MAT 168, Spring 09: HW 2 Part 3
Do problems 4.3.6, 4.3.18, and the following:
H2P3.1 Consider the regions defined by Ax =b,x≥0. Prove that an nonzero vector dis a
direction of unboundedness if and only if
Ad = 0, d ≥0.
H2P3.2 Consider S={x∈R3:x1≥x2≥x3≥0}, in other words,
−x1+x2≤0
−x2+x3≤0
xi≥0.
(i) Prove that if x∈Sthen αx ∈Sfor all α≥0.(A set with this property is called a
cone).
(ii) Find all linearly independent directions of unboundedness.
(iii) Show that the origin is the only extreme point.
Remark: This problem is true for S={x∈R3:x1≥x2≥ · · · ≥ xn≥0}for any
positive integer n.