This question comes from the exercises in the book "Introduction to linear optimization".
Consider the standard form polyhedron $\{\bf x|Ax=b,x\geq 0\}$，and assume that the rows of the matrix $\bf A$ are linear independent. Let $\bf x$ be a basic solution, and let $J=\{i\mid x_i\neq 0\}$. Show that a basis is associated with the basic solution $\bf x$ if and only if every column $\bf A_i$, $i\in J$, is in the basis.