WOMP Talk 1 , Part 1 : Algebra I Vector spaces and linear transformations

We review the notion of a vector space, basis and dimension, linear transformations between vector spaces, dual vector spaces and transformations, spectral decomposition for normal operators (which includes symmetric, Hermitian, orthogonal, and unitary operators), and determinants. Along the way we review direct-sum decompositions, bilinear forms and inner product spaces, adjoints, characteristic polynomials, and generalized-eigenspace decompositions for algebraically closed fields. The discussion of determinants includes a correct definition of the exterior (alternating) product of dual vector spaces. This talk is entirely review, except possibly for the spectral theorem (which you should at least have seen for symmetric/Hermitian or orthogonal/unitary operators).