# Linear Algebra: which of the definition of subspace of a vector space is more correct?

In a test I was asked to give a definition to a subspace to a vector space, I wrote:

A subset $V$ is a subspace of $X$ if $0 \in V$ and $\forall u,v \in V, > \exists \thinspace V$ s.t. $u+v = 0$

I was told that it is not correct. The correct definition is that the subset is closed under addition and closed under scalar multiplication.

Why is the second set of definition "more" correct than the definition I gave?

Take $X=\mathbb R^2$ and $V$ be the union of the two axes. Then $V$ satisfies your definition but is not a subspace because $(1,1)=(1,0)+(0,1)$ is not in $V$.
Your definition is entirely incorrect, it's not a question of "more" or "less". For example, according to your definition, the set $V=\{-17,0,17\}$ is a vector subspace of the real vector space $\mathbb{R}$, since $0\in V$ and every element of $V$ has an additive inverse.