Isomorphism between Dual Spaces and Annihilators in a Direct Sum

Question

Let $U$ be a finite dimensional vector space and the direct sum of two subspaces $V$ and $W$: $$U=V \oplus W$$ Give an explicit isomorphism that verifies: $$1) V^* \cong W^0$$ $$2) U^* \cong V^0 \oplus W^0$$ This is easy to prove using $\dim U = n$ , but that does not give me any clue to finding the actual isomorphism. I tried using a base for $U$ and splitting it into two bases for $V$ and $W$, but then, every functional I try to define in $V^*$ goes to zero in $W^0$. As for part 2) I have no idea on how to start. Maybe it can be proven using 1). Any suggestions?

Answer 1

Austin's suggestion works well for 1, for two consider: $$ \Phi: U^* \to V^0 \oplus W^0 $$ which sends the map $f \to f_v+f_w$ i.e since $U=V \oplus W$, we have $f(u)=f(v+w)$ (unique representation) and $f(v+w)=f(v)+f(w):=f_v(v)+f_w(w)$, by definition $f_v$ vanishes on $W$ and hence is in $W^0$, and $f_w$ vanishes on $V$ and is in $V^0$. You also should probably check that $V^0$ and $W^0$ are disjoint, so the direct sum is well-defined in the first place.