Orthogonal Complements#
Definition: Orthogonal Complement
Let \(U\) be a linear subspace of an inner product space \((V, \langle \cdot, \cdot \rangle)\).
The orthogonal complement of \(U\) is the set of all vectors in \(V\) which are orthogonal to each vector in \(U\).
\[\{v\in V \mid \langle v, u\rangle = 0, \forall u\in U\}\]
Notation
\[U^{\perp}\]
Theorem: Direct Sum of Complements
Let \((V, \langle \cdot, \cdot \rangle)\) be a finite-dimensional inner product space.
If \(U\) is a linear subspace of \(V\), then \(V\) is the direct sum of \(U\) and its orthogonal complement \(U^{\perp}\):
\[V = U \oplus U^{\perp}\]
Proof
TODO
Theorem: Orthogonal Complement is Kernel of Orthogonal Projection
If the orthogonal projection \(\pi_U\) onto a linear subspace \(U\) exists, then the orthogonal complement of \(U\) is the kernel of \(\pi_U\):
\[U^{\perp} = \ker \pi_U\]
Proof
TODO