T ( p ( x ) ) = ∫ 0 1 p ( x )...
(a) Show is a linear transformation.
(b) Compute Is one-to-one?
(c) Show that is onto.
(d) Let be the standard basis for and let be a basis for . Find .
(e) Use the matrix found in part (d) to compute
Answer & Explanation
(b) As explained in the comment above,
We see, for instance, that , so is not injective.
(c) To see that is onto, let be arbitrary and consider the constant polynomial . Then .
(d) To find , we take the basis and evaluate at each of these polynomials. We then get , so