= Annihilator of a vector subspace
{title2=$U^\circ=\{\varphi\in V^*: \varphi|_U=0\}$}
= Annihilators of vector subspaces
{synonym}
For a <vector subspace> $U\leq V$,
$$
U^\circ=\{f\in V^*:f|_U=0\}.
$$
In finite <dimension>, $\dim U^\circ=\dim V-\dim U$.
An <annihilator of a vector subspace> collects the <linear functionals> that ignore every input from that <vector subspace>. For $U=\operatorname{span}\{(1,2)\}\subseteq\mathbb R^2$, a <linear functional> $\varphi(x,y)=ax+by$ vanishes on $U$ exactly when $a+2b=0$. Hence $U^\circ$ is the one-dimensional <span> of $(x,y)\mapsto2x-y$. The zero <level set> in the <linear functional> diagram is precisely $U$.
Back to article page
