4. (a) Determine the linear transformation T: R2 R3 such that T(1,1) = (1,0,2) and T(2,3) (1,-1, 4). (b) Determine the linear transformation T: R3 R2 which maps the basis vectors {(1,0,0), (0, 1, 0), (0, 0, 1)} of R to the vectors {(1, 1), (2,3), (3,2)} respectively. (i) Find T(1,1,0) and T(6,0, -1). (ii) Find the null space N(T) and the range space R(T). (iii) Prove that T is not one-to-one but onto. 5. Find the matrices of the linear transformations with respect to the given ordered bases: : (a) D P4(R) P4(R) defined by D(p(x)) basis {1, x, x2, x3, x} for both P4(R). d3 dx3 = 3- (p(x)), with respect to the ordered (b) T: P3(R) M22(R) by [2f" (0) f(3)] T(f(x)) = f'(2) [21" with respect to the ordered basis {(1, 2, r, r} and { (6 %)], [ }]), ([1%] [89]}