MAT5283
Introduction to the theory of finite-dimensional vector spaces.
Catalog entry
Prerequisite: Prerequisite: Math 2233 Linear Algebra, Math 2243 Applied Linear Algebra or instructor approval. , or instructor consent.
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
Topics List
| 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. |
