Linear programming proof

Question

Problem:

Linear programming proof

Consider the LP:

            Min c^t x

Subject to

Ax ≥ b, x ≥ 0.

One can convert the problem to an equivalent one with equality constraints by using slack variables. Suppose that the optimal basis for the equality constrained problem is B. Prove that w = _cBB^-1 ≥ 0.

Where c_B = coefficients of Basic variables.

B^-1 = Inverse of Basis matrix.

(algebra of simplex method can be helpful in this proof)