Linear Algebra: Orthogonal basis additon

Question

Let W = span([-1,3,0], [2,2,1]), and v = [5,1,2]. Write down v as sum of two vectors v1 in W and v2 in W complement

Answer 1

W = span(w,x) with w = (–1,3,0) and and x = (2,2,1). The quick way is simply to notice that v₁ = 5 = 2•2 – (–1) = 2x₁ – w₁ and then notice that v = 2x – w. Therefore v has 0 projection in W-complement.