Question

Let T be a linear map from R3[z] to R2[z] defined as (T p)(z) = p'(z). Find the matrix of T in the basis:

4 points] Let T be a linear map from Rals] to R12] defined as (TP)(z) = p,(z). Find the matrix of T in the basis: in R2[-]; ~ _ s, r2(z) (z-s)2 in R2 [2], where t and 8 are real numbers. T1(2 Find coordinates of Tp in the basis lo, 1, 12 (if p is written in the basis qo,.., 93).

Answer 1

Given that T is a linear map from R_3[z] to R_z[z] and (T_p)z = P^�(z) T(P(z)) = P^�(z) (a) P_0(z) = 1, P_1(z) = z, P_z(z) = z^2, P_3(z) = z^3 in R_3[z] and r_0(z) = 1, r_1(z) = z, r_2(z) = z^2 in R_z[z] we have T(1) = 0 = 0.1 + 0.z + 0.z^2 T(z) = 1 = 1.1 + 0.z + 0.z^2 T(z^2) = 2z = 0.1 + 2.z + 0.z^2 T(z^3) = 3z^2 = 0.1 + 0.z + 3.z^2 therefore [T] = [0 0 0 1 0 0 0 2 0 0 0 3]_3 times 4 (b) q_0(z) = 1, q_1(z) = z - t, q_2(z) = (z - t)^2, q_3(z) = (z - t)^2 in R_3[z] and l_0(z) = 1, l_1(z) = z - s, l_2(z) = (z - s)^2 in R_2[z], t and s are real numbers \r\n we have T(1) = 0 = 0.1 + 0 middot (z - s) + 0 middot (z - s)^2