Difference between revisions of "MAT5173"
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| − | + | 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. | ||
| + | |} | ||
Revision as of 08:10, 19 March 2023
Introduction to groups rings and fields.
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) 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
| 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. |
