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The integer version of Farkas' Lemma is the statement that, for every A € Zmxn an be Zm, exactly one of the two systems of equations has a solution: Arb x = Zn and y A € Zn yTb & Z yT E Qm (a) Prove one direction of this lemma: that is, prove that it is impossible for both systems to have a solution. (Hint: again, look at y Ar.) (b) Use this statement to show that, for every pair of nonzero integers a1, a2, there exist integers x1, x2 such that a₁₁ + a₂x2 = 1 if and only if a₁ and a2 have no common factor other than ±1.

The integer version of Farkas' Lemma is the statement that, for every A € Zmxn an be Zm, exactly one of the two systems of equations has a solution: Arb x = Zn and y A € Zn yTb & Z yT E Qm (a) Prove one direction of this lemma: that is, prove that it is impossible for both systems to have a solution. (Hint: again, look at y Ar.) (b) Use this statement to show that, for every pair of nonzero integers a1, a2, there exist integers x1, x2 such that a₁₁ + a₂x2 = 1 if and only if a₁ and a2 have no common factor other than ±1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the following solution using farkas Lemma Method.

The integer version of Farkas' Lemma is the statement that, for every A E Zmxn an be Zm, exactly one of the two systems of equations has a solution: Arb r€Zn and y¹ A € Zn yTb&Z yT E Qm (a) Prove one direction of this lemma: that is, prove that it is impossible for both systems to have a solution. (Hint: again, look at y A.x.) (b) Use this statement to show that, for every pair of nonzero integers a1, a2, there exist integers x1, x2 such that a11 + a222 = 1 if and only if a₁ and a2 have no common factor other than ±1.
Transcribed Image Text:The integer version of Farkas' Lemma is the statement that, for every A E Zmxn an be Zm, exactly one of the two systems of equations has a solution: Arb r€Zn and y¹ A € Zn yTb&Z yT E Qm (a) Prove one direction of this lemma: that is, prove that it is impossible for both systems to have a solution. (Hint: again, look at y A.x.) (b) Use this statement to show that, for every pair of nonzero integers a1, a2, there exist integers x1, x2 such that a11 + a222 = 1 if and only if a₁ and a2 have no common factor other than ±1.
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