Minimal vector space over a subfield that corresponds to a given vector

Question

Let $K$ be a field and $F \subseteq K$ a subfield. Then it is possible to show that for every vector $v\in K^n$, there exists a unique minimal subspace $V < F^n $ (over $F$) with the property that $v \in \operatorname{Span}_K V$.

Does this phenomenon have a name? A reference would also be greatly appreciated.