This question concerns problem 3F 16) in Axler's (2005) "Linear Algebra Done Right".
Assuming that V and W are finite-dimensional vector spaces: how do I prove that a map that takes $T\in\mathcal{L}(V,W)$ to $T'\in\mathcal{L}(W',V')$ is an isomorphism of $\mathcal{L}(V,W)$ onto $\mathcal{L}(W',V')$.
Does it suffice to show that $\mathcal{L}(V,W)$ and $\mathcal{L}(W',V')$ have the same dimension (which is fairly easy), or do I need to show that this map is injective as well as surjective? If the latter is the case, why is showing same dimension not enough?