The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which
i) $\xi_1 = 0 $
ii) $\xi_1 = 0$ or $\xi_2 = 0 $
iii) $\xi_1 + \xi_2 = 0 $
iv) $\xi_1 + \xi_2 = 1 $
Which of these spaces is itself a vector space?
I am completely stumped on how to go about figuring this out. From looking around online, I think that any $M\subset V$ is a subspace if $aX+bY$ is in $M$ for any $a,b$ scalars.