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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
3.4 The Geometry of Linear Systems 164
3.5 Cross Product 172
Chapter 4 General Vector Spaces 183
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
4.11 Application: 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 Transformation 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
Appendix A Working with Proofs A1
Appendix B Complex Numbers A5
Answers to Exercises A13
Index I1
