Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$.
(a) Show that $W \subset V$ if and only if $V^0 \subset W^0$.
(b) Show that $(V \cap W)^0 = V^0 + W^0$. (Hint: Show $\supset$ holds, then use lots of fomulas for dimension.)
I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions $f$ in $V^*$ so that $f(s)=0$ for every $s$ in $S$. However, I don't know how to use this definition to prove (a) and (b). The hint for (b) suggests I should use formulas such as
$$\dim(V + W) = \dim V + \dim W − \dim(V \cap W)$$
but I don't know how to reach this stage.