Archetype K is the $5\times 5$ matrix \begin{equation*} K=\begin{bmatrix} 10 & 18 & 24 & 24 & -12 \\ 12 & -2 & -6 & 0 & -18 \\ -30 & -21 & -23 & -30 & 39 \\ 27 & 30 & 36 & 37 & -30 \\ 18 & 24 & 30 & 30 & -20 \end{bmatrix} \end{equation*} which is row-equivalent to the $5\times 5$ identity matrix $I_5$. So by Theorem NMRRI, $K$ is nonsingular. Then Theorem CNMB says the set \begin{equation*} \set{\colvector{10\\12\\-30\\27\\18},\,\colvector{18\\-2\\-21\\30\\24},\,\colvector{24\\-6\\-23\\36\\30},\,\colvector{24\\0\\-30\\37\\30},\,\colvector{-12\\-18\\39\\-30\\-20}} \end{equation*} is a (novel) basis of $\complex{5}$.