Tuesday 45:00. Vector spaces in general perspective on vector space as set of functions on something Dual space crucial point: modify notation < $\phi$ | v > basic idea : can think of evaluating polynomial at a point also, think of evaluating a fixed point on many different polynomials these are the same numbers objects V and V' are objects of the same kind from this perspective, V'' is V each v defines a functional on V' new notation lets us see this as linear functional Dual basis: Lemma: suppose have linear map from V to W new notation is better to show the symmetry < $\phi$ | T | v > can go from right to left, or from left to right in matrix form, get the transform of M(T) video cuts out at last discussion of dual and functionals