- Definition (Vector Space)
- Let
*F* be a field. A set *V* with two binary operations: + (addition) and $\times$ (scalar multiplication), is called a **Vector Space** if it has the following properties:

- $(V,+)$ forms an abelian group
- $v(a+b)=va+vb$ for $v\in V$ and $a,b\in F$
- $a(v+u)=av+au$ for $v,u\in V$ and $a\in F$
- $(ab)v=a(bv)$
- $1_{F}v=v$

The scalar multiplication is formally defined by $F\times V{\xrightarrow {\phi }}V$, where $\phi ((f,v))=fv\in V$.

Elements in *F* are called scalars, while elements in *V* are called vectors.

- Some Properties of Vector Spaces

- $0_{F}v=0_{V}=a0_{V}$
- $(-1_{F})v=-v$
- $av=0\iff a=0{\text{ or }}v=0$

- Proofs:

- $0_{F}v=0_{F}v+0_{V}=(0_{F}+0_{F})v=0_{F}v+0_{F}v\Rightarrow 0_{V}=0_{F}v.Also,a0_{V}=a0_{V}+0_{V}=a(0_{V}+0_{V})=a0_{V}v+a0_{V}\Rightarrow a0_{V}=0_{V}v$
- We want to show that $v+(-1_{F})v=0_{V}$, but $v+(-1_{F})v=1_{F}v+(-1_{F})v=(1_{F}+(-1_{F}))v=0_{F}v=0_{V}$
- Suppose $av=0$ such that $a\neq 0$, then $a^{-1}(av)=a^{-1}0=0\Rightarrow 1_{F}v=v=0$