Affine Space

Let $V$ be a Vector Space over a Field $K$, and let $A$ be a nonempty Set. Now define addition $p+{\bf a}\in A$ for any Vector ${\bf a}\in V$ and element $p\in A$ subject to the conditions

1. $p+{\bf0}=p$,

2. $(p+{\bf a})+{\bf b}=p+({\bf a}+{\bf b})$,

3. For any $q\in A$, there Exists a unique Vector ${\bf a}\in V$ such that $q=p+{\bf a}$.

Here, ${\bf a}$, ${\bf b}\in V$. Note that (1) is implied by (2) and (3). Then $A$ is an affine space and $K$ is called the Coefficient Field.

In an affine space, it is possible to fix a point and coordinate axis such that every point in the Space can be represented as an $n$-tuple of its coordinates. Every ordered pair of points $A$ and $B$ in an affine space is then associated with a Vector $AB$.

See also Affine Complex Plane, Affine Connection, Affine Equation, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity