Linear subspace/Dual space/1/Example

We consider the linear subspace

${}U=\langle {\begin{pmatrix}8\\6\\5\end{pmatrix}},{\begin{pmatrix}4\\7\\-3\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{3}\,.$

The orthogonal space of ${}U$ consists of all linear forms

$f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R}$

with ${}f{\begin{pmatrix}8\\6\\5\end{pmatrix}}=0$ and ${}f{\begin{pmatrix}4\\7\\-3\end{pmatrix}}=0$. Because a linear form is described by a row matrix ${}\left(a,\,b,\,c\right)$ with respect to the standard basis, we are dealing with the solution set of the linear system

${}8a+6b+5c=0\,,$

${}4a+7b-3c=0\,.$

The solution space is

${}U^{\perp }={\left\{s\left({\frac {17}{4}},\,1,\,-8\right)\mid s\in K\right\}}\,.$