Linear Algebra and Its Applications · Problem 3.1.30

The key insight is recognizing that $Ax$ lives in the column space of $A$, and the column space and left nullspace (nullspace of $A^T$) are orthogonal complements. If a vector belongs to both orthogonal subspaces simultaneously, it must be the zero vector. This orthogonality relationship is what forces $Ax = 0$ and ultimately explains why $A^TA$ and $A$ share the same nullspace.

The key insight is recognizing that $Ax$ lives in the column space of $A$, and the column space and left nullspace (nullspace of $A^T$) are orthogonal complements. If a vector belongs to both orthogonal subspaces simultaneously, it must be the zero vector. This orthogonality relationship is what forces $Ax = 0$ and ultimately explains why $A^TA$ and $A$ share the same nullspace.