Given an $n\times m$ matrix $A$ prove that $Col(A^TA)=Col(A^T)$

Question

We're studying Least-Squares in my linear algebra class; one of the theorems we've proven is that $Nul(A^TA)=Nul(A)$.

I understood that proof well-enough, however our professor also stated that $Col(A^TA)=Col(A^T)$ is a corollary of the former theorem but I'm not sure how the proof for that would go. I started by restating the problem as $(NulA)^{\perp}=(Nul(A^TA))^{\perp}$, but I'm not sure how the proof would follow or even if that's useful.

Answer 1

$Col(A^TA) \subset Col(A^T)$ because $Col(XY) \subset Col(X)$ for every martices $X,Y$

And because $dim(Col(A^TA))= n-dim(Nul(A^TA ) )=n-dim(Nul(A))= n-dim(Nul(A^T))=dim(Col(A^T))$

We have $Col(A^TA)=Col(A^T)$