Show that set of all solutions $(a, b, c)$ of the equation $a+b+2c=0$ is a subspace of a vector space $V^3(R)$

Question

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.

Answer 1

You don't need to show that it satisfies all the axioms of a vectors space explicitly, because some are inherited from the surrounding space. For example, addition is commutative and associative, scalar multiplication distributes over addition, etc.

Proving $2)$ and $3)$ are almost the same, but not quite. As pointed out by @zipirovich, the problem with $2$) is that if there are no solutions, then $2)$ holds vacuously, but the empty set is not a vector space. So, you can prove that the solution set is not empty and $2)$ holds, or you can prove $3).$ Of course, $\mathbf 0$ is clearly a solution, so in practice, proving $2)$ and $3)$ are pretty much the same.

Answer 2

To show that a subset of a vector space is a subspace, it suffices to show that the zero vector is in the subset and it closed under vector addition and scalar multiplication.

That is your option (3) in your list.