Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$? Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$?

if (3,-1,0,-1) be in span {(2,-1,3,2),(-1,1,1,-3),(1,1,9,-5)} then we must have
 (3,-1,0,-1)=$c_{1}$ (2,-1,3,2)+ $c_2$ (-1,1,1,-3)+ $c_3$ (1,1,9,-5)
so this system must have solution:
$2c_1 -c_2+c_3=3$
$-c_1 +c_2+c_3=-1$
$3c_1 +c_2+9c_3=0$
$2c_1 -3c_2-5c_3=-1$ 
since 3 first relation has unic answer and 3 end realations too and they cant have equal answer (then this system has not solution and (3-10-1) 
