Suppose that $A$ is an $m\times n$ matrix, and $B$ is a row-equivalent matrix in reduced row-echelon form. Suppose that $B$ has $r$ pivot columns, with indices given by $D=\set{d_1,\,d_2,\,d_3,\,\ldots,\,d_r}$, while the $n-r$ non-pivot columns have indices $F=\set{f_1,\,f_2,\,f_3,\,\ldots,\,f_{n-r},\,n+1}$. Construct the $n-r$ vectors $\vect{z}_j$, $1\leq j\leq n-r$ of size $n$, \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*}

Then the null space of $A$ is given by \begin{equation*} \nsp{A}=\spn{\left\{\vectorlist{z}{n-r}\right\}} \end{equation*}