Suppose that $A$ is an $m\times n$ matrix, and $B$ is a row-equivalent matrix in reduced row-echelon form with $r$ pivot columns. Let $D=\{d_1,\,d_2,\,d_3,\,\ldots,\,d_r\}$ and $F=\{f_1,\,f_2,\,f_3,\,\ldots,\,f_{n-r}\}$ be the sets of column indices where $B$ does and does not (respectively) have pivot columns. Construct the $n-r$ vectors $\vect{z}_j$, $1\leq j\leq n-r$ of size $n$ as \begin{equation*} \vectorentry{\vect{z}_j}{i}= \begin{cases} 1&\text{if $i\in F$, $i=f_j$}\\ 0&\text{if $i\in F$, $i\neq f_j$}\\ -\matrixentry{B}{k,f_j}&\text{if $i\in D$, $i=d_k$} \end{cases} \end{equation*}
Define the set $S=\set{\vectorlist{z}{n-r}}$.Then