Thread: functions and linear transformation

I'm not sure where to start with this:

Define h: P₂ --> P₃ by h(p(x)) = xp'(x) + the integral from 0 to x of p(t) dt.

a. Verify that the codomain of h is in fact P₃ and prove that h is a linear transformation.
b. Find h(p(x)) where p(x) = x² + x + 1.

Originally Posted by widenerl194

Define h: P₂ --> P₃ by h(p(x)) = xp'(x) + the integral from 0 to x of p(t) dt.
a. Verify that the codomain of h is in fact P₃ and prove that h isis a linear transformation.
b. Find h(p(x)) where p(x) = x² + x + 1.

If you are really serious then answer this question:
$\displaystyle h(ax^2+bx+c)=~?$

If you can do that then what is the generalization?