## The Orthogonal Complement of a Vector Space

Suppose we a subspace
$\left\{ \begin{pmatrix}1\\0\\1\end{pmatrix} \right\}$
of
$\mathbb{R}^3$
.
The orthogonal complement of
$S$
is the set of all vectors of
$\mathbb{R}^3$
orthogonal to every vector in
$S$
.
Let
$\mathbf{v} = \begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}$
be in the orthogonal complement of
$S$
.
Then
$\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix} \cdot \begin{pmatrix}1\\0\\1\end{pmatrix}=v_1 + v_3=0$
.
We can let
$v_1 =1, z: v_2 =-1$
so that one element of the complement of
$S$
is
$\begin{pmatrix}1\\0\\-1\end{pmatrix}$
.
The is no restriction on
$v_2$
so we can let another element of the complement of
$S$
be
$\mathbf{v} = \begin{pmatrix}0\\1\\0\end{pmatrix}$
.
The orthogonal complement to
$S$
is the space spanned by
$\left\{ \begin{pmatrix}1\\0\\-1\end{pmatrix} , \begin{pmatrix}0\\1\\0\end{pmatrix} \right\}$
.