Let $k$ be a field, let $V$ be a $k$-vector space, and let $U$ and $W$ be linear subspaces of $V$ with $U \subset W$. For a linear subspace $S \subset V$, define its annihilator in the algebraic dual space $V^*$ by
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\begin{align*}
S^0=\{\lambda \in V^* : \lambda(s)=0 \text{ for every } s \in S\}.
\end{align*}