# Affine space

over a field $k$

A set $A$ (the elements of which are called the points of the affine space) to which corresponds a vector space $L$ over $k$ (which is called the space associated to $A$) and a mapping of the set $A \times A$ into the space $L$( the image of an element $(a, b) \in A \times A$ is denoted by $\stackrel{\rightharpoonup}{ab}$ and is called the vector with beginning in $a$ and end in $b$), which has the following properties:

a) for any fixed point $a$ the mapping $x \rightarrow \stackrel{\rightharpoonup}{ax}$, $x \in A$, is a bijection of $A$ on $L$;

b) for any points $a, b, c \in A$ the relationship

$$\stackrel{\rightharpoonup}{ab} + \stackrel{\rightharpoonup}{bc} + \stackrel{\rightharpoonup}{ca} = \stackrel{\rightharpoonup}{0} ,$$

where $\stackrel{\rightharpoonup}{0}$ denotes the zero vector, is valid. The dimension of $L$ is taken for the dimension of the affine space $A$. A point $a \in A$ and a vector $l \in L$ define another point, which is denoted by $a + l$, i.e. the additive group of vectors of the space $L$ acts freely and transitively on the affine space corresponding to $L$.

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### Examples.

1) The set of the vectors of the space $L$ is the affine space $A(L)$; the space associated to it coincides with $L$. In particular, the field of scalars is an affine space of dimension 1. If $L = k ^ {n}$, then $A( k ^ {n} )$ is called the $n$- dimensional affine space over the field $k$, and its points $a = ( a _ {1} \dots a _ {n} )$ and $b = (b _ {1} \dots b _ {n} )$ determine the vector $\stackrel{\rightharpoonup}{ab} = (b _ {1} - a _ {1} \dots b _ {n} - a _ {n} )$.

2) The complement of any hyperplane in a projective space over the field $k$ is an affine space.

3) The set of solutions of a system of linear (algebraic or differential) equations is an affine space the associated space of which is the space of solutions of the corresponding homogeneous set of equations.

A subset $A ^ \prime$ of an affine space $A$ is called an affine subspace (or a linear manifold) in $A$ if the set of vectors $\stackrel{\rightharpoonup}{ab}$, $a, b \in A ^ \prime$, forms a subspace of $L$. Each affine subspace $A ^ \prime \subset A$ has the form $a + L ^ \prime = \{ {a + l } : {l \in L ^ \prime } \}$, where $L ^ \prime$ is some subspace in $L$, while $a$ is an arbitrary element of $A ^ \prime$.

A mapping $f: A _ {1} \rightarrow A _ {2}$ between affine spaces $A _ {1}$ and $A _ {2}$ is called affine if there exists a linear mapping of the associated vector spaces $\phi : L _ {1} \rightarrow L _ {2}$ such that $f(a + l) = f(a) + \phi (l)$ for all $a \in A _ {1}$, $l \in L _ {1}$. A bijective affine mapping is called an affine isomorphism. All affine spaces of the same dimension are mutually isomorphic.

The affine isomorphisms of an affine space $A$ into itself form a group, called the affine group of the affine space $A$ and denoted by ${ \mathop{\rm Aff} } (A)$. The affine group of the affine space $A( k ^ {n} )$ is denoted by ${ \mathop{\rm Aff} } _ {n} (k)$. Each element $f \in { \mathop{\rm Aff} } _ {n} (k)$ is given by a formula

$$f ( ( a _ {1} \dots a _ {n} ) ) = ( b _ {1} \dots b _ {n} ),$$

where

$$b _ {i} = \sum _ { j } a _ {i} ^ {j} a _ {j} + c _ {i} ,$$

$( a _ {i} ^ {j} )$ being an invertible matrix. The affine group ${ \mathop{\rm Aff} } (A)$ contains an invariant subgroup, called the subgroup of (parallel) translations, consisting of the mappings $f: A\rightarrow A$ for which $\phi : L \rightarrow L$ is the identity. This group is isomorphic to the additive group of the vector space $L$. The mapping $f \rightarrow \phi$ defines a surjective homomorphism of ${ \mathop{\rm Aff} } (A)$ into the general linear group GL, with the subgroup of parallel translations as kernel. If $L$ is a Euclidean space, the pre-image of the orthogonal group is called the subgroup of Euclidean motions. The pre-image of the special linear group SGL is called the equi-affine subgroup (cf. Affine unimodular group). The subgroup $G _ {a} \subset { \mathop{\rm Aff} } _ {n} (A)$ consisting of the mappings $f : A \rightarrow A$ such that $f(a+l) = a + \phi (l)$ for a given $a \in A$ and arbitrary $l \in L$ is called the centro-affine subgroup; it is isomorphic to the general linear group GL of the space $L$.

In algebraic geometry an affine algebraic set is sometimes called an affine space. A finite-dimensional affine space can be provided with the structure of an affine variety with the Zariski topology (cf. also Affine scheme).

Affine spaces associated with a vector space over a skew-field $k$ are constructed in a similar manner.

#### References

 [1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207