Uniqueness of orthogonal projector

Question

Let $S \subseteq \mathbb{R}^n$ be a subspace of $\mathbb{R}^n$, and let $P_1$, $P_2$ be arbitrary orthogonal projectors onto $S$.

How can I prove that the orthogonal projector onto $S$ is unique, i.e. that $P_1 = P_2$?

Answer 1

Hints:

A projection $P$ onto $S$ satisfies $P(x)\in S$ for every $x\in \mathbb R^n$, and if it is orthogonal, $P(x)-x\in S^\perp$ for every $x\in \mathbb R^n$.

Combine those two things to see why $P_1(x)-P_2(x)=0$ for every $x$.