For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z.
(1) V=P_3 (R), T(f(x))=f"(x), and z=x^3
(2) V=M_2x2(R), T(A)= (Matrix) [1 1]
[2 2] A,
and z=[0 1]
[1 0].
For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z.
(1) V=P_3 (R), T(f(x))=f"(x), and z=x^3
(2) V=M_2x2(R), T(A)= (Matrix) [1 1]
[2 2] A,
and z=[0 1]
[1 0].