MAT4033 - Revision history

Jose.iovino at 18:12, 20 May 2025

2025-05-20T18:12:28Z

Jose.iovino at 00:30, 25 March 2023

2023-03-25T00:30:39Z

Jose.iovino: Created page with "Introduction to the theory of finite-dimensional vector spaces. '''Sample textbook''': [1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to..."

2023-03-25T00:29:47Z

Created page with "Introduction to the theory of finite-dimensional vector spaces. '''Sample textbook''': [1] M. Thamban Nair · Arindama Singh, ''Linear Algebra'', 2008. Freely available to..."

New page

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==
{| 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.
|}

@@ Line 1: / Line 1: @@
 Introduction to the theory of finite-dimensional vector spaces.
 '''Catalog entry'''
-''Prerequisite'':  Discrete Mathematics (MAT3003), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
+''Prerequisite'':  Prerequisite: Math 2233 Linear Algebra, Math 2243 Applied Linear Algebra or instructor approval.  , or instructor consent.
 ''Contents''
-(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
+(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

@@ 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