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
Let be a linear transformation. Assume that (you need to treat the case seperately). We know that this dimension is finite (by hypothesis) and so let be a basis for . Assume that is infinite-dimensional. Then we can add vectors to so that all vectors are linearly independent, note that there is no bound on since we are assuming is infinite-dimensional. The claim is that is linearly independent - look at proof of Rank Nullity Theorem, it is similar. Thus, it means we can make this set as large as we please by keeping it linearly independent, but, then it means the range of cannot be finite dimensional since there is a limit to the number of linearly independent elements.