Linear inhomogeneous system/Elimination/Echelon form/Fact

Every (inhomogeneous) system of linear equations over a field ${}K$

can be transformed, by the manipulations described in

fact, to an equivalent linear system of the form

${\begin{matrix}b_{1s_{1}}x_{s_{1}}&+b_{1s_{1}+1}x_{s_{1}+1}&\ldots &\ldots &\ldots &\ldots &\ldots &+b_{1n}x_{n}&=&d_{1}\\0&\ldots &0&b_{2s_{2}}x_{s_{2}}&\ldots &\ldots &\ldots &+b_{2n}x_{n}&=&d_{2}\\\vdots &\ddots &\ddots &\vdots &\vdots &\vdots &\vdots &\vdots &=&\vdots \\0&\ldots &\ldots &\ldots &0&b_{m{s_{m}}}x_{s_{m}}&\ldots &+b_{mn}x_{n}&=&d_{m}\\(0&\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &0&=&d_{m+1}),\end{matrix}}$

where in each row, the first coefficient ${}b_{1s_{1}},b_{2s_{2}},\ldots ,b_{ms_{m}}$ is different from ${}0$.

Here, either ${}d_{m+1}=0$, and the last row can be omitted, or ${}d_{m+1}=0$, and then the system has no solution at all.

Proof, Write another proof