Determine the space base

Question

Determine the base space S of all solutions to the equation system =

\begin{cases} 1q +2w +7e-9r+31t =0 \\ 2q +4w+ 7e-11r+34t =0\\ 3q+6w+5e-11r+29t=0 \\ \end{cases}

$$ \left[ \begin{array}{ccccc|c} 1&2&7&-9&31&0\\ 2&4&7&-11&34&0\\ 3&6&5&-11&29&0 \end{array} \right] $$

I'm not really sure what is the question about... I've already made: $$ \left[ \begin{array}{ccccc|c} 1&2&0&-2&3&0\\ 0&0&1&-1&4&0\\ 0&0&0&-0&0&0 \end{array} \right] $$

so:

\begin{cases} q=-2w+2r-3t \\ e=r-4t\\ \end{cases}

So what should I give as answer? w,r and t?

Answer 1

So the set S = { $ \lambda * (-2, 1, 0, 0, 0) + \mu * (2, 0, 1, 1, 0) + \nu * (-3, 0, 0, 0, 1) $ } is the space of the solutions.

Now go from here and try determining a basis for S.