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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(1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector
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spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomor-
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phisms, Rank-Nullity Theorem, Product and Quotient Spaces (3) Matrices and Linear Transfor-
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mations: Matrix representation, change of basis, equivalence of matrices, similarity (4) Eigen-
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values and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan Canoni-
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cal forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz in-
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equality, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez Representa-
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tion Theorem (5) Orthogonality over finite fields (6) Multilinear algebra : Bilinear Forms and
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Functionals, Tensor product of vector spaces and Matrices.

Latest revision as of 19:48, 24 March 2026

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: (1) Algebraic Structures: rings, matrices, polynomials, fields and finite fields (2) Vector spaces: subspaces, linear independence, basis, dimension (3) Linear transformations, isomor- phisms, Rank-Nullity Theorem, Product and Quotient Spaces (3) Matrices and Linear Transfor- mations: Matrix representation, change of basis, equivalence of matrices, similarity (4) Eigen- values and Eigenvectors: The Cayley-Hamilton Theorem, Diagonalization and Jordan Canoni- cal forms (4) Real and Complex Inner Product Spaces: Norm, distances, Cauchy-Schwartz in- equality, Gram-Schmidt Orthogonalization process, The Projection Theorem, Riez Representa- tion Theorem (5) Orthogonality over finite fields (6) Multilinear algebra : Bilinear Forms and Functionals, Tensor product of vector spaces and Matrices.

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