## Elementary Linear Algebra 7th Edition

$S$ is not linearly independent.
It is eas to note that $$(1,5,-3)=(1,0,0)+\frac{5}{4}(0,4,0)+\frac{1}{2}(0,0,-6).$$ Then we have the nontrivial combination $$(1,5,-3)-(1,0,0)-\frac{5}{4}(0,4,0)-\frac{1}{2}(0,0,-6)=(0,0,0,0).$$ Which means that $S$ is not linearly independent.