Consider the map T : P2 --> P3 such that T(f(x)) = (x^2)f'(x+1)

Write the matrix representing this transformation with respect to the standard bases of P2 (1, x, x^2) and P3 (1, x, x^2, x^3).

I really don't understand how to do this because when you plug in x^2 and x^2, you get an answer that's of a degree higher than 3.

No, you don't:

For the rest of your questions give us some self work and say where did you get stuck

Tonio
If you're feeling generous, could you help with these that I also don't get concerning this problem:

a) Find a basis for the kernel of T and the image of T

b) Is T injective? Surjective?

c) Does T have a left inverse? Does it have a right inverse? Find either if it exists or prove that it does not exist.

d) Find the matrix representing the same transformation with respect to the basis (x^2 - 1, x - 1, 1) of P2 and the standard basis of P3.