2 1 point Let W be the subspace spanned by orthogonal Find t

2. [1 point Let W be the subspace spanned by orthogonal. Find the orthogonal projection of into W. Note that this basis is not

Solution

Let v = (-4,8,0,0)^T , u₁ = (1,-1,1,1)^T, u₂ = (3, -3,5,5)^T and v₃ = (3,-5,1,3)^T .Then:

Proj_u1(v) = [(v.u₁)/(u₁.u₁)]u₁ = [(-4-8)/(1+1+1+1)]u₁= (-12/4)u₁ = -3(1,-1,1,1)^T = (3,-3,3,3)^T.

Proj_u2(v) = [(v.u₂)/(u₂.u₂)]u₂ = [(-12-24)/(9+9+25+25)]u₂ = (-36/68)u₂ = -9/17(3, -3,5,5)^T= (-27/17,27/17,-45/17,-45/17)^T

Proj_u3(v) = [(v.u₃)/(u₃.u₃)]u₃ =[(-12-40)/(9+25+1+9)]u₃ = (-52/44)u₃ = -13/11(3,-5,1,3)^T = (-39/11, 65/11, -13/11, -39/11)^T . Hence, the projection of v onto W =

(3,-3,3,3)^T+(-27/17,27/17,-45/17,-45/17)^T+(-39/11, 65/11, -13/11, -39/11)^T = (-229/187,841/187,-138/187,-597/187)^T