Left Nullspace

The left nullspace of a \(m \times n\) matrix \(A\) is the following subset of \(\mathbb{R}^{m}\): \[ \{x\ |\ x^T A = 0, x \in \mathbb{R}^{m}\} \] Since \(x^TA = 0 \implies (x^TA)^T = 0 \implies A^Tx = 0\), we see that the left nullspace of \(A\) is simply the nullspace of \(A^T\). Therefore, we will denote the left nullspace \(\mathcal{N}(A^T)\) and fix the definition: \[ \mathcal{N}(A^T) = \{x\ |\ A^Tx = 0, x \in \mathbb{R}^{m}\} \] This is clearly a subspace of \(\mathbb{R}^{m}\).