Exercise 1.3.6

If $\mathit{Ax}$ equals zero, then $A^T\mathit{Ax}=0$ as well. This shows that every vector in the $N(A)$ is also in $N(A^{T}A)$, so we have $N(A)\subset N(A^{T}A)$.

On the other hand, if $A^T\mathit{Ax}=0$, then $x^T{A}^T\mathit{Ax}=0$, i.e. $\parallel \mathit{Ax}{\text{∥}}^2=0$ and we see $\mathit{Ax}=0$, so if a vector is in $N(A^{T}A)$, it’s also in $N(A)$, and we have $N(A^{T}A)\subset N(A)$.

We conclude from above that $N(A)=N(A^{T}A)$, i.e. $A^{T}A$ has the same nullspace as $A$.