Prove that $f$ is a linear combination of $f_1,f_2,...,f_n$.

Prove that $f$ is a linear combination of $f_1,f_2,...,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,...,f_n$ be linear maps from
$V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever
$f_1(x)=f_2(x)=...=f_n(x)=0$. Prove that $f$ is a linear combination of
$f_1,f_2,...,f_n$.
The solution can be found here (problem 1):
http://www.imc-math.org.uk/imc1998/prob_sol2.pdf
but I disagree that $a_k$ is guaranteed to exist. What if the set
containing all vectors $u\in V$ such that $f_1(u)=f_2(u)=...=f_{k-1}(u)$
is empty?