Question

2. Let T: P(R) + P(R) be such that Tp(x) = P(1)x2 +p(1)+ p0). a) Show that T is a linear operator. b) Find a basis for Ker(T) and a basis for Range(T). c) Is T invertible? Why? d) If possible find a basis for P(R) such that [T], is a diagonal matrix. e) Find the eigenvalues and eigenvectors of S=T* - 31.

Answer 1

3 let T: P(1) - P (IP) Tlp (*) = $(i) *° + b(u) nt bro) let p(2) g(n) t P2 (12) 2) T ( Pen+ g()) = T ((b+8)(x)) = (b+8)(132? (b+9) (1) + +9)(0) = ($(1) +901)]n?+ (501+ 901]nt brost 90) = P(n? + goin?+ bluntg 03 nt prost 9 (0) = pC1n2+ point poo) + gan?+ 9100 + 9(0) = Tibo) + T (9 (12) & pas, g(n) € Pa(IR) 2 EIR. be a scalar Now let + past pa (IR) 3) Tlab)(as) = (QA) (1) n?+(«p)(I) ** (P) (0) = 2 p (1)na td þ(in to broj ca (bron?+ P(1) ont pcos) =& T(pr.) T it linear.

let Þ(M) E Pa (IR) " p (n)= an?t butc for some a, bicFIR tato let po e kerl(T) 3 T (p (mg) = 0 (0 - Zero polynomial) Þ/0n*+plu) 3+ Þ(0) = 0 >> (9+b+c]n? + (apbtc)ntero 7) atbf so 29 a+b=0 a E-b + 0-0 23 2) þ (n) =anan = an (n-1) 2) kerl T= { pade Pesclay | ben) = an-an >> Kerl(t)= { acnen) I QELP? 2) Basis of keul (T) = { (27-2)} Pang (t) = { Tlp (m) / pin) = P2 (12)} sy pcu nºt bum + bcos | ben ) G Pale) ? ={Þ(1)(n°+a) + bcos pos E PELUR)? as Bases af Rang (11= {(272) , 1}

Inventible we know Hat T: P2(IR) + P CIR) 14 ips Mullity (T) = {0} Here Mull (T) Fo? invertible 2) t is not use Matrix Representation of T ay T (b (1) = Þ(linnt beint þ (0) Sininay be standard Basis for Palle) let = 1.71 +in+1 so T()ntnt! T (n) = 10²+1 n to - 0.1 +1.nt 1.72 T(N2) = n²th to 0.1 tintlona 6 O TIB CA Find the eigen veue of A; To solve (A-2150

ie 23-34?+80-0.50 A(2²-32+2) =0 dzot d²-32+2 o 22_2d-d+2=0 d (d-z)-1(2-2) = (2-2) (2-1)=0 So d=0,112 are eigen vale. Now find eigen veeter. By solving (A-df)x=0 *-(C) to ay par doo .m ilta Io. o Eutrutine ve Euratu 14 Sot n=0,12=-13 Įsura R.GIR? so eigen space Edro = {(6) EP03) Pre (9): mere? vs = ( 9 to eigen vector

fa d=1 nu ܘ .܀ u 0 M "₂ 23 ht"370 nith, co 5) 15-12 n=-13 EDELE (5) : 115-hey mq 5-ng? 8 (A) MER}={» (4): neke eigen veeter Consponding to dalis par d=2 11871281-13 236768 다.

-> ang=0 , ni-M2443 0 nifh2-1320 -12+1320 t 12-13=0 Ed=2 หน nicol horno * * * * * * . { (4 nex = { (i): nein so eigen vecten V₂= (3) So, if we take B- Ploolri), (1-04), (011) Basis then Itlo is diagonal ie po st 0 1 2 P'17, p = 0) when po pof{n+93, ** *4), (? so teue B= if the Required Basis