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 ⃗₁,…,e⃗ₙ\ and its dual space V^*= Hom_F-lin(V,F) with dual basis ₁,…,yₙ\⊆ V^* and let y⃗ = ∑ᵢ yᵢ⊗ e⃗ᵢ∈ V^*⊗ V. Let $d<n$ and consider the vectors q⃗₁,…,q⃗d∈ V^*⊗ V. In this note we consider the question of whether y⃗(v⃗) = v⃗ ∈ SpanF(q⃗₁(v⃗),…,q⃗d(v⃗)) for all v⃗∈ V implies that y⃗∈ SpanF(q⃗₁,…,q⃗d). 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.