I wish to find a set of basis for the kernel of the matrix
$A = \begin{bmatrix}0 & -1 & 0 & 2 \\ -1 & 0 & 2 & 0 \\ 0 & 3 & 0& -6 \\ 3 & 0 & -6 & 0 \end{bmatrix}$
I perform row reduction of $A$, in which I obtain, $ \begin{bmatrix}0 & -1 & 0 & 2 \\ -1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 0& 0 & 0 & 0 \end{bmatrix}$
This allows me to find the kernel more easily. I find two vectors $\begin{bmatrix} 0 & 2 & 0 & 1 \end{bmatrix}$ and $\begin{bmatrix} 2 & 0 & 1 & 0 \end{bmatrix}$ in the kernel of $A$.
But I know there are infinite amount of vectors in the kernel. So I am stuck.
Since I cannot compute the kernel, therefore I cannot compute the basis for the kernel.
There must be a better way to reason about this.