Question

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(10 points) Find the closest point to y in the subspace W spanned by vì and v2. -4 -2 у 0 -1 0 -1 2 3

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Answer 1

Quest VE - 2 inta OT ܝ . ܝ Soln hore, V.V2 = -4-2 tot 6 {U,,U23 is an orthogonal set. من 0. this means because neither V, nor ₂ that {V,23 is LI- Since, is spanned we have that {u, N23 is by {U, U23 orthogonal basis Projwy and an we can the answer is calculate it as Lyou,> Ly.V2) ." + V2 Luov, <V2.V₂> (1-2-2) Net (1,+4+1+4) (-4+1-3) uz (16tita) (1, -2,-1,2) + ) 4-4,1,0,3) 16 (-10 + 24 1 2-0, 26 81 130 48 130 1 130

Bues 2 is a basis for a subspace W. 880 use the Gram-Schmidt process to produce an orthogonal basis for wo Solm V = X = (1,0,0,0) X2 - {X2,V,).Vi CU, VA) = (1,1,0,0) {(1,1,0,0) (1,0,0,0)) (1,0,0,0) <(1,0,0,0 741,0,0,0) = (1,1,0,0) - +11,0,0,0) V₂ = (0,1,0,0) X3 <*3, V,) Vi (23, V2), va <V,,V2) <V2, V27 = (1,1,1,0) - +(1,0,0,0) - + (0,1,0,0) = (1,1,1,0)-(1,0,0,0) - (0,1,0,0) = 10,0,1,o) {v., V2, Va} UB are orthonormal basis for IR3