MAT5283

(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, canonical forms. (3) Inner product spaces: Norm, angles, orthogonal and orthonormal sets, orthogonal complement, best approximation and least squares, Riesz Representation Theorem. (4) Determinants: Axiomatic characterization of determinants, multilinear functions, existence and uniqueness of determinants. (5) Eigenvalues and eigenvectors: Characteristic polynomials, eigenspaces. (6) Block diagonal representation: Diagonalizability, triangularizability, block representation, Jordan normal form. (6) The Spectral Theorem.

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: Algebra and Number Systems (MAT 1313), 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 (3) Graph models: Isomorphisms, edge counting, planar graphs. (4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem (5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems. (6) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles.

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	MAT1313, 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.