Exercise: Let's $W,W',V,V'$ vector spaces over a field $F$, with $\dim W =n$ and $\dim V = n$. Suppose that $W \approx W'$ and $V \approx V'$ then $L(W,V) \approx L(W',V)$ (Show an isomorphism).
Well, I know that $\dim L(W,V) = n\cdot m = \dim L(W',V')$. Then they are isomorphics, but I need to prove showing an isomorphism.
May you give me a tip?
Let $f_V : V \to V'$ and $f_W : W \to W'$ be isomorphisms.
Show that the function $\Phi : \mathcal{L}(W,V) \to \mathcal{L}(W',V')$ defined by $$\Phi(T) = f_V \circ T \circ (f_W)^{-1}$$ is an isomorphism. Luck!