I'm working on this question as well. I thought the same but realized that it only proves some vectors in W can be generated by T(b_1*y_1 + ... + b_n*y_n). this is correct, right?
Do you have any hints? I'm pretty stuck but I've been working on it for a while.
EDIT> I was thinking of using dimV=dimT(v)+dim(KerT) (v in V), would this help at all?