Some examples related to duality theorem in linear programming

The duality problems in linear programming may read as follows. Suppose an m x n matrix A —{aij), a column vector 6 = (όi, • •-, bm) and a row vector c — (ci, , cn) are given. The primal problem: Find a column vector u = (uu •••,un) which maximizes the linear form cu subject to the conditions Au<,b and M^>0. The dual problem: Find a row vector v — (vu ,vn) which minimizes the linear form vb subject to the conditions vA^>c and v^>0. In each problem a vector satisfying the required conditions is called feasible, and if it attains the maximum or minimum it is called optimal. These problems can be represented by the following tableau: