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.