Polynomial Linear Transformation

[Needhelpzzz]

Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue?

What I did was
T(p)= (lamda) p = q (Lamda) p(t+1) = q(t) (Lamda) p(t+1) = p(t+1)
(Lamda) = 1

Eigenfunctions are nonzero constant polynomials.

Is this right though?

 

[LCKurtz]

Let V be the linear space of all real polynomials p(x) of degree < n. If p ∈ V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue?

What I did was
T(p)= (lamda) p = q (Lamda) p(t+1) = q(t) (Lamda) p(t+1) = p(t+1)
(Lamda) = 1

Eigenfunctions are nonzero constant polynomials.

Is this right though?

That may be right, but you certainly haven't proved it. What you have to work with is$$
p(t+1) = \lambda p(t)$$Surely ##\lambda = 1## and ##p(t) \equiv c## (a constant) satisfy that condition. But maybe some other ##\lambda## and ##p(t)## work too. You need to prove that ##\lambda## must equal ##1## and then show the only polynomials that work when ##\lambda =1## are constants.