Consider the vector space $\mathbb{R}^m$ with usual inner product. Let $S_1$ and $S_2$ subspaces of $\mathbb{R}^m$ , $P_1\in\mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix on subspace $S_1$ and $P_2\in\mathbb{M}_m(\mathbb{R})$ a orthogonal projection on $S_2$. If $P_1P_2=P_2P_1=0$ Show that
i)$P_1+P_2$ is a orthogonal projection matrix
ii)$S_1$ and $S_2$ are orthogonal subspaces
iii)Show that $P_1+P_2$ is a orthogonal projection matrix on $W=S_1\oplus S_2$
What I think
i)$(P_1+P_2)^2=P_1^2+P_1P_2+P_2P_1+P_2^2=P_1^2+P_2^2=P_1+P_2$
ii)Here are starting problems, I think that is just show $\forall x\in S_1$ and $\forall y\in S_2$ $<x,y>=0$. But I do not know how to prove it formally
iii)I don't know how to proof