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

Let ${}K$ be a field, and let

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

be an inhomogeneous system of linear equations over ${}K$.

Then the following manipulations on this system yield an

equivalent system.

1. Swapping two equations.
2. The multiplication of an equation by a scalar ${}s\neq 0$.
3. The omitting of an equation, if it occurs twice.
4. The duplication of an equation (in the sense to write down the equation again).
5. The omitting or adding of a zero row (zero equation).
6. The replacement of an equation ${}H$ by the equation that arises if we add to ${}H$ another equation ${}G$ of the system.

Proof, Write another proof