Difference between revisions of "MAT5173"

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|| [[Groups]]
 
|| [[Groups]]
 
|| 1.1-1.8
 
|| 1.1-1.8
|| * Isomorphisms *Normal subgroups and factor groups
+
|| * Isomorphisms  
 +
* Normal subgroups and factor groups
  
 
|| MAT1313, CS2233/2231, or instructor consent.
 
|| MAT1313, CS2233/2231, or instructor consent.

Revision as of 08:11, 19 March 2023

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.