Difference between revisions of "MAT4033"
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Latest revision as of 18:30, 24 March 2023
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. |