Question

What is the matrix P (P,) for the projection of R3 onto the subspace V spanned by the vectors 0 Pi3 12 P2 1 23 - P33 3 1 4 What is the projection p of the vector b-5 onto this subspace? Pi P2 Ps

Answer 1

Let A = [a₁,a₂] =

0	1
-2	-1
2	-1

Then, the required projection matrix for projection of R onto V is P (say) = A(ATA)-1AT =

1/3	-1/3	-1/3
-1/3	5/6	-1/6
-1/3	-1/6	5/6

P₁₁= 1/3, P₁₂= -1/3, P₁₃= -1/3

P₂₁= -1/3 , P₂₂= 5/6, P₂₃= -1/6

P₃₁=-1/3, P₃₂=-1/6, P₃₃= 5/6

Further, we have proj_a1 (b) = [(b.a₁)/(a₁.a₁)]a₁ = [(0-10+2)/(0+4+4)](0,-2,2)^T = - (0,-2,2)^T = (0,2,-2)^T and proj_a2 (b) = [(b.a₂)/(a₂.a₂)]a₂ =[ ( -4-5-1)/(1+1+1)](1,-1,-1)^T = -(10/3) (1,-1,-1)^T = (-10/3,10/3,10/3)^T.

Then, proj_V (b)= proj_a1 (b)+ proj_a2 (b)= (0,2,-2)^T +(-10/3,10/3,10/3)^T= (-10/3, 16/3, 4/3)^T.

p₁ = -10/3, p₂ = 16/3, p₃ = 4/3.