**vector space over a field $$ F$$**is a non-empty set $$ V$$ with a binary operation $$ \oplus $$ on $$ V$$ and a binary function $$ \otimes :F\times V\to V$$ such that the following hold for any $$ a,b\in F$$ and $$ u,v\in V$$

- $$ \oplus $$ is associative and commutative
- There is a identity element $$ {0}_{V}$$ with respect to $$ \oplus $$
- $$ {1}_{F}$$ is an identity element for $$ \otimes $$
- $$ \oplus $$ has inverses for $$ {0}_{V}$$
- $$ \otimes $$ distributes into $$ \oplus $$
- $$ \left(a{+}_{F}b\right)\otimes v=(a\otimes v)\oplus (b\otimes v)$$
- $$ a\otimes (b\otimes v)=\left(a{\cdot}_{F}b\right)\otimes v$$