Question

W is a rele that A linear transformation T from a vector space V into a vector space assigns to each vector 2 in V a unique vector T() in W. such that (1) Tutu = Tu+Tv for all uv in V, and (2) Tſcu)=cT(u) for all u in V and all scalar c. *** The kernel of T = {UE V , T(U)=0} The range of T = {T(U) EW , ue V } Define T :P, - R K by T(m) = PO by 1 p = 0(1) a. Show that I is a linear transformation. b. Find The kernel of T and the range of T.

Answer 1

Answer : Tip IR² defined by T(P) = (pro) P(1) where P2 ů a polynomial vector space & I of degree at most 2. het PEP2 implies that - p in a polynomial of deguee at most a. het p = ao tai x+ Az x² i ait IR T(P) = Tlaotax taqx²) = (p(ol' P(l ao tulo) +92102 Taotai(1) + a2(132 mation Taotaita (9) show that I w a linear transfor. is het p q q E P2 such that plu)- no tarda, x 2 ; ait IR q (0)= bot bu + bexi bitir

consider T(P+q), - T(P+q) = T((aotain +Q₂x2)+ (bot bine + b2x2)) = T (aotbot (atb, ) x + (Az rb2) a2) aotbo | Clotbo tautbit aetha) Now Consider T(P) + T(Q), T(P)+7 (9)- T Cauta, x + 2x2)+(bot bia+b₂x2 ao bo | 7 Got aita, I borbitba/ ( aotbo au taita, + bot bitba) ② from 0 2 , TLP+q) = T(P)+T(q), whoo Peq are any arbitrary ebent im Pa. ne a

(ii) het PEP₂ and cur any scalar un IR. Consider T(cp), T (cp) s TIC (Aotwm +Q, x)). = T( cao tea,x + ca, x²) I Cao caot catca, Now consider CTCP), CT(P) = CT Caotain +Q₂x²) a chao Taota, +A₂, from 2 , T(CP) & (TCP.) from (i) & (li), T is a linees transfor- mation. 1

I Kernel of T = { PEP2 a T(P) = 0 - 2 notara + a₂x² EP29 Pz ġ li to 10 Taotaite haot anta₂x² EP2 i varp 2 Co = and aotluta=08 2 1 an + a₂x² such that au +2₂=0 and ai ER ? Range of T = of TCP) ER? : PEP2} i ERC . GIR² Kaotatha) laotata aь & ao, ar, 92 ER al loto implies that Range of T = 1R27