A local to global question for linear functionals

Abstract: Let $F$ be an algebraically closed field and let $n\geq 3$. Consider $V=F^n$ with standard basis ${\vec{e}1,\ldots,\vec{e}_n}$ and its dual space $V^*= {\mathrm{Hom}}{F-{\mathrm{lin}}}(V,F)$ with dual basis ${y_1,\ldots,y_n}\subseteq V^*$ and let $\vec{y} = \sum_i y_i\otimes \vec{e}_i\in V^*\otimes V$. Let $d<n$ and consider the vectors $\vec{q}_1,\ldots,\vec{q}_d\in V^*\otimes V$. In this note we consider the question of whether $\vec{y}(\vec{v}) = \vec{v} \in Span_F(\vec{q}_1(\vec{v}),\ldots,\vec{q}_d(\vec{v}))$ for all $\vec{v}\in V$ implies that $\vec{y}\in Span_F(\vec{q}_1,\ldots,\vec{q}_d)$. We show this is true for $d=1$ or $d=2$, but that additional properties are needed for $d\geq 3$. We then interpret this result in terms of subspaces of $M_n(F)$ that do not contain any rank 1 idempotents.