Notice, or discover, that the following gives a nontrivial relation of linear dependence on $S$ in $C$, so by Definition LI, the set $S$ is linearly dependent.
\begin{equation*}
2(3,\,1)+(-1)(7,\,3)=(7,\,3)+(-9,\,-5)=(-1,\,-1)=\zerovector
\end{equation*}