As shown below A is a 4 times 3 matrix and b is a 4 times 1

As shown below, A is a 4 times 3 matrix and b is a 4 times 1 vector (notice that A is the 4 times 4 identity matrix with its last column removed): A = [1 0 0 0 1 0 0 0 1 0 0 0] b = [1 2 3 4] (a) What is the matrix P that projects b onto the column space of A? (b) What is the projection p of b onto the column space of A?

Solution

a) P =B(B^TB)^-1B^T

P =

1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0

b)projection of vector b on column space A is given by

b_p=Pb

b_p =

1
2
0
0