IMC 1998 · Problem 1

Let $V$ be a real vector space, and let $f, f_1, f_2, \dots, f_k$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x) = 0$ whenever $f_1(x) = f_2(x) = \dots = f_k(x) = 0$. Prove that $f$ is a linear combination of $f_1, f_2, \dots, f_k$.