Subspace and Orthogonal Complement for a Direct Sum

Theorem

• With $W$ as a finite-dimensional subspace of a finite-dimensional Inner Product Space
• Let $E$ be the orthogonal projections of $V$ on $W$
• Then:
  • $E$ is a projection of $V$ onto $W$
  • $W^{\perp }$ is the Nullspace of $E$
  • $V=W\oplus W^{\perp }$

Proof

1. Show $E$ is a projection
2. Show $E$ is a Linear Operator
3. Show $W^{\perp }$ is a nullspace
4. Show $W⊞W^{\perp }$ is a direct sum