Orthogonal Projection of Perpendicular Set Theorem

Theorem

• With $V$ as an Inner Product Space
• $W$ is a finite-dimensional Subspace of $V$
• With $E$ as the Orthogonal Projection $V$ on $W$
• The mapping $(I-E)(\beta )=\beta -{\sum }_{i=1}^n⟨\beta |{\alpha }_i⟩{\alpha }_i$ is the orthogonal projection of $V$ on $W^{\perp }$ (The $Range(I-E)=W^{\perp }$)

Proof

1. Let $\beta$ be an arbitrary vector in $V$
2. Then, it follows from the properties of Best Approximation that $\beta -E(\beta )\in W^{\perp }$
3. Additionally, $\forall \gamma \in W^{\perp }$, $\beta -\gamma =E(\beta )+(\beta -E(\beta )-\gamma )$
4. Since $E(\beta )\in W$ and $\beta -E(\beta )-\gamma \in W^{\perp }$
5. From Generalized Pythagorean Theorem, and Triangle Inequality, $||\beta -\gamma |{|}^2=||E(\beta )|{|}^2+||\beta -E(\beta )-\gamma |{|}^2\ge ||\beta -E(\beta )|{|}^2$ with strict inequality when $\gamma \ne \beta -E(\beta )$
6. Thus, $P-E(\beta )$ is the Best Approximation in $W^{\perp }$