Math 5250: Graduate Linear Algebra
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Schedule
Weeks 13–14 — Multilinear Algebra
Tensor and exterior algebras
Math 5250: Graduate Linear Algebra
Math 5250 Schedule
Week 1 — Review
Vector space structure of R^n
Linear transformations and matrices
Change of basis, kernel and range, isomorphisms
Weeks 2–3 — Abstract Vector Spaces
Abstract vector spaces and subspaces
Direct sums and quotient spaces
Linear transformations: general theory
Trace and determinant
Week 4 — Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors
Characteristic polynomials and Cayley-Hamilton theorem
Diagonalization
Weeks 5–6 — Inner Product Spaces
Inner product spaces over R
Gram-Schmidt process and orthogonal projections
Spectral theorem for self-adjoint operators (real case)
Complex inner product spaces and Hermitian operators
Weeks 7–8 — Matrix Decompositions
Jordan canonical form
Positive definite matrices
QR, SVD, and polar decomposition
Week 9 — Norms
Vector space norms and equivalence of norms
Matrix and operator norms
Weeks 10–11 — Dual Spaces
Linear functionals and the dual space
Duality and inner products
Dual bases, linear maps, and double duality
Week 12 — Bilinear Forms
Bilinear forms and matrix representations
Quadratic forms and diagonalization
Signature and definiteness
Weeks 13–14 — Multilinear Algebra
Tensor products of vector spaces
Exterior powers of vector spaces
Tensor and exterior algebras
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Weeks 13–14 — Tensor and Exterior Algebras
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Notes on the tensor algebra and exterior algebra of a vector space.