1.1 | Introduction |
1.2 | Vector Spaces |
1.3 | Subspaces |
1.4 | Linear Combinations and Systems of Linear Equations |
1.5 | Linear Dependence and Linear Independence |
1.6 | Bases and Dimension |
1.7* | Maximal Linearly Independent Subsets |
Index of Definitions |
2.1 | Linear Transformations, Null Spaces, and Ranges |
2.2 | The Matrix Representation of a Linear Transformation |
2.3 | Composition of Linear Transformations and Matrix Multiplication |
2.4 | Invertibility and Isomorphisms |
2.5 | The Change of Coordinate Matrix |
2.6* | Dual Spaces |
2.7* | Homogeneous Linear Differential Equations with Constant Coefficients |
Index of Definitions |
3.1 | Elementary Matrix Operations and Elementary Matrices |
3.2 | The Rank of a Matrix and Matrix Inverses |
3.3 | Systems of Linear Equations---Theoretical Aspects |
3.4 | Systems of Linear Equations---Computational Aspects |
Index of Definitions |
4.1 | Determinants of Order 2 |
4.2 | Determinants of order n |
4.3 | Properties of Determinants |
4.4 | Summary---Important Facts about Determinants |
4.5* | A Characterization of the Determinant |
Index of Definitions |
5.1 | Eigenvalues and Eigenvectors |
5.2 | Diagonalizability |
5.3* | Matrix Limits and Markov Chains |
5.4 | Invariant Subspaces and the Cayley-Hamilton Theorem |
Index of Definitions |
6.1 | Inner Products and Norms |
6.2 | The Gram-Schmidt Orthogonalization Process and Orthogonal Complements |
6.3 | The Adjoint of a Linear Operator |
6.4 | Normal and Self-Adjoint Operators |
6.5 | Unitary and Orthogonal Operators and Their Matrices |
6.6 | Orthogonal Projections and the Spectral Theorem |
6.7* | The Singular Value Decomposition and the Pseudoinverse |
6.8* | Bilinear and Quadratic Forms |
6.9* | Einstein's Special Theory of Relativity |
6.10* | Conditioning and the Rayleigh Quotient |
6.11* | The Geometry of Orthogonal Operators |
Index of Definitions |
7.1 | Jordan Canonical Form I |
7.2 | Jordan Canonical Form II |
7.3 | The Minimal Polynomial |
7.4* | Rational Canonical Form |
Index of Definitions |
A | Sets |
B | Functions |
C | Fields |
D | Complex Numbers |
E | Polynomials |
* Sections denoted by an asterisk are optional.