Question

DETAILS LARLINALG8 5.R.040. ASK YOUR TEACHER Apply the Gram-Schmidt orthonormalization process to transform the given basis for R" into an orthonormal basis. Use the Euclidean inner product for R" and use the vectors in the order in which they are given. B = ((0.0, 2), (0, 1, 1), (1, 1, 1)) -

Answer 1

use in the -mal have to find the orthonormal basis by Applying the Gram - Schmidt ortho normaliza- -tion process and euclidian inner product Rn and the nectors order in which in which they are ginen. B-f(0, 0,2), (0, 1, 1), (1, 1, 1) } Cet I , & Uz be three arbitra arbitrory binarly independent sectors. let 4, 143 are the required orthonor. mector & Now consider by Oprem Schmidt process les -2 Oy, 4,3", -(Ug-U2742-<0g -43342 Ug = 11 v - (Oj, 4,>", -(VJ -4274,- (Us - Ug343) there the ginen vectors are v=co, o, 27 = (0, 1, 1), V3= (1, 1, 1) "Therefore, 4, = 11 (0,0,2) (0, 0, 2)) 4. =(0,0,1) and (0,0,2) 2 U 2 i 2-(₂,4) 11 U₂ -ų, 4,7ual (0,1,1) - ((0,1,1),(0,0,1)>(0,0,1) 110,1,1) - <10,1,1),(0,0,1) > (0,0,11 (0, 1,0)

4g 43 3 - {Ugg e) 4,- (3,42 > U2 11 g < lg g4, > 4,- (3,4₂3 4211 ( 1,1,1) - (1,1,1),(0,0,1) (0,0,1)-((1,1,1), (0,1,0)(0, 1,0) [[ (1, 1, 1) - <(1,1,1),(0,0,1)>(0,0,1)-((1,1,1) (0,1,6)). (0,1,O) Il (1,1,1) - (0,0,1) -10,1,0) 11 (1, 1, 1) - (0,0,1) - (0,1,0) || (1,0,0) The required orthonormal basis for R3 px (0,0,1),(0,1,ol ,(1,0, (2,0,0)}.