Farkas' Lemma

Given a matrix Av and a column vector b,v one and only one of the following two alternatives holds. Either: (1) there exists a column vector xv ⩾ 0 with Axv = b,v or (2) there exists an unrestricted row vector yv for which yAv ⩾ 0 and ybv < 0. This lemma can be proved by defining appropriate primal and dual linear-programming problems and applying the duality theorem. Gordan's theorem; Strong duality theorem; Theorem of the alternatives.