**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Definition and examples of abstract fields.
- Review of vector spaces over abstract fields, bases and dimension.
- Subspaces, quotient spaces, sums and direct sums.
- Review of Linear transformations, matrices, change of bases, rank-nullity theorem. Dual spaces, transpose of a linear transformation.
- Properties of Determinants: determinant functions, permutations and uniqueness, additional properties, multilinear maps, exterior products.
- Eigenvectors, characteristic polynomials, invariant subspaces, simultaneous diagonalization and triangulation, primary decomposition theorem and diagonalization.
- Bilinear forms, symmetric/skew-symmetric bilinear forms, inner product spaces, positive forms, the Gram-Schmidt orthogonalization, orthogonal complements and projections, orthogonal operators.
- The adjoint of a linear operator, normal and self-adjoint operators, Spectral theorem, Hermitian forms, unitary operators.

Additional topics: Rational and Jordan Canonical Forms, Tensor products and tensor
algebra.

- K. M. Hoffmann and R. Kunze: Linear Algebra
- S. Axler: Linear Algebra done right
- Michael Artin: Algebra
- S. Lang: Linear Algebra
- Werner Greub: Multilinear Algebra.