Find the projection in the subspacing where the vector $X = (1,3,-1,3)$ originates from the vectors $A_1 (1, -1,1,1), A_2 (5,1,-3,3).$

Question

How can I find the projection in the subspacing where the vector $X = (1,3,-1,3)$ originates from the vectors $A_1 (1, -1,1,1), A_2 (5,1,-3,3).$

I have not found an example of this problem in my book. That's why I can not solve it. I'm stuck.

Answer 1

HINT

Recall that the projection matrix $P$ onto the subspace $span(A_1,A_2)$ is given by

$$P=A(A^TA)^{-1}A^T$$

where $A=[A_1\quad A_2]$.