An upper bound on the dimension of the reflexivity closure

Let ${\mathcal V},{\mathcal W}$ be linear spaces over an algebraically closed field, and let $\mathscr{S}$ be an $n$-dimensional subspace of linear operators that maps ${\mathcal V}$ into ${\mathcal W}$. We give a sharp upper bound for the dimension of the intersection of all images of nonzero operators from $\mathscr{S}$, namely $\dim ( \bigcap_{A\in\mathscr{S}\setminus\{0\}}\mathrm{Im} A ) \leq \dim{\mathcal V}-n+1$. As an application, we also give a sharp upper bound for the dimension of the reflexivity closure $\operatorname{Ref}\mathscr{S}$ of $\mathscr{S}$, namely $\dim ( \operatorname{Ref}\mathscr{S} ) \leq n(n+1)/2$.