How to find a basis for the solution space of this linear system? $$ \begin{bmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$
Solutions are $[-2t, -3t, t ]$ because $x_3$ is a free variable. So let $x_3 = t$.
But how do I find the basis for this?
This is a system $Ax=0$
$x_4$ is also free variable. So the solution should be $[-2t,-3t,t,s]$ where $t,s\in\mathbb{R}$. Therefore the solution can be written as $$[-2t,-3t,t,s]=t[-2,-3,1,0]+s[0,0,0,1].$$ That is to say, the solution is linear combination of $[-2,-3,1,0]$ and $[0,0,0,1]$, and they are linearly independent. So $\big\{[-2,-3,1,0],[0,0,0,1]\big\}$ is a basis.