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 3

Data From Exercise 3

Let T : P₂ †’ R₂ be the linear transformation defined by

(a) Which, if any, of the following polynomials are in ker(T)?
(i) 1 + x (ii) x - x² (iii) 1 + x - x²

(b) Which, if any, of the following vectors are in range (T)?

(i) 

(ii)

(iii)

(c) Describe ker(T) and range(T).

a - b b+ c. T(a + bx + cx') )