Is $\exists a \in \mathbb F \forall v \in V (av = 0)$ an axiom of vector spaces?

Question

Is the statement $0\cdot x = 0$ for all $x \in V$ an axiom of the theory of vector spaces? I understand it is implied by the distributivity laws: $$0\cdot x = (1-1)\cdot x = x - x = 0$$ But, can I say it is an axiom too?

Answer 1

You don't need this as an axiom. Since $0v=(0+0)v=0v+0v$ and $(V,+)$ is a group, it follows that $0v=0$. I usually use four axioms for the scalar multiplication: $1v=v$, $(a+b)v=av+bv$, $a(v+w)=av+aw$ and $(ab)v=a(bv)$.

Answer 2

Yes, ${0 \cdot x = 0}$ is one of the multiplicative axioms for vector spaces. Depending on the text that defines the axioms they sometimes include it already as an axiom. The approach you did is valid.