Let's say I have a non invertible system of linear equations: Ax=b
Then the pseudoinverse gives a approximate solution: x'=A^+*b
(1) Given the property: A*A^+*A = A, prove that x' is a vector which lies in the image of A and minimises the error = ||b-Ax'||
(2) show that the magnitude of the error is ||Ax'-b||^2=b^T*(I-A*A^+)*b and that the error vector lies in the kernel of A
A^+ is the pseudoinverse
The Attempt at a Solution
Let's just consider (1) for now.
I know that since b does not lie in the column space of A, we have to find a approximate solution by projecting b onto this space. Let's denote this projected as proj_b.
Thus: Ax' = proj_b.
It's kinda obvious that x' lies in the image of A. I just cannot see the link between this and the given property A*A^+*A = A.