Linear Algebra with Applications

Title......Page 2
Copyright......Page 5
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Acknowledgments......Page 26
1 Systems of Linear Equations......Page 33
1.1 Lines and Linear Equations......Page 36
1.2 Linear Systems and Matrices......Page 68
1.3 Applications of Linear Systems*......Page 99
1.4 Numerical Solutions*......Page 117
Supplementary Exercises......Page 136
2 Euclidean Space......Page 139
2.1 Vectors......Page 142
2.2 Span......Page 163
2.3 Linear Independence......Page 189
Supplementary Exercises......Page 214
3 Matrices......Page 217
3.1 Linear Transformations......Page 220
3.2 Matrix Algebra......Page 248
3.3 Inverses......Page 284
3.4 LU Factorization*......Page 309
3.5 Markov Chains*......Page 330
Supplementary Exercises......Page 347
4 Subspaces......Page 351
4.1 Introduction to Subspaces......Page 354
4.2 Basis and Dimension......Page 377
4.3 Row and Column Spaces......Page 402
4.4 Change of Basis*......Page 418
Supplementary Exercises......Page 431
5 Determinants......Page 434
5.1 The Determinant Function......Page 437
5.2 Properties of the Determinant......Page 461
5.3 Applications of the Determinant*......Page 481
Supplementary Exercises......Page 505
6 Eigenvalues and Eigenvectors......Page 508
6.1 Eigenvalues and Eigenvectors......Page 512
6.2 Diagonalization......Page 532
6.3 Complex Eigenvalues and Eigenvectors*......Page 548
6.4 Systems of Differential Equations*......Page 566
6.5 Approximation Methods*......Page 580
Supplementary Exercises......Page 591
7 Vector Spaces......Page 594
7.1 Vector Spaces and Subspaces......Page 598
7.2 Span and Linear Independence......Page 619
7.3 Basis and Dimension......Page 636
Supplementary Exercises......Page 652
8 Orthogonality......Page 654
8.1 Dot Products and Orthogonal Sets......Page 658
8.2 Projection and the Gram–Schmidt Process......Page 683
8.3 Diagonalizing Symmetric Matrices and QR Factorization......Page 702
8.4 The Singular Value Decomposition*......Page 716
8.5 Least Squares Regression*......Page 729
Supplementary Exercises......Page 748
9 Linear Transformations......Page 752
9.1 Definition and Properties......Page 756
9.2 Isomorphisms......Page 774
9.3 The Matrix of a Linear Transformation......Page 789
9.4 Similarity......Page 802
Supplementary Exercises......Page 816
10 Inner Product Spaces......Page 819
10.1 Inner Products......Page 824
10.2 The Gram–Schmidt Process Revisited......Page 845
10.3 Applications of Inner Products*......Page 866
Supplementary Exercises......Page 887
11 Additional Topics and Applications*......Page 890
11.1 Quadratic Forms......Page 894
11.2 Positive Definite Matrices......Page 909
11.3 Constrained Optimization......Page 921
11.4 Complex Vector Spaces......Page 934
11.5 Hermitian Matrices......Page 948
Supplementary Exercises......Page 959
Glossary......Page 961
Solutions to Practice Problems......Page 990
Answers to Selected Exercises......Page 1043
Index......Page 1136
Applications Index......Page 1188