Question

Exercise 14.1. For each of the following functions F, determine if it is linear by using the viewpoint of behavior with respect to vector addition and scalar multiplication. If it is not linear, either find a pair of vectors v, w for which F(v+w) + F(v)+F(w) or find a vector v and scalar c for which F(cv) Ec F(v). [2.x (a) f – 34 (b) g = my ) = (2017 (12 (6) 2 (c) h = C

Exercise 14.1. For each of the following functions F , determine if it is linear by using the viewpoint of behavior with respect to vector addition and scalar multiplication. If it is not linear, either find a pair of vectorsv,wforwhichF(v+w) ̸= F(v)+F(w)orfindavectorvandscalarcforwhichF(cv) ̸= cF(v).

Answer 1

htv-G). W= ) SR? Then f() = ((?:)) = 39] fCW) = f (05:)) - Cu toat] Now it w = {4, + ( + 1 = ( x + ( 2 ) - 343.470 (024-39-33] - 13 -3 3] f( utw)= [2ln,th - 1399)+(2k-39a) = f(7+few) flutw) = f(u) + f(5) • Let CER, J = [M] ER² 124)=107) if(()s ? (14 - 24-307) [224] [*]-440) if(CJ) =cfli f is a linear transformation. (6) g( [4]) = ny Let v = (-1)], w = (4) ER? • Let = 1:] W-(0) Then glül (1) 10/30 Then g(0) = n,Y, gLW)=k g(W) = (0) (1)=0 tw Now ū tw = putų) W =(!)+1°) =[+9)=(:) (Itz .: 9(v+w) = (UCU = 1 glutw) =(4, +4) LJ,+2) but gult g(W)=ot0=0 zny, tužtųY, +ų Ž ::g Cū+W) #glu) +90 but g (J) + 9(W) = n, %, + 2Y2 :g is not a linear transformation. gljtw) 910)+ g(W) is not a linear transformation 10

9h([7])=G) • Let u = w = ER? 2 2. Then, t(il= hlw= ūtw = 4th Sith 12, + z hlutw) = Z+22] nity I+Y =h(5)th(W) ..ni tw) = 4(5)th(W) • Let CER = ' cu then cuaca 4 z cz h(co)= "cz cu =chli) h(c)= chlo) h is a linear transformation.