Elementary Linear Algebra

Cover......Page 1
Applications and Historical Topics......Page 3
Title Page......Page 7
Copyright......Page 8
About the Authors......Page 10
Preface......Page 11
Contents......Page 14
1: Systems of Linear Equations and Matrices......Page 15
1.1 Introduction to Systems of Linear Equations......Page 16
1.2 Gaussian Elimination......Page 25
1.3 Matrices and Matrix Operations......Page 39
1.4 Inverses; Algebraic Properties of Matrices......Page 54
1.5 Elementary Matrices and a Method for Finding A-1......Page 67
1.6 More on Linear Systems and Invertible Matrices......Page 76
1.7 Diagonal, Triangular, and Symmetric Matrices......Page 83
1.8 Introduction to Linear Transformations......Page 90
1.9 Compositions of Matrix Transformations......Page 104
Network Analysis......Page 112
Electrical Circuits......Page 114
Balancing Chemical Equations......Page 117
Polynomial Interpolation......Page 119
1.11 Leontief Input-Output Models......Page 124
2.1 Determinants by Cofactor Expansion......Page 132
2.2 Evaluating Determinants by Row Reduction......Page 140
2.3 Properties of Determinants; Cramer's Rule......Page 147
3.1 Vectors in 2-Space, 3-Space, and n-Space......Page 160
3.2 Norm, Dot Product, and Distance in Rn......Page 172
3.3 Orthogonality......Page 186
3.4 The Geometry of Linear Systems......Page 197
3.5 Cross Product......Page 204
4.1 Real Vector Spaces......Page 216
4.2 Subspaces......Page 225
4.3 Spanning Sets......Page 234
4.4 Linear Independence......Page 242
4.5 Coordinates and Basis......Page 252
4.6 Dimension......Page 262
4.7 Change of Basis......Page 270
4.8 Row Space, Column Space, and Null Space......Page 277
4.9 Rank, Nullity, and the Fundamental Matrix Spaces......Page 290
5.1 Eigenvalues and Eigenvectors......Page 305
5.2 Diagonalization......Page 315
5.3 Complex Vector Spaces......Page 325
5.4 Differential Equations......Page 337
5.5 Dynamical Systems and Markov Chains......Page 343
6.1 Inner Products......Page 355
6.2 Angle and Orthogonality in Inner Product Spaces......Page 366
6.3 Gram–Schmidt Process; QR-Decomposition......Page 375
6.4 Best Approximation; Least Squares......Page 390
6.5 Mathematical Modeling Using Least Squares......Page 399
6.6 Function Approximation; Fourier Series......Page 406
7.1 Orthogonal Matrices......Page 413
7.2 Orthogonal Diagonalization......Page 422
7.3 Quadratic Forms......Page 430
7.4 Optimization Using Quadratic Forms......Page 443
7.5 Hermitian, Unitary, and Normal Matrices......Page 450
8.1 General Linear Transformations......Page 460
8.2 Compositions and Inverse Transformations......Page 473
8.3 Isomorphism......Page 485
8.4 Matrices for General Linear Transformations......Page 491
8.5 Similarity......Page 501
8.6 Geometry of Matrix Operators......Page 507
9.1 LU-Decompositions......Page 523
9.2 The Power Method......Page 533
9.3 Comparison of Procedures for Solving Linear Systems......Page 542
9.4 Singular Value Decomposition......Page 546
9.5 Data Compression Using Singular Value Decomposition......Page 554
Appendix A: Working with Proofs......Page 559
Appendix B: Complex Numbers......Page 563
Answers to Exercises......Page 571
Index......Page 601
EULA......Page 611