I am a little confused about understanding when we consider a column to have free and pivot variables. So let's say I have the following matrix:
\begin{matrix} 1 & 2 & 0 & -1 & 8\\ 0 & 0 & 3 & 6 & 18 \\ -1 & -2 & 0 & 0 & -3 \end{matrix}
This in the reduced row-echelon form would be:
\begin{matrix} 1 & 2 & 0 & 0 & 3\\ 0 & 0 & 1 & 0 & 16\\ 0 & 0 & 0 & 1 & -5 \end{matrix}
Based on this we say, that columns 1,3 and 4 are pivot columns, and the rest are free. But if I switched columns 2 and 4, or any other columns the results that we would get would be completely different. That is why it feels as though the selection of the pivot and free variables is arbitrary.
Can someone please help me?