Difference between $T \colon V \to W $ and $ T \colon V \to V $ for $\dim(V) = \dim(W)$.

Question

My professor proved rank-nullity theorem, and after that, he gave two examples for the transformations $T \colon V \to W $ and $ T \colon V \to V $ with $\dim(V) = \dim(W)$. He said something about isomorphism, but I couldn't get that.

Answer 1

What your professor said is that the vector spaces $V$ and $W$ are isomorphic.

Since this is your first linear algebra class, you are only dealing with finite-dimensional vector spaces. It turns out that if $\dim(V) = \dim(W)$, then there is in fact an isomorphism between $V$ and $W$.

This has nothing to do with the rank-nullity theorem per se. Since algebraic structures are sometimes only considered up to isomophism, your professor might have remarked "This is in fact the same example because $V$ and $W$ are isomorphic" or something along these lines.