To avoid this being an XY problem, this is my background:
I was reading theorem 2.2.2 of section 2.2.2 of this paper, which is about finding a relative interior point of a system $Ax \leq b$ efficiently.
The given proof is in a constructive style as it outlines an algorithm to find one.
I could follow the proof except for the line:
Using Gauss’ elimination algorithm we can find the index set $J \subseteq I$ of all inequalities forced to equality, given that $A_I x = b_I$.
$I$ is an already-discovered index set of inequalities forced to equality (not necessarily maximal, of course).
How do I find rest of the rows given $I$?