Condition for Linear Dependence of Linear Functionals in terms of Kernel

Theorem
Let $V$ be a vector space over a field $\GF$.

Let $f, f_1, \ldots, f_n: V \to \GF$ be linear functionals.

Suppose that:


 * $\ds \bigcap_{i \mathop = 1}^n \ker f_i \subseteq \ker f$

where $\ker f$ denotes the kernel of $f$.

Then there exist $\alpha_1, \ldots, \alpha_n \in \GF$ such that:


 * $\ds \forall v \in V: \map f v = \sum_{i \mathop = 1}^n \alpha_i \map {f_i} v$

That is:


 * $f \in \span \set {f_1, \ldots, f_n}$