Let U and W be subspaces of a finite-dimensional vector space V. Define T: U × W → V by T(u, w) = u - w.
(a) Prove that T is a linear transformation.
(b) Show that range(T) = U + W.
(c) Show that ker(T) = U ≅ W.
(d) Prove Grassman n's Identity:
dim (U + W) = dim U + dim W - dim( U ∩ W)