Farkas Lemma The following are equivalent statements of Farkas' Lemma. FARKAS LEMMA (1). Let A be an m n matrix. Then Ax = b, x 0 has a solution if and only if b is in the cone generated by the columns a1, , an of A, i.e., b cfw_ 1 a 1 + + n a n | 1 , , n
since A is Noetherian (see [1, Exercise 7, p. 126]). Thus the height of M is exactly 2, as required by (i). The proof of the theorem is complete.
ACKNOWLEDGMENTS. I wish to express my warm gratitude to Professor Ram Murty for an interesting discussion tha