2.2 Evaluating Determinants by Row Reduction 126

2.3 Properties of Determinants; Cramer’s Rule 133

3 Euclidean Vector Spaces 146

3.1 Vectors in 2-Space, 3-Space, and n-Space 146

3.2 Norm, Dot Product, and Distance in Rⁿ 158

3.3 Orthogonality 172

3.4 The Geometry of Linear Systems 183

3.5 Cross Product 190

4 General Vector Spaces 202

4.1 Real Vector Spaces 202

4.2 Subspaces 211

4.3 Spanning Sets 220

4.4 Linear Independence 228

4.5 Coordinates and Basis 238

4.6 Dimension 248

4.7 Change of Basis 256

4.8 Row Space, Column Space, and Null Space 263

4.9 Rank, Nullity, and the Fundamental Matrix Spaces 276

5 Eigenvalues and Eigenvectors 291

5.1 Eigenvalues and Eigenvectors 291

5.2 Diagonalization 301

5.3 Complex Vector Spaces 311

5.4 Differential Equations 323

5.5 Dynamical Systems and Markov Chains 329

6 Inner Product Spaces 341

6.1 Inner Products 341

6.2 Angle and Orthogonality in Inner Product Spaces 352

6.3 Gram–Schmidt Process; QR-Decomposition 361

6.4 Best Approximation; Least Squares 376

6.5 Mathematical Modeling Using Least Squares 385

6.6 Function Approximation; Fourier Series 392

7 Diagonalization and Quadratic Forms 399

7.1 Orthogonal Matrices 399

7.2 Orthogonal Diagonalization 408

7.3 Quadratic Forms 416

7.4 Optimization Using Quadratic Forms 429

7.5 Hermitian, Unitary, and Normal Matrices 436

8 General Linear Transformations 446

8.1 General Linear Transformations 446

8.2 Compositions and Inverse Transformations 459

8.3 Isomorphism 471

8.4 Matrices for General Linear Transformations 477

8.5 Similarity 487

8.6 Geometry of Matrix Operators 493

9 Numerical Methods 509

9.1 LU-Decompositions 509

9.2 The Power Method 519

9.3 Comparison of Procedures for Solving Linear Systems 528

9.4 Singular Value Decomposition 532

9.5 Data Compression Using Singular Value Decomposition 540

10 Applications of Linear Algebra 545

10.1 Constructing Curves and Surfaces Through Specified Points 546

10.2 The Earliest Applications of Linear Algebra 551

10.3 Cubic Spline Interpolation 558

10.4 Markov Chains 568

10.5 Graph Theory 577

10.6 Games of Strategy 587

10.7 Forest Management 595

10.8 Computer Graphics 602

10.9 Equilibrium Temperature Distributions 610

10.10 Computed Tomography 619

10.11 Fractals 629

10.12 Chaos 645

10.13 Cryptography 658

10.14 Genetics 669

10.15 Age-Specific Population Growth 678

10.16 Harvesting of Animal Populations 687

10.17 A Least Squares Model for Human Hearing 695

10.18 Warps and Morphs 701

10.19 Internet Search Engines 710

10.20 Facial Recognition 716

Supplemental Online Topics

• Linear Programming - A Geometric Approach

• Linear Programming - Basic Concepts

• Linear Programming - The Simplex Method

• Vectors in Plane Geometry

• Equilibrium of Rigid Bodies

• The Assignment Problem

• The Determinant Function

• Leontief Economic Models

Appendix A Working with Proofs A1

Appendix B Complex Numbers A5

Answers to Exercises A13

Index I1