This assignment has four problems, of equal weight.

1   Given any matrix A, and vectors b and c (such that the multiplications below are defined).

• Prove: If the vectors x and y satisfy Ax = b,   x > 0   and   A^Ty > c then c^Tx < b^Ty
• The following fact is known as Farkas' Lemma: Given any matrix A and vector b, the following statements are equivalent:
  • The system Ax = b,   x > 0 is consistent
  • A^Ty > 0   implies   b^Ty > 0
  What part of the Farkas' Lemma follows from the special case c = 0 ?

2   Text, p. 57, Exercise 2.

3   Text, p. 57, Exercise 4.

4   Text, p. 61, Exercise 2.