Show that set of all solutions $(a, b, c)$ of the equation $a+b+2c=0$ is a subspace of a vector space $V^3(R)$.
I wonder if I should show that
(1) the solution set satisfies all the axioms of a vector space?
(2) for the solution set $W=\lbrace (a,b,c): a, b, c \in R \rbrace$, $\alpha(a_1, b_1, c_1)+\beta(a_2,b_2, c_2)\in W$ for all $\alpha,\beta\in R$, $(a_1, b_1, c_1),(a_2,b_2, c_2)\in W$?
(3) the zero vector exists in $W$ and it is closed under vector addition and scalar multiplication?
I am unable to proceed as I don’t know which conditions would I should prove.