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m (vectors corrected)
m (\vec --> \stackrel{\rightharpoonup})
 
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''over a field  $  k $''
 
''over a field  $  k $''
  
A set  $  A $(
+
A set  $  A $ (the elements of which are called the points of the affine space) to which corresponds a vector space  $  L $
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 $)  
over  $  k $(
 
which is called the space associated to  $  A $)  
 
 
and a mapping of the set  $  A \times A $
 
and a mapping of the set  $  A \times A $
 
into the space  $  L $(
 
into the space  $  L $(
 
the image of an element  $  (a, b) \in A \times A $
 
the image of an element  $  (a, b) \in A \times A $
is denoted by  $  \vec{ab} $
+
is denoted by  $  \stackrel{\rightharpoonup}{ab} $
 
and is called the vector with beginning in  $  a $
 
and is called the vector with beginning in  $  a $
 
and end in  $  b $),  
 
and end in  $  b $),  
Line 26: Line 24:
  
 
a) for any fixed point  $  a $
 
a) for any fixed point  $  a $
the mapping  $  x \rightarrow \vec{ax} $,  
+
the mapping  $  x \rightarrow \stackrel{\rightharpoonup}{ax} $,  
 
$  x \in A $,  
 
$  x \in A $,  
 
is a bijection of  $  A $
 
is a bijection of  $  A $
Line 35: Line 33:
  
 
$$  
 
$$  
\vec{ab} + \vec{bc} + \vec{ca}  =  \vec{0} ,
+
\stackrel{\rightharpoonup}{ab} + \stackrel{\rightharpoonup}{bc} + \stackrel{\rightharpoonup}{ca}  =  \stackrel{\rightharpoonup}{0} ,
 
$$
 
$$
  
where  $  \vec{0} $
+
where  $  \stackrel{\rightharpoonup}{0} $
 
denotes the zero vector, is valid. The dimension of  $  L $
 
denotes the zero vector, is valid. The dimension of  $  L $
 
is taken for the dimension of the affine space  $  A $.  
 
is taken for the dimension of the affine space  $  A $.  
Line 58: Line 56:
 
and its points  $  a = ( a _ {1} \dots a _ {n} ) $
 
and its points  $  a = ( a _ {1} \dots a _ {n} ) $
 
and  $  b = (b _ {1} \dots b _ {n} ) $
 
and  $  b = (b _ {1} \dots b _ {n} ) $
determine the vector  $  \vec{ab} = (b _ {1} - a _ {1} \dots b _ {n} - a _ {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 $
 
2) The complement of any hyperplane in a projective space over the field  $  k $
Line 68: Line 66:
 
of an affine space  $  A $
 
of an affine space  $  A $
 
is called an affine subspace (or a linear manifold) in  $  A $
 
is called an affine subspace (or a linear manifold) in  $  A $
if the set of vectors  $  \vec{ab} $,  
+
if the set of vectors  $  \stackrel{\rightharpoonup}{ab} $,  
 
$  a, b \in A  ^  \prime  $,  
 
$  a, b \in A  ^  \prime  $,  
 
forms a subspace of  $  L $.  
 
forms a subspace of  $  L $.  

Latest revision as of 20:40, 4 April 2020


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 $.

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

Comments

An affine isomorphism is also called an affine collineation. An equi-affine group is also called a Euclidean group.

References

[a1] M. Berger, "Geometry" , I , Springer (1987) pp. Chapt. 2 MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020
How to Cite This Entry:
Affine space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_space&oldid=45141
This article was adapted from an original article by I.V. DolgachevA.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article