Let \(A\) be an \(m \times n\) real matrix. Write \(A^t\) for the transpose of \(A\), and \(N(A)\) for the nullspace of \(A\). Prove that \(N(A^t A) = N(A)\).
Solution
If \(x\in N(A)\), then \(Ax=0\), so \(A^tAx=0\) and hence \(x\in N(A^tA)\). Thus \[ N(A)\subseteq N(A^tA). \] Conversely, if \(A^tAx=0\), then \[ 0=x^tA^tAx=(Ax)^t(Ax)=\|Ax\|^2. \] Therefore \(Ax=0\), so \(x\in N(A)\). Hence \[ N(A^tA)=N(A). \]