So here is the question I have: Let $V=\mathbb{R}^4$ and $W=\operatorname{span}\{[0,1,0,2]\}$. Determine whether the set $S=\{[1,1,-2,0]+W,[1,0,-1,0]+W,[0,0,1,2]+W\}$ is linearly independent in $V/W$ and prove that your answer is correct.
I have taken the set S and multiplied each vector by a scalar, i.e. $a[1,1,-2,0]+b[1,0,-1,0]+c[0,0,1,2]=[0,0,0,0]$ and have determined that $a=b=c=0$, so it is linearly independent. I'm not sure whether this is enough though and the $+W$ in $S$ is troubling me. Thanks in advance.