Inner Product Space as Direct Sum of Subset and Orthogonal Complement Theorem

Theorem

• With $V$ as a finite Inner Product Space
• With $W$ as a finite Subspace of $V$, $W\subset V$
• With $W^{\perp }$ as the Orthogonal Complement of $W$
• Then, $V=W\oplus W^{\perp }$

Further Implications

• If we consider $E:V\to W$ as the Orthogonal Projection, then $W^{\perp }=ker(E)$
• If we consider $I-E:V\to W^{\perp }$ as the Orthogonal Projection, then $W=ker(I-E)$