MTH302: (Linear Algebra)

[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.