Let \(y\in\mathbb{R}^m_+\) be such a feasible dual vector. We have, for all feasible \(x^\star \in \mathbb{R}^n\) (i.e., \(Ax^\star = 0 \Leftrightarrow x^\star = P_A x^\star\) and \(x^\star \geq 0\)) and for any component of \(y\), \(y_j = e_j \odot y\) (the point-wise product of an elementary vector with \(0\) everywhere, except for the \(j\)th dimension where it is \(1\), and \(y\)),