Let $k$ be a field, let $V$ be a finite-dimensional $k$-vector space, and let $U \subset V$ be a linear subspace. Define the linear annihilator of $U$ in $V^*$ by
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\begin{align*}
U^0 := \{\varphi \in V^* : \varphi(u)=0 \text{ for every } u \in U\}.
\end{align*}
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Then
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\begin{align*}
\dim_k U + \dim_k U^0 = \dim_k V.
\end{align*}