MAT5173 - Revision history

Jose.iovino at 04:02, 26 March 2023

2023-03-26T04:02:46Z

Jose.iovino at 14:47, 19 March 2023

2023-03-19T14:47:28Z

Jose.iovino: /* Topics List */

2023-03-19T14:45:12Z

Topics List

Jose.iovino at 14:44, 19 March 2023

2023-03-19T14:44:33Z

Jose.iovino at 14:43, 19 March 2023

2023-03-19T14:43:20Z

Jose.iovino at 14:11, 19 March 2023

2023-03-19T14:11:40Z

Jose.iovino at 14:10, 19 March 2023

2023-03-19T14:10:56Z

Jose.iovino: Created page with "The opportunity for development of basic theory of algebraic structures. Areas of study may include monoids, semigroups, groups, isomorphism theorems, free groups, group exten..."

2023-03-10T02:38:32Z

Created page with "The opportunity for development of basic theory of algebraic structures. Areas of study may include monoids, semigroups, groups, isomorphism theorems, free groups, group exten..."

New page

The opportunity for development of basic theory of algebraic structures. Areas of study may include monoids, semigroups, groups, isomorphism theorems, free groups, group extensions and group actions, Sylow theorems, group chains and composition series, nilpotent and solvable groups, cohomology of groups.

@@ Line 1: / Line 1: @@
 Introduction to groups and rings.
-'''Sample textbook''':
+== 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].
@@ Line 7: / Line 7: @@
-'''Catalog entry'''
+== Catalog entry ==
 ''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

← Older revision		Revision as of 14:47, 19 March 2023
Line 1:		Line 1:
−	Introduction to groups rings ~~and fields~~.	+	Introduction to groups and rings.

	'''Sample textbook''':		'''Sample textbook''':

@@ Line 46: / Line 46: @@
 * Permutations
 *Cayley's Theorem
+|-
 |  7-8
 || [[Cosets and Lagrange's Theorem]]

@@ Line 12: / Line 12: @@
 ''Contents''
-(1) Groups: Cyclic groups, permutation groups and Cayleys theorem, group homomorphisms, normal subroups, quotient groups and Lagrange's theore, the theorems of Euler and Fermat.
+(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:
+(3) Rings of polynomials: The Division Algorithm and irreducible polynomials.

@@ Line 12: / Line 12: @@
 ''Contents''
-(1) Finite-dimensional vector spaces: Vector space axioms, subspaces, linear independence and bases, dimension, sums and quotients of vector spaces
+(1) Groups: Cyclic groups, permutation groups and Cayleys theorem, group homomorphisms, normal subroups, quotient groups and Lagrange's theore, the theorems of Euler and Fermat.
-(2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis.
+(2) Rings: Ring homomorphisms, integral domains and fields, maximal and prime ideals.
-(3) Gauss-Jordan elimination: Row operations, echelon forms, determinants.
+(3) Rings of polynomials:
@@ Line 26: / Line 23: @@
 ==Topics List==
 {| class="wikitable sortable"
-! Week !! Topic !! Sections from the Nair-Singh book !! Subtopics !! Prerequisite
+! Week !! Topic !! Sections from the Judson-Beezer book !! Subtopics !! Prerequisite
 |-
-|  1-3
+|  1-2
 || [[Groups]]
-|| 1.1-1.8
+|| 3
-|| * Isomorphisms
+||
-* Normal subgroups and factor groups
+* Definitions and classical examples
+* Subgroups
 || MAT1313, CS2233/2231, or instructor consent.
 |-
 |  4-5
-|| [[Linear transformations]]
+|| [[Cyclic groups]]
-|| 2.1-2.6
+|| 4
-|| Rank and nullity, matrix representation, the space of linear transformations.
+||
-|-
+* Classification of cyclic groups.
 |-
 |  7-8
-|| [[Inner product spaces]]
+|| [[Cosets and Lagrange's Theorem]]
-|| 4.1-4.8
+|| 10
-|| Projections, orthogonal bases and Gram-Schmidt, least squares approximation, Riesz representation.
+||
 |-
 |  9
-|| [[Eigenvalues and eigenvectors]]
+|| [[Homomorphisms]]
-|| 5.1-5.5
+|| 11
-|| Eigenspaces, characteristic polynomials
+|| The Isomorphism Theorem
 |-
-|  10
+|  10-11
-|| [[Canonical forms]]
+|| [[Rings]]
-|| 6.1-6.5
+|| 16
-|| Jordan form
+||
 |-
-|  11-13
+|  12-end
-|| [[Spectral representation]]
+|| [[Rings of Polynomials]]
-|| 7.1-7.6
+|| 17
-|| Singular value and polar decomposition.
+||
 |}

← Older revision		Revision as of 14:10, 19 March 2023
Line 1:		Line 1:
−	~~The opportunity for development~~ of ~~basic theory~~ of ~~algebraic structures~~. ~~Areas of study may include monoids~~, ~~semigroups~~, groups, ~~isomorphism theorems~~, ~~free groups~~, ~~group extensions~~ and ~~group actions~~, ~~Sylow theorems~~, ~~group chains~~ and ~~composition series~~, ~~nilpotent~~ and ~~solvable groups~~, ~~cohomology of groups~~.	+	Introduction to groups rings and fields.
		+
		+	'''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].
		+
		+
		+
		+	'''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 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
		+	\|\| [[Groups]]
		+	\|\| 1.1-1.8
		+	\|\| * Isomorphisms *Normal subgroups and factor groups
		+
		+	\|\| 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.
		+	\|}