I have a set of 3 vectors, E = { [1, 1, -3], [2, -1, 3], [-6, 3, -9] }
and I want to know if this previous set is a subset of F = { [1, 2, 3], [3, 2, 1], [1, 0, 1] }
$ \left[ \begin{array}{ccc|ccc} 1 & 3 & 1 &1 &2 &-6\\ 2 & 2 & 0 &1 &-1 &3 \\ 3 & 1 & 1 &-3&3&-9\end{array} \right] Gauss-> \left[ \begin{array}{ccc|ccc} 1 & 3 & 1 &1 &2 &-6\\ 0 & -4 & -2 &-1 &-5 &15 \\ 0 & 0 & 2 &-4&7&-21\end{array} \right] $
Now my answer : Yes, because there is no contradiction.
b) if F is a subset of E
$ \left[ \begin{array}{ccc|ccc} 1 & 2 & -6 &1 &3 &1\\ 1 & -1 & 3 &2 &2 &0 \\ -3 & 3 & -9 &3&1&1\end{array} \right] Gauss-> \left[ \begin{array}{ccc|ccc} 1 & 2 & -6 &1 &3 &1\\ 0 & -3 & 9 &1 &-1 &-1 \\ 0 & 0 & 0 &9&7&1\end{array} \right] $
Answer : No, because there is a contradiction.
Should I do something with the free variable (c3) here ?
Am I missing something here or the only thing I have to do is, gauss and check for a contradiction ?
Thanks