How do I show that $S=\{u_1,u_2\}$ is a basis for the solution space $Ax=0$ without doing Elementary.Row.Operations?
Given the solution space of $Ax=0$ has dimension 2.
$A=\begin{pmatrix} 3 & -1 & -2 & -4\\ 0 & 1 & -1 & 1\\ -1 & 1 & 0 & 2\\ -2 & 1 & 1 & 3 \end{pmatrix}$
$u_1=\begin{pmatrix} 1\\ 1\\ 1\\ 0 \end{pmatrix} ,u_2=\begin{pmatrix} 1\\ -1\\ 0\\ 1 \end{pmatrix}$
Ok, so my guess is that, since $u_1$ and $u_2$ are linearly independent, and there are two vectors in the basis, that fulfils one of the conditions for S to be a basis for V.
Now I need to prove that S spans V, but how do I do that without doing any E.R.Os?