Let $V, W$ be two finite-dimensional vector spaces, $f: V\rightarrow W$ a linear map, and $U \subseteq W$ a vector subspace. I'm trying to show that $(f^{-1}(U))^0 = f^*(U^0)$, i.e. that the annihilator of the inverse image of $U$ is the image of the annihilator under the the dual $f^*$ of $f$. $(f^{-1}(U))^0 \supseteq f^*(U^0)$ is easy to prove, but I'm having troubles with the other direction…
(the annihilator for $Y \subseteq X$ vector spaces is defined here as $Y^0 := \{x^* \in X^*\ |\ \forall y \in Y: x^*(y) =0 \}$, with $X^*$ the dual space of $X$; for inner product spaces, $X^0 \cong X^\perp$).