Difference between revisions of "MAT5173"

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Introduction to groups rings and fields.  
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Introduction to groups and rings.  
  
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'''Sample textbook''':
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== Sample textbook ==
  
 
[1] Thomas W. Judson and Robert A. Beezer, ''Abstract Algebra: Theory and Applications'', 2008. [http://abstract.ups.edu/aata/aata.html Freely available online].
 
[1] Thomas W. Judson and Robert A. Beezer, ''Abstract Algebra: Theory and Applications'', 2008. [http://abstract.ups.edu/aata/aata.html Freely available online].
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'''Catalog entry'''
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== Catalog entry ==
  
 
''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
 
''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
  
 
''Contents''
 
''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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(1) Groups: Cyclic groups, permutation groups and Cayley's theorem, group homomorphisms, normal subroups, quotient groups and Lagrange's theore, the theorems of Euler and Fermat.
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(2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis.  
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(2) Rings: Ring homomorphisms, integral domains and fields, maximal and prime ideals.  
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(3) Gauss-Jordan elimination: Row operations, echelon forms, determinants.
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(3) Rings of polynomials: The Division Algorithm and irreducible polynomials.
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(3) Inner product spaces: Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
 
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(4) Eigenvalues and eigenspaces: Characteristic polynomials, diagonalization.
 
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(5) Jordan form, spectral representation.
 
  
  
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==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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! Week !! Topic !! Sections from the Nair-Singh book !! Subtopics !! Prerequisite
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! Week !! Topic !! Sections from the Judson-Beezer book !! Subtopics !! Prerequisite
 
|-
 
|-
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|  1-3    
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|  1-2    
 
|| [[Groups]]
 
|| [[Groups]]
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|| 1.1-1.8
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|| 3
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|| * Isomorphisms
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||  
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* Normal subgroups and factor groups
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* Definitions and classical examples
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* Subgroups
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* Isomorphisms
 
|| MAT1313, CS2233/2231, or instructor consent.
 
|| MAT1313, CS2233/2231, or instructor consent.
 
|-
 
|-
 
|  4-5   
 
|  4-5   
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|| [[Linear transformations]]
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|| [[Cyclic groups]]
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|| 2.1-2.6
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|| 4
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|| Rank and nullity, matrix representation, the space of linear transformations.
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||  
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* Classification of cyclic groups.
 
|-
 
|-
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|  6   
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|  5-6   
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|| [[Gauss-Jordan elimination]]
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|| [[Permutation groups]]
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|| 3.1-3.7
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|| 5
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|| Row operations, echelon form and reduced echelon form, determinants.
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||  
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* Permutations
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*Cayley's Theorem
 
|-
 
|-
 
|  7-8   
 
|  7-8   
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|| [[Inner product spaces]]
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|| [[Cosets and Lagrange's Theorem]]
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|| 4.1-4.8
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|| 10
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|| Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
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||  
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* Normal subgroups
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* Factor Groups
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* The theorems of Euler and Fermat
 
|-
 
|-
 
|  9   
 
|  9   
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|| [[Eigenvalues and eigenvectors]]
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|| [[Homomorphisms]]
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|| 5.1-5.5
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|| 11
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|| Eigenspaces, characteristic polynomials
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|| The Isomorphism Theorem
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|
 
|-
 
|-
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|  10   
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|  10-11    
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|| [[Canonical forms]]  
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|| [[Rings]]  
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|| 6.1-6.5
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|| 16
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|| Jordan form
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||  
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* Ring homomorphisms
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*Integral domains and fields
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*Maximal and Prime Ideals
 
|-
 
|-
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|  11-13    
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|  12-end    
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|| [[Spectral representation]]  
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|| [[Rings of Polynomials]]  
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|| 7.1-7.6
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|| 17
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|| Singular value and polar decomposition.
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||  
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* The Division Algorithm
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* Irreducible Polynomials
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* Solving cubic and quartic equations
 
|}
 
|}

Latest revision as of 22:02, 25 March 2023

Introduction to groups and rings.

Sample textbook

[1] Thomas W. Judson and Robert A. Beezer, Abstract Algebra: Theory and Applications, 2008. Freely available online.


Catalog entry

Prerequisite: Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

Contents (1) Groups: Cyclic groups, permutation groups and Cayley's theorem, group homomorphisms, normal subroups, quotient groups and Lagrange's theore, the theorems of Euler and Fermat. (2) Rings: Ring homomorphisms, integral domains and fields, maximal and prime ideals. (3) Rings of polynomials: The Division Algorithm and irreducible polynomials.




Topics List

Week Topic Sections from the Judson-Beezer book Subtopics Prerequisite
1-2 Groups 3
  • Definitions and classical examples
  • Subgroups
  • Isomorphisms
 MAT1313, CS2233/2231, or instructor consent.
4-5 Cyclic groups 4
  • Classification of cyclic groups.
5-6 Permutation groups 5
  • Permutations
  • Cayley's Theorem
7-8 Cosets and Lagrange's Theorem 10
  • Normal subgroups
  • Factor Groups
  • The theorems of Euler and Fermat
9 Homomorphisms 11 The Isomorphism Theorem
10-11 Rings 16
  • Ring homomorphisms
  • Integral domains and fields
  • Maximal and Prime Ideals
12-end Rings of Polynomials 17
  • The Division Algorithm
  • Irreducible Polynomials
  • Solving cubic and quartic equations
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