Homework SourseFind the orthogonal projection of v 7 4 18 16 onto the subs
Find the orthogonal projection of v = [7 4 18 -16] onto the subspace W of R^4 spanned by [-3 -2 -1 3] and [2 4 -14 0] ![Find the orthogonal projection of v = [7 4 18 -16] onto the subspace W of R^4 spanned by [-3 -2 -1 3] and [2 4 -14 0] SolutionLet u1=(-3,-2,-1,3)T and u2=(2,4,](/WebImages/47/find-the-orthogonal-projection-of-v-7-4-18-16-onto-the-subs-1147993-1761617497-0.webp "Find the orthogonal projection of v = [7 4 18 -16] onto the subspace W of R^4 spanned by [-3 -2 -1 3] and [2 4 -14 0] SolutionLet u1=(-3,-2,-1,3)T and u2=(2,4,")
Solution
Let u1=(-3,-2,-1,3)T and u2=(2,4,-14,0)T.Then,Proju1(v)=[(v.u1)/(u1.u1)]u1=[(-21-8-18-48)/(9+4+1+9)]u1= -95/23(-3,-2,-1,3)T =( 285/23,190/23,95/23,-285/23)T . Also, Proju2(v)=[(v.u2)/(u2.u2)]u2= [(14+16-252)/(4 +16+196)]u2 = (-222/216)u2 = (-37/36)( 2,4,-14,0)T = (-37/18, -37/9, 259/18, 0)T.
Then the orthogonal projection of v onto W is (285/23,190/23,95/23,-285/23)T+(-37/18, -37/9, 259/18, 0)T = (4279/414, 859/207, 7667/414, -285/23)T.
