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 formerly 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.