A local to global question for linear functionals

Let $F$ be an algebraically closed field and let $n\ge 3$. Consider $V=F^n$ with standard basis $\{{\vec{e}}_1,\dots ,{\vec{e}}_n\}$ and its dual space $V^{\ast }={\mathrm{H}\mathrm{o}\mathrm{m}}_{F-\mathrm{l}\mathrm{i}\mathrm{n}}(V,F)$ with dual basis $\{y_1,\dots ,y_n\}\subseteq V^{\ast }$ and let $\vec{y}={\sum }_i{y}_i\otimes {\vec{e}}_i\in V^{\ast }\otimes V$. Let $d<n$ and consider the vectors ${\vec{q}}_1,\dots ,{\vec{q}}_d\in V^{\ast }\otimes V$. In this note we consider the question of whether $\vec{y}(\vec{v})=\vec{v}\in Spa{n}_F({\vec{q}}_1(\vec{v}),\dots ,{\vec{q}}_d(\vec{v}))$ for all $\vec{v}\in V$ implies that $\vec{y}\in Spa{n}_F({\vec{q}}_1,\dots ,{\vec{q}}_d)$. We show this is true for $d=1$ or $d=2$, but that additional properties are needed for $d\ge 3$. We then interpret this result in terms of subspaces of $M_n(F)$ that do not contain any rank 1 idempotents.