# MAT5283

Introduction to the theory of finite-dimensional vector spaces.

## Sample textbook

[1] M. Thamban Nair · Arindama Singh, *Linear Algebra*, 2008. Freely available to UTSA students.

## Catalog entry

*Prerequisite*: Discrete Mathematics (MAT3003), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

*Contents*
(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
(2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis.
(3) Gauss-Jordan elimination: Row operations, echelon forms, determinants.
(3) Inner product spaces: Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
(4) Eigenvalues and eigenspaces: Characteristic polynomials, diagonalization.
(5) Jordan form, spectral representation.

## Topics List

Week | Topic | Sections from the Nair-Singh book | Subtopics | Prerequisite |
---|---|---|---|---|

1-3 | Finite-dimensional vector spaces | 1.1-1.8 | Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces | MAT3003, CS2233/2231, or instructor consent. |

4-5 | Linear transformations | 2.1-2.6 | Rank and nullity, matrix representation, the space of linear transformations. | |

6 | Gauss-Jordan elimination | 3.1-3.7 | Row operations, echelon form and reduced echelon form, determinants. | |

7-8 | Inner product spaces | 4.1-4.8 | Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation. | |

9 | Eigenvalues and eigenvectors | 5.1-5.5 | Eigenspaces, characteristic polynomials | |

10 | Canonical forms | 6.1-6.5 | Jordan form | |

11-13 | Spectral representation | 7.1-7.6 | Singular value and polar decomposition. |