Answer
$S$ is linearly independent.
Work Step by Step
Assume the combination
$$a(1,-4,1)+b(6,3,2)=(0,0,0).$$
We have the system
\begin{align*}
a+6b&=0\\
-4a+3b&=0\\
a+2b&=0.
\end{align*}
The above system has a unique solution, that is, $$a=0, \quad b=0.$$
Consequently, $S$ is linearly independent.