## Elementary Linear Algebra 7th Edition

$S$ is linearly independent.
Assume the combination $$a(4,-3,6,2)+b(1,8,3,1)+c(3,-2,-1,0)=(0,0,0,0).$$ We have the system \begin{align*} 4a+b+3c&=0\\ -3a+8b-2c&=0\\ 6a+3b-c&=0\\ 2a+b&=0 \end{align*} Since the determinant of the coefficient matrix is given by $$\left| \begin{array} {ccc} -4&1&6\\ -3&-2&0 \\ 4&3&0 \end{array} \right|=-6\neq 0$$ one can see that the above system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$ Consequently, $S$ is linearly independent.