(5) Let X, Y, Z, W be vector spaces. (a) 0)) Let S. T, i, j be linear transformations of vector spaces tting into the below commutative diagram X i; Y il ii 3 Z > W i.e. j o T = S o 2'. Suppose that both vertical maps 2', j are isomorphisms. Prove that rank(8) = rank(T) and nullity(3) = nullity(T). Suppose T : X > Y is a linear transformation. Let n = dim(Y). Let A = [TE be the matrix representing T with respect to arbitrary bases :1 of X and B of Y. Using (5)(a), prove that rank(T) equals the dimension of the subspace of F \" spanned by the columns of A