[Def]: Vector Space

Let $X$ be a nonempty set over a field $\mathbb{F}$ with two operations:

1. $+$: $X \times X \rightarrow X$: $(a, b) \mapsto a+b$
2. $*:$ $\mathbb{F} \times X \rightarrow X$: $(r, a) \mapsto r \cdot a$

If

1. $a+b=b+a$, $\forall a, b \in X$
2. $(a+b)+c=a+(b+c)$, $\forall a, b, c \in X$
3. $\exist 0 \in X$ s.t. $a+0=0+a=a$, $\forall a \in X$
4. $\forall x\in X$, $\exist -x \in X$, s.t. $x+(-x)=-x+x=0$
5. $r \cdot (a+b)=r \cdot a + r \cdot b$, $\forall r \in \mathbb{R}$
6. $r\cdot (s\cdot a)=(r\cdot s)\cdot a$, $\forall r,s\in \mathbb{R}$
7. $(r+s)\cdot a=r\cdot a+s\cdot a$, $\forall r,s\in \mathbb{R}$
8. $\forall a\in X$, $1\cdot a=a$

Then $X$ is called a vector space over $\mathbb{F}$