System of linear equations/Set of variables/Equivalent system/Manipulations/Fact/Proof

Proof

Most of the statements are immediately clear. (2) follows from the fact that if

${}\sum _{i=1}^{n}a_{i}\xi _{i}=c\,$

holds, then also

${}\sum _{i=1}^{n}(sa_{i})\xi _{i}=sc\,$

holds for every ${}s\in K$. If ${}s\neq 0$, then this implication can be reversed by multiplication with ${}s^{-1}$.

(6). Let ${}G$ be the equation

${}\sum _{i=1}^{n}a_{i}x_{i}=c\,,$

and ${}H$ be the equation

${}\sum _{i=1}^{n}b_{i}x_{i}=d\,.$

If a tuple ${}(\xi _{1},\ldots ,\xi _{n})\in K^{n}$ satisfies both equations, then it also satisfies the equation ${}H'=G+H$. And if the tuple satisfies the equations ${}G$ and ${}H'$, then it also satisfies the equation ${}G$ and ${}H=H'-G$.

To fact