MAT5283 - Revision history

Jose.iovino at 01:48, 25 March 2026

2026-03-25T01:48:28Z

Jose.iovino at 23:06, 24 March 2026

2026-03-24T23:06:09Z

Jose.iovino at 18:13, 20 May 2025

2025-05-20T18:13:56Z

Jose.iovino at 18:13, 20 May 2025

2025-05-20T18:13:36Z

Jose.iovino at 04:04, 26 March 2023

2023-03-26T04:04:39Z

Jose.iovino at 00:31, 25 March 2023

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Jose.iovino at 00:29, 25 March 2023

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Jose.iovino at 02:00, 19 March 2023

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Jose.iovino at 01:55, 19 March 2023

2023-03-19T01:55:05Z

Jose.iovino at 20:47, 9 March 2023

2023-03-09T20:47:55Z

@@ Line 5: / Line 5: @@
 ''Prerequisite'':  Prerequisite: Math 2233 Linear Algebra, Math 2243 Applied Linear Algebra or instructor approval.  , or instructor consent.
-''Contents''
+''Contents'':
 (1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector
 spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomor-

← Older revision		Revision as of 23:06, 24 March 2026
Line 6:		Line 6:

	''Contents''		''Contents''
−	(This course has been designed for advanced undergraduate students and first year graduate students. The subjects have been developed to include material that is fundamental for contemporary applications of linear algebra like Error-Correcting Codes, Cryptography, Quantum Computing and Quantum Algorithms. Topics include: (1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, ~~isomorphisms~~, Rank-~~Nulity~~ Theorem, Product and Quotient Spaces (3) Matrices and Linear ~~Transformations~~: Matrix representation, change of basis, equivalence of matrices, similarity (4) ~~Eigenvalues~~ and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan ~~Canonical~~ forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz ~~inequality~~, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez ~~Representation~~ Theorem (5) Orthogonality over finite fields	+	(1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector
		+	spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomor-
		+	phisms, Rank-Nullity Theorem, Product and Quotient Spaces (3) Matrices and Linear Transfor-
		+	mations: Matrix representation, change of basis, equivalence of matrices, similarity (4) Eigen-
		+	values and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan Canoni-
		+	cal forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz in-
		+	equality, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez Representa-
		+	tion Theorem (5) Orthogonality over finite fields (6) Multilinear algebra : Bilinear Forms and
		+	Functionals, Tensor product of vector spaces and Matrices.

← Older revision		Revision as of 18:13, 20 May 2025
Line 7:		Line 7:
	''Contents''		''Contents''
	(This course has been designed for advanced undergraduate students and first year graduate students. The subjects have been developed to include material that is fundamental for contemporary applications of linear algebra like Error-Correcting Codes, Cryptography, Quantum Computing and Quantum Algorithms. Topics include: (1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomorphisms, Rank-Nulity Theorem, Product and Quotient Spaces (3) Matrices and Linear Transformations: Matrix representation, change of basis, equivalence of matrices, similarity (4) Eigenvalues and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan Canonical forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz inequality, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez Representation Theorem (5) Orthogonality over finite fields		(This course has been designed for advanced undergraduate students and first year graduate students. The subjects have been developed to include material that is fundamental for contemporary applications of linear algebra like Error-Correcting Codes, Cryptography, Quantum Computing and Quantum Algorithms. Topics include: (1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomorphisms, Rank-Nulity Theorem, Product and Quotient Spaces (3) Matrices and Linear Transformations: Matrix representation, change of basis, equivalence of matrices, similarity (4) Eigenvalues and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan Canonical forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz inequality, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez Representation Theorem (5) Orthogonality over finite fields
−
−
−
−
−
−
−
−
−
−
−	~~==Topics List==~~
−	~~{\| class="wikitable sortable"~~
−	~~! 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.~~
−	\|}

@@ Line 1: / Line 1: @@
 Introduction to the theory of finite-dimensional vector spaces.
-== Sample textbook ==
+'''Catalog entry'''
-[1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to UTSA students.
+''Prerequisite'':  Prerequisite: Math 2233 Linear Algebra, Math 2243 Applied Linear Algebra or instructor approval.  , or instructor consent.

@@ Line 1: / Line 1: @@
 Introduction to the theory of finite-dimensional vector spaces.
-'''Sample textbook''':
+== Sample textbook ==
 [1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to UTSA students.
@@ Line 7: / Line 7: @@
-'''Catalog entry'''
+== Catalog entry ==
 ''Prerequisite'':  Discrete Mathematics (MAT3003), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

@@ Line 32: / Line 32: @@
 || 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.
+|| MAT3003, CS2233/2231, or instructor consent.
 |-
 |  4-5

@@ Line 1: / Line 1: @@
 Introduction to the theory of finite-dimensional vector spaces.
@@ Line 16: / Line 13: @@
 ''Contents''
 (1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
-(2) Linear
+(2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis.
-(3) Graph models: Isomorphisms, edge counting, planar graphs.
+(3) Gauss-Jordan elimination: Row operations, echelon forms, determinants.
-(4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem
+(3) Inner product spaces: Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
-(5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems.
+(4) Eigenvalues and eigenspaces: Characteristic polynomials, diagonalization.
-(6) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles.
+(5) Jordan form, spectral representation.
@@ Line 44: / Line 40: @@
 |-
 |  6
-|| [[Gauss-jordan elimination]]
+|| [[Gauss-Jordan elimination]]
 || 3.1-3.7
 || Row operations, echelon form and reduced echelon form, determinants.

← Older revision		Revision as of 01:55, 19 March 2023
Line 1:		Line 1:
−	(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.	+	(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==
		+	{\| class="wikitable sortable"
		+	! 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.
		+	\|}

← Older revision		Revision as of 20:47, 9 March 2023
Line 1:		Line 1:
−	~~A study~~ of ~~linear algebraic structures that may include linear transformations~~, ~~inner~~ product spaces, ~~eigenvalues~~, ~~Cayley-Hamilton theorem~~, ~~similarity~~, ~~the~~ Jordan ~~canonical~~ form~~, spectral theorem for normal transformation and applications~~.	+	(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.