The general way to look for kernels is to use the elementary row operations and try to reduce those rows as much as possible. But, now I'm trying to solve a bit less straight-forward problem.
$$\vec x, \vec y, \vec z \in \mathbb{K}^{2013}$$ $$A=[\vec x + \vec y, \vec x - \vec y, \vec x + \vec z, \vec x - \vec z, \vec x +\vec y + \vec z]$$
What's the kernel basis for $A$? I've found the image basis using el. column operations and, unless I've made a mistake, it's simply $\vec y, \vec x, \vec z$. This was easier, since $A$ has 5 columns. Now, what can I do with its 2013 rows to find the kernel?
Thanks!