Answer
$S$ spans $R^2$.
Work Step by Step
Assume the combination
$$a(2,0)+b(0,1)=(0,0).$$
We have the system
\begin{align*}
2a&=0\\
b&=0.
\end{align*}
The above system has only the trivial solution, that is, $a=0, b=0$.
Consequently, $S$ is linearly independent and since $R^2$ has the dimension $2$ then $S$ spans $R^2$.