But for linear programming a much stronger result, called strong duality, holds. We may show it by using the notions we introduced before.
Let's consider a feasible basic solution of problem P corresponding to the basic matrix B and define a dual vector u by taking the product of the costs of the basic variables for the inverse of B. It is immediate to verify the the value cx is equal to ub. Moreover u is dual feasible if and only if all the reduced costs are non positive, that is if and only if x is an optimal solution of the primal problem. In this case from the weak duality it follows that also u has to be optimal for the Dual problem. This argument proves the strong duality theorem, which states that if the primal problem has an optimal solution then also the dual problem has an optimal solution and the two problems have the same optimal value.