Consider the Euclidean space in two dimensions E2 and the following system of equations 3x1 − 4x2 = 2 2x1 + 5x2 = 9

a) Introduce an auxiliary basis (unit columns) and auxiliary variables z1, z2 and solve the auxiliary problem using the pivot method and choose pivots on the main diagonal of the relevant matrix.

b) Call B the relevant basis. Verify that the inverse of the basis B corresponds to B−1 = T2T1, where Ti, i = 1, 2, represents the transformation matrix at the ith iteration.

c) Verify that BB−1 = I.

d) Verify that, given the inverse matrix B−1, the solution can be obtained as B−1b, where b = [2, 9].

e) Introduce an auxilliary basis (unit columns) and auxilliary variables z1, z2 and solve the auxilliary problem using the pivot method and choose pivots off the main diagonal of the relevant matrix. Comment on your findings.