Linear Programming Exercise 4.19 Let P = {x \in Rn I Ax = b, x \geq 0} be a nonempty polyhedron, and let m be the dimension of the vector b. We call xj a null variable if xj = 0 whenever IEP. ONLY PART B AND C (a) Suppose that there exists some p \in Rm for which p' A \geq 0 ' , p'b = 0, and such that the jth component of p' A is positive. Prove that xj is a null variable. (b) Prove the converse of (a) : if xj is a null variable, then there exists some p \in Rm with the properties stated in part (a). (c) If xj is not a null variable, then by definition, there exists some y \in P for which yj > 0. Use the results in parts (a) and (b) to prove that there exist x \in P and p \in Rm such that: p'A \geq 0' , p'b = 0 , x + A ' p > 0.