Let $V$ be a vector space such that $\dim V < \infty $ with an inner product. Let $W_1,W_2 $ be two subspaces of $V$ , such that $W_1 \cap W_2 = \{0\}$. Let $P_{W_1},P_{W_2} $ be the orthogonal projection on $W_1$ and $W_2$.
My first question, does it means that $P_{W_1}P_{W_2} = 0 $? I think so because, let $v \in V$ so $P_{W_2}(v) = v $ with only $W_2$ component. So $P_{W_1}P_{W_2}(v) = 0 $ cause $W_1\cap W_2 = \{0\} $ , am I right?
which means , $P_{W_1}P_{W_2}= P_{W_2}P_{W_1} = 0$ ?
Now I want to prove that $P_{W_1}P_{W_2} = P_{W_2}P_{W_1} \iff W_1\bot W_2$.
What I tried:
$$P_{W_1}P_{W_2} = 0 \iff W_2 \subseteq \ker (P_{W_1})$$ and $\operatorname{im}(P_{W_2}) = W_2 $ something I proved, and $\ker (P_{W_1}) = W_1^\perp$ which means $ W_1 \perp W_2$
If all that I tried is right, why they gave me in the question the data - $P_{W_1}P_{W_2} = P_{W_2}P_{W_1}$ I didn't use it in my proof, thanks a lot.