Difference between revisions of "MAT5283"

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(1) Vector spaces: Abstract vector spaces, subspaces, bases, dimension, sums and direct sums. (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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(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.
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'''Sample textbook''':
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[1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to UTSA students.
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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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''Contents''
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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
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(3) Graph models: Isomorphisms, edge counting, planar graphs.
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(4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem
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(5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems.
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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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==Topics List==
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{| class="wikitable sortable"
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! Week !! Topic !! Sections from the Nair-Singh book !! Subtopics !! Prerequisite
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|-
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|  1-3 
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|| [[Finite-dimensional vector spaces]]
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|| 1.1-1.8
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|| 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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|-
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|  4-5 
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|| [[Linear transformations]]
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|| 2.1-2.6
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|| Rank and nullity, matrix representation, the space of linear transformations.
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|-
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|  6 
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|| [[Gauss-jordan elimination]]
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|| 3.1-3.7
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|| Row operations, echelon form and reduced echelon form, determinants.
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|-
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|  7-8 
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|| [[Inner product spaces]]
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|| 4.1-4.8
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|| Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
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|-
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|  9 
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|| [[Eigenvalues and eigenvectors]]
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|| 5.1-5.5
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|| Eigenspaces, characteristic polynomials
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|-
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|  10 
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|| [[Canonical forms]]
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|| 6.1-6.5
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|| Jordan form
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|-
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|  11-13 
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|| [[Spectral representation]]
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|| 7.1-7.6
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|| Singular value and polar decomposition.
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|}

Revision as of 19:55, 18 March 2023

(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.