Elementary Linear Algebra 7th Edition

$S$ is linearly independent set.
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.