Question

In the following transformations:

a)Find the Kernel and Image

b)Find dimK(T) and dimI(T) and show that dimK(T)+dimi(T)=dimV

c)say if the transformations are injective, suprajective or bijective

i) such that:

T:R? → R

ii), such that:

iii) , such that:

Answer 1

T: IR3 R² be a Linear Transformation such that T(x19 12) = (x-4, 22) - @ For Kennel as By the Definition of Hull space (I) NOT : {XEIR3 TCX)=0} Hence By 6 - T (248 12) = 0 3 (x-4, 22) = 2010) x-go, 2220312zo] re-gro [x = g = k lear) H = {(x19,2) } = {(K 16,0)} KERHELI) = M = {(2,2,0): k=1} DIM M) = Dim kori Fon Range space of it) € Image of (t)- By the Definition RO) = { tex) EIR2 :} T(x19,2) = (x-4, 22) consider the std - Basis of 1R3 I B {es = (doo), (012,0), (o oid) Tles) = 110,0) = (210) Tezl = T (onio) = (adio) image Al Tez) = T10 old) = (012) Bats of Ro - is vectors (210) (-10), 1012) to 1. 106 02 J 02 R₂4P₂ thi By @ I do? -2 b 1 W

de Pange (II = {(310), 1012) , 6 Dim Kent) = 1 2 Dim KEROT & Dim RJ = = Din(n) d-12 = 3 DIMRO = 2 3:3) o lat T is the one Tally. T(X). T(X₂) Tlxs, Y; ,Zs) = - 1 I (X2 192 122) (g-Ya. 222) = (x2-G, 922) ! By O- xs-yd = x2 - y2 222 =222 By and it is not clear that (xs Gd, 2) = (x2, 22, 20) 2 X & X2 so t is not one-one ne Hance T is surjective - T: Max (IR) Mz x2 R) be a Transformation such that Tlal 912 913 ) = 1911-912 913-913) - (921 922 923 @ For Kennel It) We knew that pos = 8X E May (1) TX)=0} kenct = 1911-912 913-913 -100 1921 922 923 911-912 = 0 913-9130 Tall= 9126 KERT) = MOJ = San 912 913 ) an-912:07 (as an 923) - lain 912 913 ) 10

draget) Range (11= {(210), 2012) ] 6 Dim kentt) = d 2 Dim KERCtr Dim R = 3-DINN) KIRIT) = Mo Sran 9, 913 921 922 923 sid tol lood looo loool Dim KOT & DIM MOT = 5 For Rt bet consider Basis of Maxg (IR) Bassool lololo OOLOOO dol, lood) By D T (ed) = ( 10o= (tool= ( ) = Tesla Todo) (0-10)= Lo J) = 62 Mes) = T (002) = 0o 2-2) = (60) = 23 They) = T (000) = 1000) = Coo) - vin Thes) = 1180) : .00)- 10%) = us Meo) = T1000) = 1000 = 100) = Ro- & ravectors of es, U2, U3, U9, Usi Vo } hrayıl 00) 0 0 2 0 2 0 0 o-do o oooo DOO 0 70.000 OOOOOO70 Range (I) RT = 6 DIM RG = 2 DIM KERU = 5 DIMR t = 2 DI KO + DIMRI) : 512 - Dimlv)

bet tis the one Taking tixol = H2) = T( 914 912 1913 ) = T111 biz baz - (911-912 913-913) = 1 big-b12 big-b13) 911-912 = big-biz this does not show that 91 - bly912 = biz. --- - So X & X2 Hence I is het one one Sot is surjective T! IP (IR) P(R) such that - TP) = nple) + Plex) - 4 © MM = RERO - WIT Mot t o Mos = {X E HEUR) & TX)=0} By G - TLPLK) = 0 xpe) + Ple) = 0 qPP - - 4p(x) da d Pire) = -xdx Play Integraty both side. In plu = -x7c 1 pers: Gera re Ple) = Gent is not a polynomial of Degree less than equal to a 9 So plezo Hence Mit = sol as DIM MOJ = 0. Range ty - Corolder std. Basis for IRR) - Bi & dixix2} T(4= wlto oltlixton 240-23 Mr) & nonat & ut outdone toon3 Tn2l = 2.n2+ 2a= ond + 2-x+ 622 t 1.23 Basis ROM = Sla vectors of (a, Ha?, mp3) } = Sx, 6132, 2x7x33

6 DM Mo Dim penal son Dim Ro = 3 Dim Mo + Pro Rt) = DIM)=3 0+3=3 = 3 - 3 6 AS DIM M = 0 Hence T is one one and on to so I ts Bijective