Problems:
For each statement, state whether it is true or false. Be sure to justify your answer.
a) Suppose you are given a linear program in Rn with mE equality constraints and mI inequality constraints. Let x be an element of the polyhedron at which n - mE inequality constraints are active. Then x must be an extreme point of the polyhedron.
b) If an LP has more than one optimal solution, and has an optimal extreme point, then it must have at least two optimal extreme points.
c) Let S be a subspace in Rn and {x1, x2, ..., xn} be a set of vectors whose span is S. The only vector d such that the cross product of d and xi is 0 for all i = 1, 2, ..., n is the zero vector.