can someone solve it please Find the orthogonal projection o

can someone solve it please

Find the orthogonal projection of v = [3 4 0 15] onto the subspace W spanned by {[4 -2 1 -5], [-4 1 -2 -5]}. proj_w(v) = [24/23 -59/46 95/46 -385/23]

Solution

Let u₁ = (4,-2,1,-5)^T and u₂ = (-4,1,-2,-5)^T. We have proj_u1 (v) = [(v.u₁)/(u₁.u₁)]u₁ =                                                [( 12-8+0-75)/(16+4+1+25)]u₁ = -(71/46)(4,-2,1,-5)^T = (-284/46,142/46,-71/46,355/46)^T. Also, proj_u2 (v) = [(v.u₂)/(u₂.u₂)]u₂=[(-12+4+0-75)/(16+1+4+25)]u₂=-(83/46)(-4,1,-2,-5)^T= (332/46,-83/46,166/46,415/46)^T. Hence, proj_W (v)=proj_u1 (v)+proj_u2 (v)=(-284/46,142/46,-71/46,355/46)^T+(332/46,-83/46,166/46,415/46)^T = ( 48/46, 59/46,95/46,60/46)^T= (24/23, 59/46, 95/46,30/23)^T.