Answer
$S$ is a basis for $R^2$.
Work Step by Step
The set $S=\{(1,2),(1,-1)\}$ is a linearly independent set of vectors. Indeed,
assume that
$$a(1,2)+b(1,-1)=(0,0), \quad a,b\in R.$$
Then, we have the following system of equations
\begin{align*}
a+b&=0\\
2a-b&=0.
\end{align*}
Solving the above equations, we find that $a=0,b=0$ and hence $S$ is linearly independent. Now, since $R^2$ is vector space of dimension $2$, then by Theorem 4.12, $S$ is a basis for $R^2$.
