Find the orthogonal projection of v rightarrow 19 18 9 8 on

Find the orthogonal projection of v rightarrow = [19 18 -9 8] onto the subspace W spanned by {[-1 1 -5 4], [-1 -1 4 -2], [-4 -3 2 -1]}. proj_w(v rightarrow) = []

Solution

Let u₁ = (-1,1,-5,4)^T , u₂ = (-1,-1,4,-2)^T and u₃ = (-4,-3,2,-1)^T. Then

Proj_u1(v) = [(v.u₁)/(u₁.u₁)]u₁= [(-19+18+45+32)/(1+1+25+16)]u₁= 76/43(-1,1,-5,4)^T=                                              (-76/43,76/43,-380/43,304/43)^T.

Proj_u2(v) = [(v.u₂)/(u₂.u₂)]u₂= [(-19-18-36-16)/(1+1+16+4)]u₂ =( -89/22) (-1,-1,4,-2)^T = (89/22,89/22,-178/11,89/11)^T.

Proj_u3(v) = [(v.u₃)/(u₃.u₃)]u₃= [(-76-54-18-8)/(16+9+4+1)]u₃= (-156/30)u₃= -26/5(-4,-3,2,-1)^T = (104/5,78/5,-52/5,26/5)^T.

Then the orthogonal projection of v onto W is (-76/43,76/43,-380/43,304/43)^T+                             (89/22,89/22,-178/11,89/11)^T + (104/5,78/5,-52/5,26/5)^T =                                               (109159/4730,101283/4730,-83766/2365,47703/2365)^T