# Prove there exists unique partition of a matrix

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

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