| Chapter 1: Systems of Linear Equations |
| 1.1 | Solutions and Elementary Operations |
| 1.2 | Gaussian Elimination |
| 1.3 | Homogeneous Equations |
| 1.4 | An Application to Network Flow |
| 1.5 | An Application to Electrical Networks |
| 1.6 | An Application to Chemical Reactions |
| Supplementary Exercises for Chapter 1 |
Chapter 2: Matrix Algebra |
| 2.1 | Matrix Addition, Scalar Multiplication, and Transposition |
| 2.2 | Matrix Multiplication |
| 2.3 | Matrix Inverses |
| 2.4 | Elementary Matrices |
| 2.5 | LU-Factorization |
| 2.6 | An Application to Input-Output Economic Models |
| 2.7 | An Application to Markov Chains |
| Supplementary Exercises for Chapter 2 |
Chapter 3: Determinants and Diagonalization |
| 3.1 | The Laplace Expansion |
| 3.2 | Determinants and Matrix Inverses |
| 3.3 | Diagonalization and Eigenvalues |
| 3.4 | An Application to Polynomial Interpolation |
| 3.5 | An Application to Linear Recurrences |
| 3.6 | An Application to Population Growth |
| 3.7 | Proof of the Laplace Expansion |
| Supplementary Exercises for Chapter 3 |
Chapter 4: Vector Geometry |
| 4.1 | Vectors and Lines |
| 4.2 | The Dot Product and Projections |
| 4.3 | Planes and the Cross Product |
| 4.4 | An Application to Least Squares Approximation |
| Supplementary Exercises for Chapter 4 |
Chapter 5: The Vector Space Rn |
| 5.1 | Subspaces and Dimension |
| 5.2 | Rank of a Matrix |
| 5.3 | Similarity and Diagonalization |
| 5.4 | Linear Transformations |
| Supplementary Exercises for Chapter 5 |
Chapter 6: Vector Spaces |
| 6.1 | Examples and Basic Properties |
| 6.2 | Subspaces and Spanning Sets |
| 6.3 | Linear Independence and Dimension |
| 6.4 | Existence of Bases |
| 6.5 | An Application to Polynomials |
| 6.6 | An Application to Differential Equations |
| Supplementary Exercises for Chapter 6 |
Chapter 7: Orthogonality |
| 7.1 | Orthogonality in Rn |
| 7.2 | Orthogonal Diagonalization |
| 7.3 | Positive Definite Matrices |
| 7.4 | QR-Factorization |
| 7.5 | Computing Eigenvalues |
| 7.6 | Complex Matrices |
| 7.7 | An Application to Quadratic Forms |
| 7.8 | An Application to Best Approximation and Least Squares |
| 7.9 | An Application to Systems of Differential Equations |
Chapter 8: Linear Transformations |
| 8.1 | Examples and Elementary Properties |
| 8.2 | Kernel and Image of a Linear Transformation |
| 8.3 | Isomorphisms and Composition |
| 8.4 | The Matrix of a Linear Transformation |
| 8.5 | Change of Basis |
| 8.6 | Invariant Subspaces and Direct Sums |
| 8.7 | Block Triangular Form |
| 8.8 | More on Linear Recurrences |
Chapter 9: Inner Product Spaces |
| 9.1 | Inner Products and Norms |
| 9.2 | Orthogonal Sets of Vectors |
| 9.3 | Orthogonal Diagonalization |
| 9.4 | Isometries |
| 9.5 | An Application to Fourier Approximation |
Appendix A: Complex Numbers |
| Appendix B: Mathematical Induction |
2003 McGraw-Hill Higher Education