A local to global question for linear functionals

Let F be an algebraically closed field and let n 3. Consider V=Fⁿ with standard basis \e₁, , eₙ\ and its dual space V^*= Hom₅-₋₈₍ (V, F) with dual basis \y₁, , yₙ\ V^* and let y = ᵢ yᵢ eᵢ V^* V. Let d<n and consider the vectors q₁, , qd V^* V. In this note we consider the question of whether y (v) = v SpanF (q₁ (v), , qd (v) ) for all v V implies that y SpanF (q₁, , qd). We show this is true for d=1 or d=2, but that additional properties are needed for d 3. We then interpret this result in terms of subspaces of Mₙ (F) that do not contain any rank 1 idempotents.