Originally Posted by

**MacstersUndead** Alright, let's us definition as you laid out. If v is a linear combination of vectors in S, then there exists constants $\displaystyle c_i$ such that

$\displaystyle c_1(-1,2,1) + c_2(3,1,2) + c_3(1,5,4) + c_4(-6,5,1) = (-5,3,0)$

by comparison we get the simultaneous system of equations

$\displaystyle -c_1 + 3c_2 + 1c_3 + -6c_4 = -5$

$\displaystyle 2c_1 + c_2 + 5c_3 + 5c_4 = 3$

$\displaystyle 1c_1 + 2c_2 + 4c_3 + 1c_4 = 0$

You can solve this system using a matrix. If such $\displaystyle c_i$ exist, then you have show v is in a span of S. If no such $\displaystyle c_i$ exist then v is not in the span of S.