Answer
$S$ does not span $R^3$.
Work Step by Step
The set $S=\{(1,1,2),(0,2,1)\}$ is not a basis for $R^3$ because $S$ does not span $R^3$. For example, suppose the vector $(0,0,1)\in R^2$ and writing it as a linear combination of the elements of $S$, we get
$$(0,0,1)=c_1(1,1,2)+c_2(0,2,1), \quad c_1,c_2\in R.$$
Which yields the following system of equations
\begin{align*}
c_1&=0\\
c_1+2c_2&=0\\
2c_1+c_2&=1.
\end{align*}
It is easy to note that the above system is in-consistent, since the first and the second equations lead to $c_1=0,c_2=0$ and these values do not satisfy the third equation.