A post on Quroa:
stated that the orthogonal projection of a point $\vec{\text{b}}$ to the "subspace" defined by $A\vec{\text{x}}=\vec{\text{b}}$ is given by $$A^T(A\vec{\text{x}_0}-\vec{\text{b}})=0$$
I'm not clear how this equation can be useful since $A\vec{\text{x}_0}-\vec{\text{b}}=\vec{\text{0}}$ and any vector dotted with a zero vector will yield zero. isn't the above equation redundant?
Also, second question, is it correct that, for $\vec{\text{x}_0}$ to exist, $A$ must not be a full rank matrix? if so, how to formally prove that?