Math 314
## Math 314

** ADVANCED LINEAR ALGEBRA **

This is semester long proof based course in linear algebra. We will
cover the following topics:

- fields and vector spaces
- spans, linear independence
- basis and dimension
- linear transformations, matrix representation
- spaces of linear transformations, invertibility
- linear systems, row reduction, echelon forms
- fundamental spaces associated with a linear transformation, rank
and nulity
- determinants, formal definition, properties and computations
- minors and cofactor expansions, Cramer rules
- eigenvalues, eigenvectors, characteristic polynomials
- invariant subspaces, diagonalizability, Jordan canonical form
- Cayley-Hamilton theorem, spectral mapping theorem
- structure of nilpotent operators
- inner products and norms
- orthogonality, orthogonal projections, Gram-Schmidt algorithm
- dual spaces and tensors

A basic goal of the course is to use the
abstract theory to develop an intuition in concrete examples and learn
to understand and produce sound mathematical arguments.