We have a matrix $A \in \mathbb{R}^{m \times n}$ and rank$(A)=m$.

Prove that there exists unique partition $[B(A), N(A)]$ of $\{1,2, \dots, n\}$ (the columns of $A$) such that there exist $\bar{x} \in \mathbb{R}^n, \bar{y} \in \mathbb{R}^m$ satisfying

$A\bar{x}=0$, $\bar{x} \geq 0$, $\bar{x}_{B(A)} > 0$, $\bar{x}_{N(A)} = 0$

and $A^T\bar{y} \geq 0$, $(A^T\bar{y})_{B(A)} = 0$, $(A^T\bar{y})_{N(A)} > 0$

  • $\begingroup$ What does the notation $\bar{x}_{B(A)}$ mean? $\endgroup$ – Jack's wasted life Nov 1 '19 at 17:28

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