Consider the Euclidean space in three dimensions E3 and the following system of equations 3x1 − 4x2 + x3 = −2 2x1 + 5x2 − 3x3 = 3

−x1 + 6x2 − 2x3 = 5

a) Introduce an auxiliary basis (unit columns) and auxiliary variables z1, z2, z3 and solve the auxiliary problem using the pivot method.

b) Call B the relevant basis. Verify that the inverse of the basis B corresponds to B−1 = T3T2T1, where Ti, i = 1, 2, 3, represents the transformation matrix at the ith iteration.

c) Verify that, given the inverse matrix B−1, the solution can be obtained as B−1b, where b = [−2, 3, 5].