Find the orthogonal projection of upsilon 12 5 25 11 onto t

Find the orthogonal projection of upsilon = [12 -5 25 -11] onto the subspace W spanned by {[3 0 5 -4], [-5 3 3 1]} proj_W (upsilon) = []

Solution

Span(v, w)

v1 = v = (3, 0 , 5, -4)

v2 = w - ((w.v)/v.v)v = (-5, 3 , 3, 1) - ((-4/50))(3, 0 , 5, -4)

=( -5, 3 , 3, 1) - (-12/50 , 0 , -4/10 , 16/50 ) = (-238/12 , 3 , -26/10 , -34/50) = (-119/6 ,3, -13/5 , -17/25)

projw(u) = [(u.v1)/v1.v1)]v1 + ((u.v2)/v2.v2)v2

= 205/40(3, 0 , 5, -4) +

[(12, -5,25,11)(-119/6 ,3, -13/5 , -17/25)/(-119/6 ,3, -13/5 , -17/25).(-119/6 ,3, -13/5 , -17/25)](-119/6 ,3, -13/5 , -17/25)