Difference between revisions of "MAT5283"

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(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
 
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(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.
 
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Introduction to the theory of finite-dimensional vector spaces.  
 
Introduction to the theory of finite-dimensional vector spaces.  
  
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'''Sample textbook''':
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== Sample textbook ==
  
 
[1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to UTSA students.
 
[1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to UTSA students.
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'''Catalog entry'''
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== Catalog entry ==
  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
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''Prerequisite'': Discrete Mathematics (MAT3003), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
  
 
''Contents''
 
''Contents''
 
(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
 
(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
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(2) Linear
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(2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis.  
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(3) Graph models: Isomorphisms, edge counting, planar graphs.
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(3) Gauss-Jordan elimination: Row operations, echelon forms, determinants.
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(4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem
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(3) Inner product spaces: Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
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(5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems.
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(4) Eigenvalues and eigenspaces: Characteristic polynomials, diagonalization.
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(6) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles.  
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(5) Jordan form, spectral representation.
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|| 1.1-1.8
 
|| 1.1-1.8
 
|| Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
 
|| Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
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|| MAT1313, CS2233/2231, or instructor consent.
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|| MAT3003, CS2233/2231, or instructor consent.
 
|-
 
|-
 
|  4-5   
 
|  4-5   
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|-
 
|-
 
|  6   
 
|  6   
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|| [[Gauss-jordan elimination]]
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|| [[Gauss-Jordan elimination]]
 
|| 3.1-3.7
 
|| 3.1-3.7
 
|| Row operations, echelon form and reduced echelon form, determinants.
 
|| Row operations, echelon form and reduced echelon form, determinants.

Latest revision as of 22:04, 25 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.
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