Question on projection operator and direct sum Here is question 10,section 6.6,Hoffman and Kunze:
Let $F$ be a field of characteristic 0.Let $V$ be a finite dimensional vector space over $F$.Suppose that $E_1,..,E_k$ are projections of V such that $E_1+..+E_k=I$.Prove that $E_i E_j$=0.
My attempt:
Let $W_i=E_i(V)$.
Then we get $V=W_1+...+W_k$.
Again from matrix representation on both sides of $E_1+..+E_k=I$ we get $dim W_1+..dim W_k=dim V$(since trace of matrix rep.of $E_i$ is dim $W_i$).Hence we have $V$ is the direct sum of $W_1,..W_k$.
How do i proceed after this? $V$ being the direct sum of $W_i$,we will obtain k projections,but they may not equal $E_i$,right?

Hint: 
Assume that for some $\alpha\in V$, $\ E_iE_j\alpha\neq 0$, and then conclude that $E_j\alpha\in W_i\cap W_j$ which contradicts $V$ being the direct sum of $W_1,..., W_k$.
