Find the least squares error vector e=b-Ax resulting from the least squares solution x and verify that it is orthogonal to the column space of A?

Question

A=(3, 1, 1), (2, -4, 10), (-1, 3, -7)

and b=(2, -2, 5)

I found that A*A^T is not invertible so Ax=0 and that therefore e is equal to b, but the answer is not correct. I just can't find where I made a mistake

Answer 1

Verifying that $e$ is orthogonal to the column space of $A$ means that $A_i^T \cdot e = 0$ for each column, $A_i$ of $A$. That means you should instead compute $A^T e$ and see that it is equal to zero.

Since you have given a particular $A$ and $b$ I suppose the above is what you want rather than a proof of the statement in general.

Answer 2

$AA^T$ is singular because $A$ doesn’t have full rank: one of its rows/columns is a linear combination of the other two. Delete the redundant one and you should be good to go.