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

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