Introduction to Linear Algebra: My student and I will be covering such topics as inner product spaces, the theory of determinants, eigenvalues, diagonalization of real symmetric matrices, the spectral theory of real unitary and hermitian matrices, and the triangulation of matrices over algebraically closed fields. We will also touch upon the major theorems in linear algebra: Cayley-Hamilton's Theorem, Primary Decomposition, Jordan Normal Form, and possibly the Cyclic Decomposition Theorem. The book we are using use is Serge Lang's 2004 edition of "Linear Algebra". This book covers almost all of the topics enumerated above.

1) Here are some notes on vector spaces, linear maps, matrices, and inner product spaces.