In each case verily that PD ← b is the inverse of PB ← D and that PE ← DPD ←B = PE ← B, where B, D, and E are ordered bases of V.
(a) K = R3, fl = {(l, 1, 1), (1, -2, 1), (1, 0, -1)),
D = standard basis,
E = {(1, 1, 1), (1,-1, 0), (-1, 0, 1)}
(b) V = P2, B = {1, x, x2},
D = {1 + x + x2, 1 - x , - 1 + a2 }, E = {x2, x, 1}