Halfway through Example SSP4, we need to show that the system of equations \begin{equation*} \linearsystem{ \begin{bmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & -8\\ 0 & 1 & -6 & 24\\ 1 & -4 & 12 & -32\\ -2 & 4 & -8 & 16\\ \end{bmatrix} } {\colvector{a\\b\\c\\d\\e}} \end{equation*} is consistent for every choice of the vector of constants satisfying $16a+8b+4c+2d+e=0$.

Express the column space of the coefficient matrix of this system as a null space, using Theorem FS. From this use Theorem CSCS to establish that the system is always consistent. Notice that this approach removes from Example SSP4 the need to row-reduce a symbolic matrix.