I will prove the forward direction, you try proving the other direction.
Say that is a linear transformation. Then by rank-nullity theorem we know . But we are told that thus . Putting this together we have that . But so this means . Finally, .
I really need your help with this:
Suppose that V and W are finite-dimensional and that U is a subspace of V. Prove that there exists a T – linear map from V to W such that null(T)=U if and only if dim(U)>=dim(V)-dim(W).
Thank you in advance!