# 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.