Math 5250 Schedule

Schedule

Date	Topic	Content
Tu 09/01	Review	Vector space structure of R^n, subspaces, span, linear independence, basis, dimension
Th 09/03	Review	Matrices and linear transformations, change of basis; kernel and range, similarity, isomorphisms
Tu 09/08	Abstract Vector Spaces	Abstract vector spaces and subspaces
Th 09/10	Abstract Vector Spaces	Direct sums and quotient spaces
Tu 09/15	Abstract Vector Spaces	Linear transformations: general theory
Th 09/17	Abstract Vector Spaces	Trace and determinant
Tu 09/22	Eigenvalues and Eigenvectors	Eigenvalues and eigenvectors
Th 09/24	Eigenvalues and Eigenvectors	Characteristic polynomials and Cayley-Hamilton theorem; diagonalization
Tu 09/29	Inner Product Spaces	Inner product spaces over R, orthogonality, orthogonal bases
Th 10/01	Inner Product Spaces	Gram-Schmidt process, orthogonal projections, orthogonal complement
Tu 10/06	Inner Product Spaces	Adjoint operators; spectral theorem for self-adjoint operators (real case)
Th 10/08	Inner Product Spaces	Complex inner product spaces; spectral theorem for Hermitian operators
Tu 10/13	Matrix Decompositions	Jordan canonical form
Th 10/15	Matrix Decompositions	Positive definite matrices
Tu 10/20	Matrix Decompositions	QR decomposition, SVD, and polar decomposition
Th 10/22	Matrix Decompositions	Applications and examples
Tu 10/27	Norms	Vector space norms, equivalence of norms in finite dimensions
Th 10/29	Norms	Matrix and operator norms, Frobenius norm, operator norm formula
Tu 11/03	Dual Spaces	Linear functionals and the dual space
Th 11/05	Dual Spaces	Duality and inner products
Tu 11/10	Dual Spaces	Dual bases and linear maps, transpose matrix, double duality
Th 11/12	Dual Spaces	Applications and examples
Tu 11/17	Bilinear Forms	Bilinear forms, matrix representation, change of basis
Th 11/19	Bilinear Forms	Quadratic forms, symmetric bilinear forms, equivalence, diagonalization; signature and definiteness
Tu 11/24	Thanksgiving Break
Th 11/26	Thanksgiving Break
Tu 12/01	Multilinear Algebra	Tensor products of vector spaces and linear maps
Th 12/03	Multilinear Algebra	Exterior powers of vector spaces
Tu 12/08	Multilinear Algebra	Tensor and exterior algebras
Th 12/10	Multilinear Algebra	Applications and review