Homework SourseGiven v find the closest point to nu in the subspace W spa
Given v = [], find the closest point to nu in the subspace W spanned by matrix.![Given v = [], find the closest point to nu in the subspace W spanned by matrix.SolutionLet us denote the vectors that span w are v1 and v2 Dot product of v1.v2](/WebImages/30/given-v-find-the-closest-point-to-nu-in-the-subspace-w-spa-1084893-1761570300-0.webp "Given v = [], find the closest point to nu in the subspace W spanned by matrix.SolutionLet us denote the vectors that span w are v1 and v2 Dot product of v1.v2")
Solution
Let us denote the vectors that span w are v1 and v2
Dot product of v1.v2 = -4 -12 +14 +2 =0 They are orthogonal set and v1 and v2 are non zero.So, they
are linearly independent.
ProjectionWv = {(v.v1)/(v1.v1)}v1 + {(v.v2)/(v2.v2)}v2
={( 2+24 +12 - 1)/42 }(-1,6,-2 ,-1) + {( -8 -8+42 -2)/(73)}(4, -2 , -7, -2)
=0.88( -1 , 6 , -2 , -1) + 0.33(4, -2 , -7 , -2)
= (0.44 , 4.62 ,-4.07 , -1.54)
= (0.44 , 4.62 , -4.07 , -1.54)
