Question

Question 2. 20 marks Determine whether the following sets form subspaces of the standard 2-dimensional Euclidean space (R%. +. :): (a) A = {(11.12)|*1 = -2). (b) B = {(21,12) 2119 = 0). (e) C = {(1, 12) 12 = 2x;}. (d) D = {(2,12) ||*|-|-2), where x. denotes the absolute value of x, for i=1,2. (e) E = {(*1,22) | } = =3)

Answer 1

2) VECIR, Z_) $ Euclidean rector space. Note that a non-empty subset W of ris : subspace of r if in OEW i) x-y EW, Nx; YEW S lii LHEW, K&Ewf scalare. I @ A= {(24, sis) / 4 =-xi ? cv. AS 05-0 .coo) EA - let cau, ness & 9,92) E A A 24 =-Hs of y=-72 consider, cau, 22)-(G.,Y) = (x4-91, 2012) - Nou, suay, anh (Y) = -x + y = - (25/2) - cay, and (9,72) G K 2 let a be any scalar. consider, a co4, x) = (224, das j. Now Lau = 4(-247 - - (das). = 2(4, E 4 - 6 Hence, by 0.®*®, His subspace of V. I RE 1 B = 6 cap, as) / 34 sz=0} svi - Af0 .000 - 10.02 € B it con, 2) of (77) E B 24h.20 £9, 42=0 I consider cau, 227- (9.,Y2) = (24h , th-Y2) I Nocy, fault (24-4) (22-Ye) = x xes-2972-9, tylla co-4f2-4, auta - cau, as) - CJ, Y) & B . Hences R is not subspace of v

© C = d cau, as) / xq = 234 2 cv I AS o-2.0 t (0,0) E C 14 causes) & (9,72 EC ) 22=224 4 I consider, cay, so-cy, 72) = (249, 227) Nou Ces - 12) = 224-24, = 2 (24-17 1. cay, a)-(Y,,Y2) E C @ lct a be any scalari Il consides, acaq, as) = (224, dres) Now, dxi - 26284) = 2(arul 7 a cau, as) E C - Hence, by co, fw, ċ is subspace of Y | @ Daf buma) / 1001 1201 ci vo LAS 1012101 = (0,0) ED O I let cau, sef cy, Y) E D = fal = lalf Tilaly consider csu, xes) - (41,42) = (4-y,, -72). I now, 14-yils 1041 tly,l = lnal+ 1121 & 122-12 :ice: 1 24-YA1 E104-724 For eng. (au, Ho2 = (2, 2) st. 12/=/214 (91,42) = (1-12 Sit. 111=1017 а (4, 4 ) , (94,2 е but c24,2)-(41,42) =(2, 2):-61,-1)=(1,3) bud H +131 Hence, com.au)- 9,72 ED. .. D is not selbspace of v -

I @ E = {(21., 22) / 2 = x2 } E V. AS 0²=02. 2 (0,0) E E.. let cau, sez) & (Y1,727 €. E = ai= x2 49% = 42 (4 862) - CYLY2) = (24-Y a -72) now, exy-y2= sela2a44 ty, ? = 22-2xy + y 2² ... Ł coe-72)2. a. ( 29, 2) -4,42) & Ei- for eig. (2,+2) E & f c -12E E . (2,22-(1-1)=(1,32 & 1²832 d (2, 2)-(1,-1) - (1,3) & E Hence, & is not subspace of v C E = L cap, sez) I or = x2 } . ed caq,xe) / xu = I x } & cxe, xz) / 109/=lasi ED D is not subspace, so & Nis not. UN N I so