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 = (](/WebImages/39/find-the-orthogonal-projection-of-v-7-4-18-16-onto-the-subs-1119803-1761595669-0.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 = (")
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++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, -4247/414, -285/23)T
