Linear inhomogeneous system/Elimination/Echelon form and triangular form/Fact

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 is different from .

Here, either , and the last row can be omitted, or , and then the system has no solution at all.

With the help of renaming the variables, we get an equivalent system of the form

with diagonal elements .