MAT5173

Introduction to groups rings and fields.

Sample textbook:

[1] Thomas W. Judson and Robert A. Beezer, Abstract Algebra: Theory and Applications, 2008. Freely available online.

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 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	Groups	1.1-1.8	* Isomorphisms *Normal subgroups and factor groups	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.