Es sei
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{\displaystyle {}K}
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{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&c_{2}\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&c_{m}\end{matrix}}}
ein inhomogenes lineares Gleichungssystem über
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{\displaystyle {}K}
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{\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}}}
das zugehörige homogene Gleichungssystem. Wenn
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{\displaystyle {}{\left(z_{1},\ldots ,z_{n}\right)}}
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{\displaystyle {}{\left(y_{1}+z_{1},\ldots ,y_{n}+z_{n}\right)}}
