Question

Let V = P1(R) and W = R2. Let B = (1,x) and y=((1,0), (0, 1)) be the standard ordered bases for V and W respectively. Define a linear map T:V + W by T(P(x)) = (p(0) – 2p(1), p(0) + p'(0)). (a) Let FEW* be defined by f(a,b) = a – 26. Compute T*(f). (b) Compute [T]y,ß and (T*]*,y* using the definition of the matrix of a linear transformation.

Answer 1

T:P,(IR) R is giren. T is defined as TCP cao) = (pio) - 2PLU , plo) + pros ..pCal. E P.CR... ß = {1, 2} is a basss for v = P(R) and y=-211;0) » (01)} is a basis for W.R la f€wie f: WR is a functional, defined f(a,b) (W=RRY /(a,b) EWART by -a-2b Now it is a definition lie Tif): fot T² (f) f) : foT -it So, T*: W* → V T* :-(17)* ( P. (R))* is defined as = fotot few V f€ w* = (R)* p(a) EP, (R) simply take P(X)= ax+b E P. (IR) Where a, b ER Tal (pca)) = (607) (ax.+6) f (T can+b So, for any 2

or, T*(f)p(~) = f ( Þ(O) -2p0), 60+1 HCo)+ p'cov) f(b-2(a+b)-, -b+) pin) = anto f(<2a-b), (a+b P(0) = b Þ(1) - lat.... 62a-b) – 2 (a+b) or Tl) (an+b) - 42 -36, P(x)= a sv can+b) EP (IR) (Ams) = atb b Now T (1) 25, (1-2x1, ito) (1,1) (-1).(1,0) + 1,(0,1) (0-2x1 , 0+1) (-2,1) (2).(190) + 1. (0,1) Se Tin) = So, ܟܝܙ ته [77 - J* *). com Since [T*) 13 DES that meet you so, [T*] [T I shall it for this problem elaim t Bit -3)* A 1 62 +1 (Ans)

Now proof of claim for this problem ß = {l, n} - basis for p = P, (IR) ४ {(1,0)..,(0,1).} basis for W=; het ß* = {1* , x*} & 8 * = { 8*, & } be y* respectively, where 3) = (190) 82=(0,1), Now to determine & x*, let axtbey be ang and then [can+b)] basis for and W B (6) a .Sc, * b a. 1* (antb) & 2 [antb) Similarly let (2,4) € W = RP be any Then [com byly = 2 So, 3*(x,y) , 77" (ayy) - 22 by definition Now T*(3*) 8&T and lienee T*(0,*) (1) = (0,76 T) (1) (8,*T)(I) = **(T(11) = 7*(-1,1) = 7,* (1-1.8, + laimez !) =(-1) Y* (%).+1.3 (42) =(-1). 14.100 fem*+<**} = 1,7* + (-21*) (1) 1 8 9 26 27 28 29 30 = -1 *

Again T*(74*) (*) = (367)() x*(T)) = 2^(-2, 1) 3,* (t2). 7, +1.82) (69.1* +(32)3*) () 61.1**.(-29.71 -2 Hence T*(**) 1 Again, T²(%) = 80T 7*(0,*) (1) = ( 2T) (1) - 0*(TU).) = 3 *-1,1) (1.5* + 1.8*J (0) & T*(**)(x) = (0,467 )(x) = 8,1 100) = Y(-2,1) (1.19 + 1.2* ) ) So, T**) 1. 1*+ 1.3* ܝܙ دهه -2 Therefore [T" 18 T - - -2 This prove ores our claim [T*]** ([1])