Orthogonal Complement

Definition: Orthogonal Complement

Let $(U,F,+,\cdot )$ be a subspace of an inner product space $(V,F,+,\cdot )$.

The orthogonal complement of $U$ is the set of vectors in $V$ which are orthogonal to all vectors in $U$.

$$\{\mathbf{v}\in V\mid \mathbf{v}\perp \mathbf{u},\forall \mathbf{u}\in U\}$$

NOTATION

$U^{\perp }$

THEOREM

The orthogonal complement $(U^{\perp },F,+,\cdot )$ is also a subspace of $(V,F,+,\cdot )$ and its dimension is

$$\dim (U^{\perp })=\dim (V)-\dim (U)$$

PROOF

TODO