if means , so Ker(T) =
Since Dim( ) = Dim(V) + Dim(W)
Rank(T) = Dim(V)
so Rank(T) + Dim(Ker(T)) = Dim(V) + Dim(W)
Dim(Ker(T)) = Dim(W)
All finite dimensional vector spaces of equal dim are isomorphic.
additionally, you can set up a linear map from Ker(T) (since its a subspace of your vector space),
Its easy to see this is both a linear map and a bijection, thus they are isomorphic.