TITLE: orthogonal projection onto orthogonal complement
QUESTION [6 upvotes]: If $V=M \oplus M^{\perp}$. For any $v\in V$, the orthogonal projection of $v$ onto $M$ along $M^{\perp}$ is well defined. 
Can we take the orthogonal projection of $v$ onto $M^{\perp}$ along $M$?

REPLY [6 votes]: For your case: If $P_{M}$ is the orthogonal projection of $X$ onto $M$, then $I-P_{M}$ is the orthogonal projection of $X$ onto $M^{\perp}$. This is because $x=x_{M}+x_{M^{\perp}}$ implies $x_{M^{\perp}}=x-x_{M}$. The assumption $X=M\oplus M^{\perp}$ is all that is required, regardless of the subspace $M$ and regardless of dimension.