Linear Algebra, finite-dimensional null space and range

Hi everyone,

I'm trying to show that given a Linear Map T from V to V and that the range and null space of T are both finite-dimensional, then V must be finite-dimensional.

My start at the solution is as follows:

Because the range and null space are f.d., we write a basis for them and note that since the range is a subset of V, this basis is in V. Applying the map hasn't seemed to help much, although I strongly suspect the fact that the map maps back into the domain is key to this problem. Could you point me in the right direction please? (No blatant solutions, however please; I'd just like a hint.)

Best,

Alex