Let $E$ be a vector space over $k$ of finite dimension $n$. Let $f_1, \ldots, f_p \in \mathcal{L}(E)$ with $p \leq n$ and assume that the $f_i$ are simultaneously diagonalizable (put it simply, there exists a basis of $E$, where the matrices representing the $f_i$ are all diagonal).
Assume that for all $x \in E$, the vectors $f_1(x), \ldots, f_p(x)$ are linearly dependent. Then the vectors $f_1, \ldots, f_p$ are linearly dependent in $\mathcal{L}(E)$.
I have a complicated proof of that, and I'd like to know if there is a simple argument proving this result. Furthermore, I feel like the diagonalizability assumption could be withdrawn and one could only assume that for any $i,j$ we have $f_i \circ f_j = f_j \circ f_i$, but I have no clue on how to prove that last assertion.