Let $E_1,E_2$ be orthogonal projection on an inner product space V. Prove that $E_1E_2=0$ if and only if the range subspaces of $E_1$ and $E_2$ are orthogonal.
I can prove one direction pretty easily, if the range subspaces of $E_1$ and $E_2$ are orthogonal, we have that $range(E_1)=W_1,range(E_2)=W_2$ are linearly independent, so $E_1E_2=0$.
I am having trouble with the other direction. How do you start with the assumption that $E_1E_2=0$?
Suppose that $E_1E_2=0$, and let $v_j$ be an element of the range of $E_j$ for $j=1,2$. Since $E_jv_j=v_j$ and the $E_j$ are self-adjoint, we have $$ (v_1,v_2)=(E_1v_1,E_2v_2)=(v_1,E_1E_2v_2)=(v_1,0)=0 $$ Therefore the ranges of $E_1$ and $E_2$ are orthogonal.