Definition:Affine Space
Definition
Associativity Axioms
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which two mappings are defined:
- $+ : \EE \times V \to \EE$
- $- : \EE \times \EE \to V$
satisfying the following associativity conditions:
\((\text A 1)\) | $:$ | \(\ds \forall p, q \in \EE:\) | \(\ds p + \paren {q - p} = q \) | ||||||
\((\text A 2)\) | $:$ | \(\ds \forall p \in \EE: \forall u, v \in V:\) | \(\ds \paren {p + u} + v = p + \paren {u +_V v} \) | ||||||
\((\text A 3)\) | $:$ | \(\ds \forall p, q \in \EE: \forall u \in V:\) | \(\ds \paren {p - q} +_V u = \paren {p + u} - q \) |
Then the ordered triple $\struct {\EE, +, -}$ is an affine space.
Group Action
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set.
Let $\phi: \EE \times V \to \EE$ be a free and transitive group action of $\struct {V, +_V}$ on $\EE$.
Then the ordered pair $\struct {\EE, \phi}$ is an affine space.
Weyl's Axioms
Let $K$ be a field.
Let $\struct{V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which a mapping is defined:
- $- : \EE \times \EE \to V$
satisfying the following associativity conditions:
\((\text W 1)\) | $:$ | \(\ds \forall p \in \EE: \forall v \in V: \exists ! q \in \EE:\) | \(\ds v = q - p \) | ||||||
\((\text W 2)\) | $:$ | \(\ds \forall p, q, r \in \EE:\) | \(\ds \paren{r - q} +_V \paren{q - p} = r - p \) |
Then the ordered pair $\tuple {\EE, -}$ is an affine space.
Addition
Let $\tuple {\EE, +, -}$ be an affine space.
Then the mapping $+$ is called affine addition.
Subtraction
Let $\tuple {\EE, +, -}$ be an affine space.
Then the mapping $-$ is called affine subtraction.
Tangent Space
Let $\tuple {\EE, +, -}$ be an affine space.
Let $V$ be the vector space that is the codomain of $-$.
Then $V$ is the tangent space of $\EE$.
Vector
Let $\EE$ be an affine space.
Let $V$ be the tangent space of $\EE$.
An element $v$ of $V$ is called a vector.
Point
Let $\EE$ be an affine space.
Any element $p$ of $\EE$ is called a point.
Also see
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): affine space