Includes index.

Linear Systems and Matrices 1

1.1 Linear Systems of Equations 1
1.2 Elementary Operations and Gauss Elimination Method 5
1.3 Homogeneous Linear Systems 11
1.4 Introduction to Matrices and the Matrix of a Linear System 16
1.5 Elementary Row Operations on a Matrix 21
1.6 Proofs of Facts 28
1.7 Chapter Review Questions and Project 29

Algebra of Matrices 33

2.1 Scalar Multiplication and Addition of Matrices 33
2.2 Matrix Multiplication and Its Properties 38
2.3 Transpose 49
2.4 Proofs of Facts 52
2.5 Chapter Review Questions and Projects 55

Subspaces 61

3.1 Linear Combination of Vectors 61
3.2 Vector Subspaces 66
3.3 Linear Dependence, Linear Independence, and Basis 75
3.4 Proofs of Facts 84
3.5 Chapter Review Questions and Project 87

Rank 91

4.1 Elementary Operations and Rank 91
4.2 Null Space and Nullity of a Matrix 97
4.3 Elementary Matrices 103
4.4 Proofs of Facts 108
4.5 Chapter Review Questions and Project 111

Inverse, Rank Factorization, and LU-Decomposition 115

5.1 Inverse of a Matrix and Its Properties 115
5.2 Further Properties of Inverses 121
5.3 Full-Rank Factorization 122





5.4 LU-Decomposition of a Matrix 126
5.5 Proofs of Facts 129
5.6 Chapter Review Questions and Projects 130

Determinants 133

6.1 Determinant 133
6.2 Properties of the Determinant 139
6.3 Cofactors and Inverse of a Matrix 143
6.4 Cramer's Rule 146
6.5 Chapter Review Questions and Projects 148

Eigenvalue Problems 151

7.1 Eigenvalues and Eigenvectors 151
7.2 Characteristic Polynomial 152
7.3 Calculating Eigenvalues and Eigenvectors (Another Approach) and
The Cayley-Hamilton Theorem 159
7.4 Applications of the Cayley-Hamilton Theorem 165
7.5 Properties of Eigenvalues, Diagonalizability,
And Triangularizability 173
7.6 Proofs of Facts 179
7.7 Chapter Review Questions and Projects 183

Inner Product Spaces 187

8.1 Gram-Schmidt Orthogonalization Process 187
8.2 Diagonalization of Symmetric Matrices 193
8.3 Application of the Spectral Theorem 200
8.4 Least-Squares Solution 202
8.5 Generalized Inverse and Least-Squares Solution 209
8.6 Proofs of Facts 212
8.7 Chapter Review Questions and Projects 215

Vector Spaces and Linear Mappings 221

9.1 Vector Spaces 221
9.2 Linear Dependence and Linear Independence 225
9.3 Linear Mappings 226
9.4 Some Properties of Linear Mappings: Image and Kernel 233
9.5 Linear Mappings and Matrices 238
9.6 Chapter Review Questions and Projects 243




Determinants (Revisited) 247

10.1 Permutations 247
10.2 Determinants 251
10.3 Cofactor Expansion 257
10.4 Adjoint of a Matrix 261
10.5 Cramer's Rule 263
10.6 Product Theorem of Determinants 264

Answers and Hints to Selected Exercises 269
Drill Solutions Using Matlab 299
Some Basic Matlab Operations 399
Index 413