A Brief Course in Linear Algebra [Source Code]

Preface i
Chapter 1. Linear Algebra, Basic Notions 1
1. Introduction 1
2. Matrix Algebra 4
3. Formal Rules 12
4. Linear Systems of Algebraic Equations 14
5. Singularity, Pivots, and Invertible Matrices 22
6. Gauss-Jordan Reduction in the General Case 33
7. Homogeneous Systems and Vector Subspaces 42
8. Linear Independence, Bases, and Dimension 47
9. Calculations in R n 57
10. Review Problems 62
Chapter 2. Determinants and Eigenvalues 65
1. Introduction 65
2. Definition of the Determinant 68
3. Some Important Properties of Determinants 76
4. Eigenvalues and Eigenvectors 83
5. Diagonalization 93
6. The Exponential of a Matrix 97
7. Review 100
Chapter 3. Applications 103
1. Real Symmetric Matrices 103
2. Repeated Eigenvalues, The Gram–Schmidt Process 105
3. Change of Coordinates 109
4. Classification of Conics and Quadrics 116
5. Conics and the Method of Lagrange Multipliers 124
6. Normal Modes 129
7. Review 136
Solutions to Problems 139
Index 165
Appendix A. GNU Free Documentation License 167