On "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra" by David Avis & Komei Fukuda Section 2
Let $A$ be an $m \times n$ matrix, with columns indexed by the set $E = \{1, 2. . . . , n\}$. Fix distinct indices f and 9 of E. Consider the system of equations $$Ax = 0, x_g = 1. (2.1)$$ For any $J \subseteq E$, $x_J$ denotes the subvector of $x$ indexed by $J$, and $A_J$ denotes the submatrix of $A$ consisting of columns indexed by $J$. A basis $B$ for (2.1) is a subset of $E$ of cardinality $m$ containing $f$ but not $g$, for which $A_B$ is nonsingular. We are only concerned with systems (2.1) that have at least one basis, and assume this for the rest of the paper. Given any basis B, we can transform (2.1) into the dictionary $$x_B=-A_B^{-1} A_N x_N = \bar{A} x_N. (2.2)$$ where $N = E - B$ is the co-basis and $\bar A$ denotes $-A_B^{-1} A_N$. $\bar A$ is called the coefficient matrix of the dictionary, with rows indexed by $B$ and columns indexed by $N$, so that $\bar{A} = (\bar{a}_{ij}:i\in B , J\in N)$. Note that the co-basis always contains the index $g$.
I don't quite understand what the paper was talking about. I tried to google the "subvector" which seemed to suggest multiple different definitions: Is sub-vector an established mathematical entity?. The submatrix was also not defined.
Could you explain a bit what the subvector or submatrix was defined and why it was the case, please? Also, how does $x_g=1$, how does it exist for a random matrix?