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.

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