Question

2. Let T: P2 P2 be given by T (p(x)) = x2p"(x) – S p(x)dx a. Show that T is a linear transformation b. Find Ker(T) and its basis. Is T one-to-one? c. Find Range(T) and its basis. Is T onto? Verify the dimension theorem.

Answer 1

7 T: puuppl be given by T($)) = xp'a) -> [bros da Let Þ(a), a (a) Epl Then T (tos) +20) Dop (e) -t x (ptasa da +((P +1)(w) (p+q3'te = {A}"co) - a fhosen Jacoon = T(m) + T (ecm)) + 2q" (1) - X Let CEF. Then T (epen) are f'(x) - a jetragen S'teoses] -e T(hr) [~p"Ca) flence T is a linear transformation.

Let bra) =adtbatc € kert: Thus T (bra) Tlax" +64 +c) = 0 = =u, Hence 2aa - a (ax + bx+c) dx] a [ [18++ button Zaar a z? -) Zaxy an an ea t t + b2 ba 2 = ax = an + + 을 ا

an a + Herce A te 2)2 = equating the Coefficient both sides we get aso, and 블을 b b=-20 This bate -2ente Thus Kert = N and bou) ce(1-21) ((1–22) : 647) Hence kent = 4(1–2a) ? Hence ((1-2x)) is basis of kert 1. Since kert 7 Col. Hence Tisnut Since {e, a, at} is a basis of h. Then T(1) = -x, T(2) = -1 T(x) = 2x - 72 Hence Range T = L {TO T(1), T(X), Tay? = 4-2, 22, 220 - 24 = Lex, 22% - 7 / = {x, 22_1 dimkort one ore

Now we shall show that indefendent Set {a, 2 x 33} is a finearly eiat G. (22 - 2 > G- Equating the coefficient of 3 x +2012 SO a anda in both sides we get e,= C2=0, e, 5o. Herce {x, 2 - ) is basis of Imt Sina dim Im T = 2 Hence T is not onto. Now dimkert t din Imtadion 1+2=3=dimP₂. Hence proved tdim P2