Let $F$ be a sub-field of the complex numbers (i.e. a field of characteristic zero). Let $V$ be a finite dimensional vector space over $F$. Suppose that $E_1, E_2, \dots , E_k$ are projection of $V$ and that $E_1 + E_2 + \dots + E_K = I$. Prove that $E_iE_j = 0$ $\forall$ $i \neq j$.
I have tried this problem for for $n = 2$ as follows:
As $E_1$ and $E_2$ are projection operators, $E_1^2 = E_1$ and $E_2^2 = E_2$. Also since $E_1 + E_2 = I$, multiplying this relation by $E_1$ we get $$ E_1 (E_1 + E_2) = E_1 I = E_1 \Rightarrow E_1^2 + E_1 E_2 = E_1 \Rightarrow E_1 + E_1E_2 = E_1 \Rightarrow E_1E_2 = 0 $$
But what to do for the general case?
Can you break a vector $x=x_1+\dots+x_k$
$E_1(x)=x_1$ and $E_i(x_j)=0,i\neq j$ and what happen $E_iE_j(x),i\neq j $ ?