# Math Help - linear programming proofs

1. ## linear programming proofs

i have these two questions that i am having trouble understanding.

1. show that the linear kuhn-tucker theorem implies the complementary slackness conditions.

2. show that if A is a linear transformation and A' is the dual transformation, then the matrices of A and A' in the standard bases are transposes of each other.

i'm not sure if i'm understanding what the linear kuhn-tucker theorem is saying, because i have found many versions of it, but none of them were comprehensible for me.

2. just as information, the linear kuhn tucker thm states that x* in R^m is a solution to the linear programming problem <c_f|x>=f(x) ->min and Ax<=b if and only if there exists a vector y* in R^n, y*>=0 such that L(x*,y)<=L(x*,y*)<=L(x,y*) for all x in R^m and all y in R^n with y>=0

the complementary slackness condition says that such (x*,y*) satisfying this condition also satisfies y_k*(g_k(x*)-b_k)=0 for 1<=k<=n

*note L(x,y) is the lagrange function L(x,y)=<c_f|x>+<Ax-b|y>