If you evaluate this
$ \begin{pmatrix} 0 && 0 \\ 2 && 0 \end{pmatrix} \cdot \{x_1, x_2\}=0$
You get that $x_1 = 0$ and $x_2$ can be anything. Therefor any vector with $x_1 = 0$ is an eigenvector including $\{0, 1\}$ and any multiple of it (they all point in the same direction).
