Zero vector in a vector space.

Question

Let $X$ a vector space on $\mathbb{R}$ and denote with $\theta$ the zero vector, i must prove that $$0x=\theta$$ for all $x\in X$

Now,

$0x=(0+0)x=0x+0x\Rightarrow$ $0x=\theta$

Correct?

Answer 1

Yes, essentially this is correct. However, the implication could be (arguably) made clearer by adding a short explanation along the lines "$-0x$ is the (unique) additive inverse of $0x$ and therefore..."

But this is the common proof of this fact correctly executed (also, see here).