Projections On Subspaces
Why
Given a vector $x \in \R ^n$ and a subspace $S \subset \R ^n$, what is the closest point in $S$ to $x$.
Result
Suppose $U \in \R ^{n \times k}$ with
$U^\top U = I$.
Then
\[ \textstyle \proj_{U}(x) = UU^\top x \]