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 119
2.1 Determinants by Cofactor Expansion 119
2.2 Evaluating Determinants by Row Reduction 127
2.3 Properties of Determinants; Cramer’s Rule 134
3 Euclidean Vector Spaces 149
3.1 Vectors in 2-Space, 3-Space, and n-Space 149
3.2 Norm, Dot Product, and Distance in R n 161
3.3 Orthogonality 175
3.4 The Geometry of Linear Systems 186
3.5 Cross Product 193
4 General Vector Spaces 205
4.1 Real Vector Spaces 205
4.2 Subspaces 214
4.3 Spanning Sets 223
4.4 Linear Independence 231
4.5 Coordinates and Basis 241
4.6 Dimension 251
4.7 Change of Basis 259
4.8 Row Space, Column Space, and Null Space 266
4.9 Rank, Nullity, and the Fundamental Matrix Spaces 279
5 Eigenvalues and Eigenvectors 295
5.1 Eigenvalues and Eigenvectors 295
5.2 Diagonalization 305
5.3 Complex Vector Spaces 315
5.4 Differential Equations 327
5.5 Dynamical Systems and Markov Chains 333
6 Inner Product Spaces 347
6.1 Inner Products 347
6.2 Angle and Orthogonality in Inner Product Spaces 358
6.3 Gram–Schmidt Process; QR-Decomposition 367
6.4 Best Approximation; Least Squares 382
6.5 Mathematical Modeling Using Least Squares 391
6.6 Function Approximation; Fourier Series 398
7 Diagonalization and Quadratic Forms 407
7.1 Orthogonal Matrices 407
7.2 Orthogonal Diagonalization 416
7.3 Quadratic Forms 424
7.4 Optimization Using Quadratic Forms 437
7.5 Hermitian, Unitary, and Normal Matrices 444
8 General Linear Transformations 455
8.1 General Linear Transformations 455
8.2 Compositions and Inverse Transformations 468
8.3 Isomorphism 480
8.4 Matrices for General Linear Transformations 486
8.5 Similarity 496
8.6 Geometry of Matrix Operators 502
9 Numerical Methods 519
9.1 LU-Decompositions 519
9.2 The Power Method 529
9.3 Comparison of Procedures for Solving Linear Systems 538
9.4 Singular Value Decomposition 542
9.5 Data Compression Using Singular Value Decomposition 550
10 Applications of Linear Algebra 555
10.1 Constructing Curves and Surfaces Through Specified Points 555
10.2 The Earliest Applications of Linear Algebra 561
10.3 Cubic Spline Interpolation 568
10.4 Markov Chains 578
10.5 Graph Theory 587
10.6 Games of Strategy 597
10.7 Forest Management 605
10.8 Computer Graphics 612
10.9 Equilibrium Temperature Distributions 620
10.10 Computed Tomography 629
10.11 Fractals 639
10.12 Chaos 655
10.13 Cryptography 668
10.14 Genetics 679
10.15 Age-Specific Population Growth 688
10.16 Harvesting of Animal Populations 697
10.17 A Least Squares Model for Human Hearing 705
10.18 Warps and Morphs 711
10.19 Internet Search Engines 720
10.20 Facial Recognition 726
Appendix A Working with Proofs A1
Appendix B Complex Numbers A7
Answers to Exercises A15
Index i1
