Vector space

Definition

Let $F$ be a field. A vector space over $F$ is a set $V$ with two maps: an addition $V \times V \to V$, $(u, v) \mapsto u + v$, and a scalar multiplication $F \times V \to V$, $(a, v) \mapsto av$. These axioms must hold for all $u, v, w \in V$ and $a, b \in F$:

1. $u + (v + w) = (u + v) + w$.
2. $u + v = v + u$.
3. There is an element $0 \in V$ with $v + 0 = v$ for all $v \in V$.
4. For each $v \in V$ there is an element $-v \in V$ with $v + (-v) = 0$.
5. $a(bv) = (ab)v$.
6. $1v = v$, where $1$ is the unit of $F$.
7. $a(u + v) = au + av$.
8. $(a + b)v = av + bv$.

The elements of $V$ are vectors, and the elements of $F$ are scalars.