# Thread: Change of Basis Help

1. ## Change of Basis Help

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.

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.

2. Originally Posted by letitbemww
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: $T(x^2):=(x^2)(x^2+1)'=x^2(2x)=2x^3\in P_3$

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.

.

3. We help those who help themselves. Now that you know that this really is a valid linear transformation, what are T(1), T(x), and T(x^2)? How would you write those in terms of the basis {1, x, x^2, x^3}?

If you honestly don't know how to do any of (a) thorugh (d), you have more problems than we can help you with! Talk to your teacher about it.