Is there a relationship between the column space of $A$ and the column space of the projection matrix of $A$

Question

Given $A$ which is mxn with linearly independent columns, and projection matrix $P=A(A^TA)^{-1}A^T$ which would be mxm. Are the column spaces of the two related? Do the two share the same column space?

Answer 1

Let $x$ be in the orthogonal complement of the column space of $P$, i.e., $x^TA(A^TA)^{-1}A=0$. This implies $(A^Tx)(A^TA)^{-1}(A^Tx)=0$. Now $(A^TA)^{-1}$ is positive definite, which implies $A^Tx=0$, or $A^Tx=0$, hence $x$ is in the orthogonal complement of the column space of $A$. This proves $im(P)^\perp \subset im(A)^\perp$ and $im(A)\subset im(P)$. Since $im(P)\subset im(A)$ trivially, the claim follows.