Active constraint ($a_i^{T}x=b_i$)

An inequality constraint which holds with equality at the point under consideration. For $Ax\ge b$, a direction annihilating every active row permits sufficiently small feasible moves of either sign. If the active rows have rank $n$, their common solution is a basic feasible solution; the independent equalities uniquely determine it.

At an extreme point of $\{x:Ax\ge b\}$, the active rows span the ambient dimension: otherwise a nonzero direction orthogonal to all active rows permits small feasible moves of both signs. Select $n$ independent active rows to form an integer matrix $B$. Its nonzero determinant has absolute value at least one. Cramer's rule and the determinant expansion bound each coordinate by $n!U^n$, where $U$ bounds the entries of $A$ and $b$. The vector $b$ may be real; only the denominator matrix needs to be integer.