Homework SourseList Let x 3 6 2 and y 5 3 8 Use the GramSchmidt process to
List Let x = [3 6 2] and y =[5 3 8] Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R^3 spanned by x and y. ![List Let x = [3 6 2] and y =[5 3 8] Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R^3 spanned by x and y. SolutionWe have x = (](/WebImages/32/list-let-x-3-6-2-and-y-5-3-8-use-the-gramschmidt-process-to-1094339-1761576779-0.webp "List Let x = [3 6 2] and y =[5 3 8] Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R^3 spanned by x and y. SolutionWe have x = (")
Solution
We have x = (3,6,2)T and y = (5,3,8)T . Then Proj x (y) = [(y.x)/(x.x)] x = [( 15+18+16)/(9+36+4)] x = (49/49)(3,6,2)T = (3,6,2)T . Further y - Proj x (y) = (5,3,8)T - (3,6,2)T = (2,-3,6)T= x1 (say). Then {x,x1} is an orthogonal basis for the subspace of R3 spanned by x and y. Now, we only need to normalize x and x1. Since ||x||= (32+62+22) = (9+36+4)= 49 = 7 and since ||x1||=[22+(-3)2+62] = (4+9+36) = 49 = 7, hence x’ = x/||x||= 1/7(3,6,2)T = (3/7, 6/7,2/7)T and x1’ = 1/7(2,-3,6)T = (2/7,-3/7,6/7)T . An orthonormal basis for the given subspace of R3 is {x’, x1’}.
