Affine space

Definition

Let $K$ be a field and let $V$ be a vector space over $K$. An affine space over $V$ is a nonempty set $A$ with a map $A \times V \to A$, written $(p, v) \mapsto p + v$, that has these properties. For each point $p$ of $A$: $p + 0 = p$. For each point $p$ of $A$ and all vectors $u$ and $v$ of $V$: $(p + u) + v = p + (u + v)$. For all points $p$ and $q$ of $A$, there is exactly one vector $v$ of $V$ with $p + v = q$. This vector is written $q - p$ or $\overrightarrow{pq}$. The elements of $A$ are the points, and $V$ is the space of translations of $A$. The dimension of $A$ is the dimension of $V$.