Vector Space over a Field

Let $K$ be a field, with additive identity $0$ and multiplicative identity $1$.

A vector space over $K$ is a set $V$ together with two operations: an addition, assigning to each pair $u,v$ of elements of $V$ an element $u+v$ of $V$, and a scalar multiplication, assigning to each $\lambda\in K$ and each $v\in V$ an element $\lambda v$ of $V$, subject to the following conditions for all $u,v,w\in V$ and all $\lambda,\mu\in K$.

1. $(u+v)+w=u+(v+w)$.
2. $u+v=v+u$.
3. There is an element $0_{V}\in V$ such that $v+0_{V}=v$ for every $v\in V$.
4. For every $v\in V$ there is an element $w\in V$ with $v+w=0_{V}$.
5. $\lambda(\mu v)=(\lambda\mu)v$, the product $\lambda\mu$ being formed in $K$.
6. $1v=v$.
7. $\lambda(u+v)=\lambda u+\lambda v$.
8. $(\lambda+\mu)v=\lambda v+\mu v$, the sum $\lambda+\mu$ being formed in $K$.

The elements of $V$ are called vectors, the elements of $K$ scalars, and an element $0_{V}$ as in condition 3 a zero vector of $V$.

A vector space over the field of complex numbers is called a complex vector space, and a vector space over the field of real numbers a real vector space.