System of linear equations/Set of variables/Equivalent system/Manipulations/Fact/Proof
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Proof
Most of the statements are immediately clear. (2) follows from the fact that if
holds, then also
holds for every . If , then this implication can be reversed by multiplication with .
(6). Let be the equation
and be the equation
If a tuple satisfies both equations, then it also satisfies the equation . And if the tuple satisfies the equations and , then it also satisfies the equation and .