Difference between revisions of "MAT5283"

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(1) Vector spaces: Abstract vector spaces, subspaces, bases, dimension, sums and direct sums. (2) Linear transformations: Rank and nullity, isomorphisms, bases, change of basis, canonical forms. (3) Inner product spaces: Norm, angles, orthogonal and orthonormal sets, orthogonal complement, best approximation and least squares, Riesz Representation Theorem. (4) Determinants: Axiomatic characterization of determinants, multilinear functions, existence and uniqueness of determinants. (5) Eigenvalues and eigenvectors: Characteristic polynomials, eigenspaces. (6) Block diagonal representation: Diagonalizability, triangularizability, block representation, Jordan normal form. (6) The Spectral Theorem.
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Introduction to the theory of finite-dimensional vector spaces.
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'''Catalog entry'''
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''Prerequisite'': Prerequisite: Math 2233 Linear Algebra, Math 2243 Applied Linear Algebra or instructor approval.  , or instructor consent.
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''Contents''
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(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

Latest revision as of 12:13, 20 May 2025

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

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