The Orthogonal Complement of a Vector Space
\[\left\{ \begin{pmatrix}1\\0\\1\end{pmatrix} \right\}\]
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 \[\left\{ \begin{pmatrix}1\\0\\-1\end{pmatrix} , \begin{pmatrix}0\\1\\0\end{pmatrix} \right\}\]
.
