As discussed above, any matrix \(A \in \mathbb{R}^{m \times n}\) divides up the domain and co-domain into two orthogonal subspaces respectively. \(\mathcal{R}(A^T)\) and \(\mathcal{N}(A)\) together span the domain. \(\mathcal{R}(A)\) and \(\mathcal{N}(A^T)\) together span the co-domain. Each of these four subspaces can be visualized in the co-domain - in terms of columns of the matrix \(A\) and also in the domain - in terms of the rows of the matrix \(A\).