Answer
S is not linearly independent.
Work Step by Step
Since $(3,3,3)=(1,1,1)+(2,2,2)$, then we have the non trivial combination
$$(3,3,3)-(1,1,1)-(2,2,2)=(0,0,0).$$
Which means that the vectors are linearly dependent and hence $S$ is not linearly independent.