Axioms of a vector space I have been taught that in order to check whether something is a vector space I need to check that the following 9 axioms hold:


*

*$v+w=w+v$

*there exists a $0 \in V$ with $v+0=v$

*for every $v \in V$ there is an inverse

*associativity 

*$\lambda (v+w)=\lambda v + \lambda w$

*$(\lambda + \mu )v=\lambda v + \mu v$

*$(\lambda \mu) v=\lambda(\mu v)$

*$1v=v$

*$0v=0$


Do I need to check all of these axioms every time or is there some sort of shortcut?
 A: With experience, it becomes easy to detect vector spaces. Also, as noted in the comments, if some $W$ is a subset of a vector space $V$ over a field $\Bbb F$, all you need to do to prove that $W$ is a vector space is to check whether it is a subspace of $V$; this boils down to verifying that:


*

*For all $x,y \in W$, $x - y \in W$.

*For all $\alpha \in \Bbb F$ and all $x \in W$, $\alpha x \in W$.
Which is equivalent to checking that:


*

*For all $\alpha, \beta \in \Bbb F$ and all $x,y \in W$, $\alpha x + \beta y \in W$.


Which is also equivalent to checking that:


*

*For all $\alpha \in \Bbb F$ and all $x,y \in W$, $\alpha x + y \in W$.


In general, most of the time you are lucky enough that the thing you are trying to verify that it is a vector space is already a subset of a well-known vector space.
A: The last can be deduced:
from $1\vec v= \vec v$ we have $(-1) \vec v+\vec v=(-1+1)\vec v=0$ so $(-1)\vec v= -\vec v $ ( unicity of the opposite)
and
$$
0\vec v=(a-a)\vec v=a \vec v-a \vec v=1a\vec v -1a\vec v=a(\vec v - \vec v)=0
$$
But you have to prove all the other.
