Polynomial linear operator problems

Question

Question:

Polynomial Linear Operator

Let T: P₃→P₃ be defined bu (Tp)(t) = p(t + 1).

a) Show that T is a linear operator.

b) Find the nullspace and range of T.

c) Let β= (1, 1+t, 1+t+t², 1+t+t²+t³). Show that Beta is a basis for P₃.

d) Find M(T,β, β).

e) Find the eigenvalues and eigenvectors of T and give the characteristic and minimal polynomials.

f) Exhibit a Jordan basis for P₃.