The projection of a vector $v$ onto the column space of A is
$$A(A^T A)^{-1}A^T v$$
If the columns of $A$ are orthogonal, does the projection just become $A^Tv$? I think it should because geometrically you just want to take the dot product with each of the columns of $A$. But how can I show this is true?
Best Answer
No. If the columns of $A$ are orthonormal, then $A^T A=I$, the identity matrix, so you get the solution as $A A^T v$.