Find the closest point to y in the subspace W spanned by v1

Find the closest point to y in the subspace W spanned by v_1 and v_2 where y = [3 -1 1 13], v_1 = [1 -2 -1 2], v_2 = [-4 1 0 3]. Justify all steps.

Solution

We have v_i.v₂ = -4 -2 + 0 + 6 = 0. Therefore, { v₁, v₂} is an orthogonal set. Further since none of these 2 vectors is zero, these are linearly independent and hence form an orthogonal basis for W. Hence the point closest in W to y is proj_W y which can be computed as under:

[(y.v₁ )/(v₁. v₁)]v₁ +[(y.v₂ )/(v₂. v₂ )]v₂ =[ (3 +2 – 1+26)/(1+4 + 1 +4)]v₁ + [(-12 – 1 +0 + 39)/ (16+1+0+9)]v₂ = (30/10) v₁ + (26/ 25) v₂= 3 v₁ + v₂ = ( -1, -5, -3, 9)