Columns space is orthogonal to the left null space

Since we are dealing with the column space, let us represent \(A\) in terms of columns: \[ A = \begin{bmatrix} \vert & & \vert\\ c_1 & \cdots & c_n\\ \vert & & \vert \end{bmatrix} \] Let \(x \in \mathcal{N}(A^T)\). Then: \[ \begin{aligned} A^T x &= 0\\\\ \begin{bmatrix} \large— & c_1^T & \large—\\ & \vdots &\\ \large— & c_n^T & \large— \end{bmatrix}x &= 0 \end{aligned} \] It follows that \(c_i^T x = 0\) for \(1 \leqslant i \leqslant n\). This means that the left nullspace of \(A\) is orthogonal to the column space of \(A\).