Linear system/Solution space/Orthogonal space/Example

Let a homogeneous linear system

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

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

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

Let

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

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