This is a small piece of article from http://sbseminar.wordpress.com/
<...> So, first of all, consider a vector arrangement $S=\{v_1,\ldots,v_n\}\subset V$ where V is your favorite vector space over your favorite coefficient field $\mathbb{K}$. This is just an arbitrary set of non-zero vectors (with repetition allowed), though, for simplicity, I’ll assume it spans all of V. You could also think of this as a hyperplane arrangement on $V^*$, by looking at the zero set of $v_i$ thought of as a function on $V^*$.
Now, let $W$ be the vector space of solutions to the equation $\sum_{i=1}^nw_iv_i=0$, where $w_i\in \mathbb{K}$. This is not just a vector space, but one equipped with a set distinguished functions $w_i:W\to \mathbb{K}$. That is, $S^\vee = \{w_1,\ldots, w_n\}\subset W^*$ is a new vector arrangement, which we call the Gale dual to S. <...>
Can you explain me, how functions $w_i \in W^*$ act on W.