The full problem statement is as follows,
Assume that vector space $E$ is finite dimensional, and let $f_i:E \to E$ be any $p \geq 2$ linear maps such that $f_1 + \ldots + f_p = \operatorname{id}_E$. Prove that $f_i^2 = f_i, 1 \leq i \leq p$ implies $f_j \circ f_i = 0$, for all $i\neq j, 1\leq i,j \leq p$.
I have tried to apply $f_j$ on both sides, $$f_j \circ (f_1 + \ldots +f_p) = f_j \circ \operatorname{id}_E $$ $$f_j \circ f_1 + \ldots + f_j \circ f_{j-1} + f_j \circ f_{j+1} + \ldots + f_j \circ f_p = 0$$ but don't know how to make every term go to zero.
How can I proceed from here? Any help is highly appreciated!