Answer
$S$ is linearly independent set.
Work Step by Step
Assume the combination
$$a(-2,2)+b(3,5)=(0,0).$$
We have the system
\begin{align*}
-2a+3b&=0\\
2a+5b&=0.
\end{align*}
Since the determinant of the coefficient matrix is given by
$$\left| \begin{array} {cc} -2&3\\2&5 \end{array} \right|=-16\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0.$$
Consequently, $S$ is linearly independent set.