Answer
$S$ is linearly independent.
Work Step by Step
Assume the combination
$$a(-4,-3,4)+b(1,-2,3)+c(6,0,0)=(0,0,0).$$
We have the system
\begin{align*}
-4a+b+6c&=0\\
-3a-2b&=0\\
4a+3b&=0.
\end{align*}
Since the determinant of the coefficient matrix is given by
$$\left| \begin{array} {ccc} -4&1&6\\ -3&-2&0
\\ 4&3&0
\end{array} \right|=-6\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$
Consequently, $S$ is linearly independent.