Rank of $A^T A$ where $A$ has a full column rank

Question

I'm having some trouble proving this: Suppose $A$ is a rectangular $m\times n$ matrix ($m > n$),

If $A$ has a full column rank ($\text{r}(A)=n$) then so does $A^TA$ (= it is invertible)

I read a proof that tries to show that $\mathcal{N}(A)=\{\mathbf 0\}$ implies $\mathcal{N}( A^TA)=\{\mathbf 0\}$, but I can't understand why that is right.

Thanks for your time.

Answer 1

Let $x \in \mathcal{N}(A^TA)$. Hence $A^TAx=0_{n \times 1}$, thus $$ \|Ax\|^2=x^TA^TAx=x^T0_{n \times 1} = 0, $$ therefore $Ax=0$, thus $x \in \mathcal{N}(A)$, but since $\text{r}(A)=n$ we must have that $\mathcal{N}(A)=\{0\}$, and then $x=0$. So indeed $\mathcal{N}(A^TA)=\{0\}$, which easily implies that $\text{r}(A^TA)=n$ as wanted.