Question

4. Let R3 be equipped with the dot product and let B = {(2,1,1),(0,4,-2),(3,5, -1)}. B is a basis for R?. Use the Gram-Schmidt process to convert B into an orthonormal basis.

Answer 1

Grom - Schmidt The Olthogonalization Docess Subspace Given basis {x1, x2 .xp} for of Rn detine (X2, V.) VI224 V2 = x2 V, (20V) ( 343, 2,) (23,602) 12 V3 : 23 در را ) (02₂) -(XP, VP-I? VAI ( 34,00) -V - (XP, Va) v2 ap Vp (200,27) (p-, VP-1) (12,1) orthogonal Set Lei, U2, ... Op} is Then the Ne ) sat / scet Wup!! the บ Il vill and ۱۴۱۱ basis. is ortho normal

3 - Hare (- ری (3) ر( 2- را ره) (ارا (2) (1- ,5 رو) - ره (2-h.0) = 72 ,(2/۱/۱) : )8 (!ا (2) (۱ را (2) : ع - 1 عر Lact ) لا (* (8) - ره د را (۷۱) 1 ) (ا را (2) : (o, -2) - (0) 4-2 )( 211,1), دارا21) 12 را (2) (21٫1 ). 2- به + - ہے - 4 (0) - + + ط 2 (ا را (2) (ا رہا ,o) : 43 ( را | 3 3) - (2- ما (0) - (3) 12 ولا ( 3 ) : ا ) " 3 3 141 140م :(1) v 2 //vi/| 5Y

19, 13 (xs ,۱) ۱, اور ولا ) 2 V₂ (۱۱۷) ( 192 19) | (.را (2) (را:2) (ارد3) - (1- رو رد) - (1,(2) (آرا 2) ( ا ) (2,51) . ا (-* - )(33) ا) 3 - (3,5, -1) - (را21) ا- ك+ ) ++ہا - + 55 + 1 3 3 3 (- 3) 3 و4 + 2 + 4 ه) g

(35,-1) ~ 1 (211,1) - 56 x 9 - 구 3 6 3 14 2. (3)S, I ) - 5 (2,1,1) - 28 1 ( 29 2 4 =3응 †56 196 | 5-5 - 308 3 87 3 87 ) 66 -1 - 308 +10 196 120) ( 3 8 8구 6. -12 13 - 5 [ 29. 29 29 12 -12 ) V3 ( ) 2 29 29 29 3 () 옮 81+ 36+144 29 8u . The ortho normal Set is Coor 13 (2,1,1) 3 - { ge) 110211 3 | 5% ( ) 9 20