For example, the matrix is $$ \left [\begin{matrix} 1 & 3 & -2 & 2 & 5 \\ 0 & 3 & -3 & 0 & 2 \\ -1 & 1 & -2 & -2 & 0 \\ 2 & 1 & 1 & 4 & 1 \\ \end{matrix}\right ], $$
and after elimination, it is $$ \left [\begin{matrix} 1 & 3 & -2 & 2 & 5 \\ 0 & 3 & -3 & 0 & 2 \\ 0 & 0 & 0 & 0 & \frac{7}{3} \\ 0 & 0 & 0 & 0 & 0 \\ \end{matrix}\right ]. $$
The 3rd and 4th columns are free columns, and they are definately linear combinations of left columns.
How to prove it?