Subspace Orthogonal Complements

Main result

The orthogonal complement of a subspace is a subspace.

Let $L \subset \R ^n$ be a subspace. Then

\[ \dim L + \dim L^{\perp } = n. \]

Let $b_1, \dots , b_m$ be a basis for a subspace $L \subset \R ^n$. Then $x \perp L$ if and only if $x \perp b_i$ for $i = \upto{m}$.