Every
(inhomogeneous)
system of linear equations over a field
can be transformed, by the manipulations described in
fact,
to an
equivalent linear system
of the form
-
where in each row, the first coefficient
![{\displaystyle {}b_{1s_{1}},b_{2s_{2}},\ldots ,b_{ms_{m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6100c1a1cc3e4e7992bf33beeaa2892239f040)
is different from
![{\displaystyle {}0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5428e3b06006771c083bd17ed8fce8f3be334b2)
.
Here, either
,
and the last row can be omitted, or
,
and then the system has no solution at all.