To provide an in-depth review and to fill in gaps in some background material in Abstract Linear Algebra, which is often presumed in standard first year graduate courses. The material to be covered is also part of the syllabus of the qualifying exams on Algebra.
Below is a tentative list of topics to be covered; the actural coverage may vary depending on the instructor.
- Vector spaces, isomorphism, linear transformations: basis, dimension, quotient spaces, direct sums, rank and nullity. Coordinatization.
- Examples from various places: geometry, linear ODE, quantum mechanics, graph theory, etc.
- Similarity, eigenvalues, diagonalization, Jordan canonical form, application to ODE's and other areas, Rational canonical form.
- Role of the ground field (or extended ground field): In particular applications involving linear operators on vector spaces over the complex field (E.g. Jordan canonical form)
- Bilinear forms, sesquilinear forms, nondegeneracy, Euclidean and Unitary inner products.
- Some detailed study of Hermitian and unitary matrices, in particular, diagonalization involving Hermitian and unitary matrices. Basic properties of orthogonal and unitary groups. Self-adjoint linear transformations.
- Duality, esp. finite-dimensional case.
- Additional topics, if time permits: tensor product defined by naming basis, symmetric and wedge square, higher powers, determinants, Kronecker product, \(V^*\otimes W\), differential forms, Schur duality.