Let S and T be linear transformations V → W, where dim V = n and dim W = m.
Show that im S = im T if and only if T = SR for some isomorphism R : V → V. [Show that dim(ker S) = dim(ker T) and choose bases {e1,..., er..., en] and (f1 ..., fr ,..., fn] of V where {er+1 ..., en] and {fr+1 ..., fn] are bases of ker S and ker T, respectively. If 1 < I < r, show that S(ei) = T(gi) for some g1 in V, and prove that (g1 .. ,gr, fr+1,..., fn} is a basis of V.]