Question

Please attempt both.

1. Perform the marquis de Laplace process in both possible ways (remember, you choose the lead vector) on the basis 3 -2 -1 to create an orthogonal basis for R2. Geometrically represent each process in 3 plots: The first two vectors, the projection and perpendicular, and finally the new basis. 2. Perform the marquis de Laplace process on the basis 3 -2 1 3 -1 3 5 to create an orthogonal basis for R3.

Answer 1

ANSWER

0 0 Given 604[:] [:]}={u, ugy **01011 (-2)*(-1)-2)x(1) = 2-3 = -1 0 so, the sets is linearly independent and hence normal basis of R². but (-2,1). (3,-1) =(-2)*(340)x61) = -6-|--7 so sis not orthogonal basis U. Uz to We will now use Gram schmidt ooth ogoolization to get orthogonal basis of R2 corresponding tos let s'= {0, Val is the required thogonal basis of R². Vi= u = [7]a dogo la= [] vi Vivi 13,-1)-(-2,01 (-2,1).(-2,11/ V2=U2 - UaVi - (891 + 2) 7 / 1

*- 444 3 = 14 5 7/5 13-14 15-14 t+7/5) -5+7 So, we can check (-2,1). () - 2 + 2 = 0 59-41:] . So tha vezuinol orthogonal basis of R² - 내 90' V2 = 42 - Prosv, U21 U2 g let ish s={(3) [:] [:]} = twowyway > let we say. D- 3 3 1 - 1 3 5 - 1

3 (-) + (-2)/- 1 - 1 / +-) (-1) 425/3=114 +(-3)??? = 60°/-231+5)*8-1.3(-1+3)+1-3 (5+3) with respect to istrow = 42)(6)-3()+318) =-12-6+ 24 .6 to ind (By de place 50, Sis a lineally ependent set process) containing three vectors so s is a basis of R3 but (-2,1,3)-(3,-1,5) = -6-1715-800 sos is not an orthogonal basis let ,st=. Mi Vaivg] is cooresponding orthogonal basis of R3 corresponding s=44, 42, 43) Vis U, a wie Meno ug 3 3 5 V2 = U2 U2. Vi Vi.V - 13,-1,5). (-2,1,3) (-2, 1,3).(-2, 1,3) [3 -2

二 -H+15 - 2 +1+9 3 3 (3) 1-11-11) 又十 colit dle ele 29 乡 介 5 | 1 4-4 1 35-12 四4 =15 g 23 59 v. - 29/4 -V4 Q911

V = -1 12913 ,M3= 4 231 VB =U3-U3V VINI V) - La V2 V 벗 (1,1)-213) 13 (31,3) (31,-) () ) (끝) t 29 It 구 -6-1- 3 -- HET t8 4+ +9 ! - 23 3 3 ㅋ 29 ㅋ (44123) * 231 2 In 4 - 75 - x1_ 29. 3 t 1491 25 73 X 씀 23키 9 110 t - 2 24 25 키 4 명 3 t lt 14 tlɛe

100 tho + 5그 19키 It 23] 3- 10 블 be XSI 71 + 25X-11 tx it 네 5 구 25X23 txit 3 10 125 497 1411 - 귀 0-725 497 It + 275 497 - 497 1355+275 497 It + 595 SES- 5901 ttb5- 497 49 | 141-10-425 T tbD 99 133 | 449 43554 295 L-444 +1065-545 49 t- 5 8 부 TE 13314 It 19 bl It so vz= 키키 이키 191 (t 1/11

so, Required orthogona basis for R3 ૧૧h ૬), ૧૧,