Thisproblemsetgivesuspracticewithcomputingthedual,andwiththeoremsofthealter-native.Duality theory is one of the most powerful aspects of convex optimization and it is usefulformanyappliedandtheoreticalpurposes.It illustrates (depending on howyou solve it) the power of duality and complementary slackness, as the hint suggests.It's a deepstatement about geometry of linear inequalities, and I encourage you to imagine applications ofthis (or a similar)result.

For a given matrix A, show that the following two statements are equivalent:

(i)Ax 0 and x 0 implies that x = 0.

(ii)There exists some vector p 0 such that p^TA 0, and p^TA < 0 (strict inequality), where A denotes the first column of A.