Find bases for the kernel and range of the linear transformations T in the indicated exercises. In each case, state the nullity and rank of T and verify the Rank Theorem.

Exercise 4

Data From Exercise 4

Let T : P₂ → P₂ be the linear transformation defined by T(p)x)) = xp′(x).

(a) Which, if any, of the following polynomials are in ker(T)?
(i) 2

(i) x²
(iii) 1 - x

(b) Which, if any, of the polynomials in part (a) are in range(T)?
(c) Describe ker(T) and range(T).