Orthogonal Projection onto Orthogonal Complement

Theorem

1. With $V$ as a Inner Product Space
2. With $W$ as a finite Subspace $W\subset V$
3. With $E$ as the Orthogonal Projection of $V$
4. With Orthogonal Basis $\beta \subset W$
5. $(I-E):\beta \to \beta -E(\beta )$ is the orthogonal projection of $V$ onto $W^{\perp }$

Intuition

$(I-E)$ is the orthogonal projection of $V$ onto $W^{\perp }$ $range(I-E)=W^{\perp }$