Q:

Why a real Vector Space has either one or infinitely many vectors?

I have a hard time understanding the above question, so could someone help?

A:

if $v \in V$ is a vector in the vector space, then for every $\lambda \in \mathbb R$ also $\lambda v \in V$ (This is an axiom)
So either is $v = 0$ is the only vector in the vector space,  so all the vectors in the form $\lambda v$ are equal to $0$, or there are infinitel many, depending on $\lambda$
What I mean is: if for example $v = (1,1)$ is in out space, then also $2v = (2,2)$ has to be in the vector space (because of our axiom). But then also $3v = (3,3)$ and so on, for every $\lambda$: $(\lambda, \lambda)$ has to be in $V$

