Linear system/Solution space/Orthogonal space/Example

Let a homogeneous linear system

${\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}}$

over a field ${\displaystyle {}K}$ be given. We consider the ${\displaystyle {}i}$-th equation as a kernel condition for the linear form

${\displaystyle L_{i}\colon K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{j=1}^{n}a_{ij}x_{j}.}$

Let

${\displaystyle {}F=\langle L_{1},\ldots ,L_{m}\rangle \,}$

denote the linear subspace of the dual space ${\displaystyle {}{K^{n}}^{*}}$ generated by these linear forms. Then ${\displaystyle {}F^{\perp }}$ is the solution space of this linear system.