Exercise 4.2.27

If $A^{T}Ax=0$ then $Ax$ is in the nullspace of $A^T$. But $Ax$ is always in the column space of $A$. To be in both of those perpendicular spaces, $Ax$ must be zero. So $A$ and $A^{T}A$ have the same nullspace $:A^{T}Ax=0$ exactly when $Ax=0$.