In

vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$\backslash overrightarrow$, and a transitive and free Group action (mathematics), action of the additive group of $\backslash overrightarrow$ on the set . The elements of the affine space are called ''points''. The vector space $\backslash overrightarrow$ is said to be ''associated'' to the affine space, and its elements are called ''vectors'', ''translations'', or sometimes ''free vectors''.
Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,
: $\backslash begin\; A\; \backslash times\; \backslash overrightarrow\; \&\backslash to\; A\; \backslash \backslash \; (a,v)\backslash ;\; \&\backslash mapsto\; a\; +\; v,\; \backslash end$
that has the following properties.
# Right identity:
#: $\backslash forall\; a\; \backslash in\; A,\backslash ;\; a+0\; =\; a$, where is the zero vector in $\backslash overrightarrow$
# Associativity:
#: $\backslash forall\; v,w\; \backslash in\; \backslash overrightarrow,\; \backslash forall\; a\; \backslash in\; A,\backslash ;\; (a\; +\; v)\; +\; w\; =\; a\; +\; (v\; +\; w)$ (here the last is the addition in $\backslash overrightarrow$)
# Free and transitive action:
#: For every $a\; \backslash in\; A$, the mapping $\backslash overrightarrow\; A\; \backslash to\; A\; \backslash colon\; v\; \backslash mapsto\; a\; +\; v$ is a bijection.
The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above:
#Existence of one-to-one
#:For all $v\; \backslash in\; \backslash overrightarrow\; A$, the mapping $A\; \backslash to\; A\; \backslash colon\; a\; \backslash mapsto\; a\; +\; v$ is a bijection.
Property 3 is often used in the following equivalent form.
#Subtraction:
#:For every in , there exists a unique $v\backslash in\backslash overrightarrow\; A$, denoted , such that $b\; =\; a\; +\; v$.
Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are by definition endowed with a transitive group action, and for a principal homogeneous space such a transitive action is by definition free.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, an affine space is a geometric structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A ...

that generalizes some of the properties of Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensi ...

s in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segment
250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B''
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

s.
In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vector
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s'', also called ''translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.
Any vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. In this case, the elements of the vector space may be viewed either as ''points'' of the affine space or as ''displacement vectors'' or ''translations''. When considered as a point, the zero vector is called the ''origin''. Adding a fixed vector to the elements of a linear subspace
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

produces an ''affine subspace''. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector. In finite dimensions, such an ''affine subspace'' is the solution set of an inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences
Science (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally s ...

linear system. The displacement vectors for that affine space are the solutions of the corresponding ''homogeneous'' linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.
The ''dimension'' of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine planeIn geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
*Euclidean planes, which are affine planes over the real number, reals, equipped with a metric (mathematics), metric, the Euclidean distan ...

. An affine subspace of dimension in an affine space or a vector space of dimension is an affine hyperplane
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

.
Informal description

The followingcharacterization
Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative
A narrative, story or tale is any account of a series of related events or experiences, whether nonfictional ( memoir, biography, ...

may be easier to understand than the usual formal definition: an affine space is what is left of a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

after you've forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger
Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...

, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translation (geometry), translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it —is the origin. Two vectors, and , are to be added. Bob draws an arrow from point to point and another arrow from point to point , and completes the parallelogram to find what Bob thinks is , but Alice knows that he has actually computed
: .
Similarly, Alice and Bob may evaluate any linear combination of and , or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer.
If Alice travels to
:
then Bob can similarly travel to
: .
Under this condition, for all coefficients , Alice and Bob describe the same point with the same linear combination, despite using different origins.
While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.
Definition

An ''affine space'' is a set together with atranslation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

sSubtraction and Weyl's axioms

The properties of the group action allows for the definition of subtraction for any given ordered pair of points in , producing a vector of $\backslash overrightarrow$. This vector, denoted $b\; -\; a$ or $\backslash overrightarrow$, is defined to be the unique vector in $\backslash overrightarrow$ such that : $a\; +\; (b\; -\; a)\; =\; b.$ Existence follows from the transitivity of the action, and uniqueness follows because the action is free. This subtraction has the two following properties, called Hermann Weyl, Weyl's axioms: # $\backslash forall\; a\; \backslash in\; A,\backslash ;\; \backslash forall\; v\backslash in\; \backslash overrightarrow$, there is a unique point $b\; \backslash in\; A$ such that $b\; -\; a\; =\; v.$ # $\backslash forall\; a,b,c\; \backslash in\; A,\backslash ;\; (c\; -\; b)\; +\; (b\; -\; a)\; =\; c\; -\; a.$ In Euclidean geometry, the second Weyl's axiom is commonly called the ''parallelogram rule''. Affine spaces can be equivalently defined as a point set , together with a vector space $\backslash overrightarrow$, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first Weyl's axioms.Affine subspaces and parallelism

An ''affine subspace'' (also called, in some contexts, a ''linear variety'', a flat (geometry), flat, or, over the real numbers, a ''linear manifold'') of an affine space is a subset of such that, given a point $a\; \backslash in\; B$, the set of vectors $\backslash overrightarrow\; =\; \backslash $ is alinear subspace
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of $\backslash overrightarrow$. This property, which does not depend on the choice of , implies that is an affine space, which has $\backslash overrightarrow$ as its associated vector space.
The affine subspaces of are the subsets of of the form
: $a\; +\; V\; =\; \backslash ,$
where is a point of , and a linear subspace of $\backslash overrightarrow$.
The linear subspace associated with an affine subspace is often called its ', and two subspaces that share the same direction are said to be ''parallel''.
This implies the following generalization of Playfair's axiom: Given a direction , for any point of there is one and only one affine subspace of direction , which passes through , namely the subspace .
Every translation $A\; \backslash to\; A:\; a\; \backslash mapsto\; a\; +\; v$ maps any affine subspace to a parallel subspace.
The term ''parallel'' is also used for two affine subspaces such that the direction of one is included in the direction of the other.
Affine map

Given two affine spaces and whose associated vector spaces are $\backslash overrightarrow$ and $\backslash overrightarrow$, an ''affine map'' or ''affine homomorphism'' from to is a map : $f:\; A\; \backslash to\; B$ such that : $\backslash begin\; \backslash overrightarrow:\; \backslash overrightarrow\; \&\backslash to\; \backslash overrightarrow\backslash \backslash \; b\; -\; a\; \&\backslash mapsto\; f(b)\; -\; f(a)\; \backslash end$ is a well defined linear map. By $f$ being well defined is meant that implies . This implies that, for a point $a\; \backslash in\; A$ and a vector $v\; \backslash in\; \backslash overrightarrow$, one has : $f(a\; +\; v)\; =\; f(a)\; +\; \backslash overrightarrow(v).$ Therefore, since for any given in , for a unique , is completely defined by its value on a single point and the associated linear map $\backslash overrightarrow$.Vector spaces as affine spaces

Every vector space may be considered as an affine space over itself. This means that every element of may be considered either as a point or as a vector. This affine space is sometimes denoted for emphasizing the double role of the elements of . When considered as a point, thezero vector
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is commonly denoted (or , when upper-case letters are used for points) and called the ''origin''.
If is another affine space over the same vector space (that is $V\; =\; \backslash overrightarrow$) the choice of any point in defines a unique affine isomorphism, which is the identity of and maps to . In other words, the choice of an origin in allows us to identify and up to a canonical isomorphism. The counterpart of this property is that the affine space may be identified with the vector space in which "the place of the origin has been forgotten".
Relation to Euclidean spaces

Definition of Euclidean spaces

Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces. Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form . The inner product of two vectors and is the value of the symmetric bilinear form : $x\; \backslash cdot\; y\; =\; \backslash frac\; 12\; (q(x\; +\; y)\; -\; q(x)\; -\; q(y)).$ The usual Euclidean distance between two points and is : $d(A,\; B)\; =\; \backslash sqrt.$ In older definition of Euclidean spaces through synthetic geometry, vectors are defined as equivalence classes of ordered pairs of points under equipollence (geometry), equipollence (the pairs and are ''equipollent'' if the points (in this order) form a parallelogram). It is straightforward to verify that the vectors form a vector space, the square of the Euclidean distance is a quadratic form on the space of vectors, and the two definitions of Euclidean spaces are equivalent.Affine properties

In Euclidean geometry, the common phrase "affine property" refers to a property that can be proved in affine spaces, that is, it can be proved without using the quadratic form and its associated inner product. In other words, an affine property is a property that does not involve lengths and angles. Typical examples are parallelism, and the definition of a tangent. A non-example is the definition of a normal (geometry), normal. Equivalently, an affine property is a property that is invariant under affine transformations of the Euclidean space.Affine combinations and barycenter

Let be a collection of points in an affine space, and $\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n$ be elements of the ground field. Suppose that $\backslash lambda\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; =\; 0$. For any two points and one has : $\backslash lambda\_1\; \backslash overrightarrow\; +\; \backslash dots\; +\; \backslash lambda\_n\; \backslash overrightarrow\; =\; \backslash lambda\_1\; \backslash overrightarrow\; +\; \backslash dots\; +\; \backslash lambda\_n\; \backslash overrightarrow.$ Thus this sum is independent of the choice of the origin, and the resulting vector may be denoted : $\backslash lambda\_1\; a\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; a\_n\; .$ When $n\; =\; 2,\; \backslash lambda\_1\; =\; 1,\; \backslash lambda\_2\; =\; -1$, one retrieves the definition of the subtraction of points. Now suppose instead that the field (mathematics), field elements satisfy $\backslash lambda\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; =\; 1$. For some choice of an origin , denote by $g$ the unique point such that : $\backslash lambda\_1\; \backslash overrightarrow\; +\; \backslash dots\; +\; \backslash lambda\_n\; \backslash overrightarrow\; =\; \backslash overrightarrow.$ One can show that $g$ is independent from the choice of . Therefore, if : $\backslash lambda\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; =\; 1,$ one may write : $g\; =\; \backslash lambda\_1\; a\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; a\_n.$ The point $g$ is called the centroid, barycenter of the $a\_i$ for the weights $\backslash lambda\_i$. One says also that $g$ is an affine combination of the $a\_i$ with coefficients $\backslash lambda\_i$.Examples

* When children find the answers to sums such as or by counting right or left on a number line, they are treating the number line as a one-dimensional affine space. * Any coset of a subspace of a vector space is an affine space over that subspace. * If is a matrix (mathematics), matrix and lies in its column space, the set of solutions of the equation is an affine space over the subspace of solutions of . * The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. * Generalizing all of the above, if is a linear mapping and lies in its image, the set of solutions to the equation is a coset of the kernel of , and is therefore an affine space over . * The space of (linear) complementary subspaces of a vector subspace in a vector space is an affine space, over . That is, if is a short exact sequence of vector spaces, then the space of all split exact sequence, splittings of the exact sequence naturally carries the structure of an affine space over .Affine span and bases

For any subset of an affine space , there is a smallest affine subspace that contains it, called the affine span of . It is the intersection of all affine subspaces containing , and its direction is the intersection of the directions of the affine subspaces that contain . The affine span of is the set of all (finite) affine combinations of points of , and its direction is the linear span of the for and in . If one chooses a particular point , the direction of the affine span of is also the linear span of the for in . One says also that the affine span of is generated by and that is a generating set of its affine span. A set of points of an affine space is said to be or, simply, independent, if the affine span of any strict subset of is a strict subset of the affine span of . An or barycentric frame (see , below) of an affine space is a generating set that is also independent (that is a minimal generating set). Recall the ''dimension'' of an affine space is the dimension of its associated vector space. The bases of an affine space of finite dimension are the independent subsets of elements, or, equivalently, the generating subsets of elements. Equivalently, is an affine basis of an affine space if and only if is a linear basis of the associated vector space.Coordinates

There are two strongly related kinds of coordinate systems that may be defined on affine spaces.Barycentric coordinates

Let be an affine space of dimension over a field (mathematics), field , and $\backslash $ be an affine basis of . The properties of an affine basis imply that for every in there is a unique -tuple $(\backslash lambda\_0,\; \backslash dots,\; \backslash lambda\_n)$ of elements of such that : $\backslash lambda\_0\; +\; \backslash dots\; +\; \backslash lambda\_n\; =\; 1$ and : $x\; =\; \backslash lambda\_0\; x\_0\; +\; \backslash dots\; +\; \backslash lambda\_n\; x\_n.$ The $\backslash lambda\_i$ are called the barycentric coordinates of over the affine basis $\backslash $. If the are viewed as bodies that have weights (or masses) $\backslash lambda\_i$, the point is thus the Centroid, barycenter of the , and this explains the origin of the term ''barycentric coordinates''. The barycentric coordinates define an affine isomorphism between the affine space and the affine subspace of defined by the equation $\backslash lambda\_0\; +\; \backslash dots\; +\; \backslash lambda\_n\; =\; 1$. For affine spaces of infinite dimension, the same definition applies, using only finite sums. This means that for each point, only a finite number of coordinates are non-zero.Affine coordinates

An affine frame of an affine space consists of a point, called the ''origin'', and a linear basis of the associated vector space. More precisely, for an affine space with associated vector space $\backslash overrightarrow$, the origin belongs to , and the linear basis is a basis of $\backslash overrightarrow$ (for simplicity of the notation, we consider only the case of finite dimension, the general case is similar). For each point of , there is a unique sequence $\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n$ of elements of the ground field such that : $p\; =\; o\; +\; \backslash lambda\_1\; v\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; v\_n,$ or equivalently : $\backslash overrightarrow\; =\; \backslash lambda\_1\; v\_1\; +\; \backslash dots\; +\; \backslash lambda\_n\; v\_n.$ The $\backslash lambda\_i$ are called the affine coordinates of over the affine frame . Example: In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an orthonormal frame, that is an affine frame such that is an orthonormal basis.Relationship between barycentric and affine coordinates

Barycentric coordinates and affine coordinates are strongly related, and may be considered as equivalent. In fact, given a barycentric frame : $(x\_0,\; \backslash dots,\; x\_n),$ one deduces immediately the affine frame : $(x\_0,\; \backslash overrightarrow,\; \backslash dots,\; \backslash overrightarrow)\; =\; \backslash left(x\_0,\; x\_1\; -\; x\_0,\; \backslash dots,\; x\_n\; -\; x\_0\backslash right),$ and, if : $\backslash left(\backslash lambda\_0,\; \backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n\backslash right)$ are the barycentric coordinates of a point over the barycentric frame, then the affine coordinates of the same point over the affine frame are : $\backslash left(\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n\backslash right).$ Conversely, if : $\backslash left(o,\; v\_1,\; \backslash dots,\; v\_n\backslash right)$ is an affine frame, then : $\backslash left(o,\; o\; +\; v\_1,\; \backslash dots,\; o\; +\; v\_n\backslash right)$ is a barycentric frame. If : $\backslash left(\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n\backslash right)$ are the affine coordinates of a point over the affine frame, then its barycentric coordinates over the barycentric frame are : $\backslash left(1\; -\; \backslash lambda\_1\; -\; \backslash dots\; -\; \backslash lambda\_n,\; \backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n\backslash right).$ Therefore, barycentric and affine coordinates are almost equivalent. In most applications, affine coordinates are preferred, as involving less coordinates that are independent. However, in the situations where the important points of the studied problem are affinity independent, barycentric coordinates may lead to simpler computation, as in the following example.Example of the triangle

The vertices of a non-flat triangle form an affine basis of the Euclidean plane. The barycentric coordinates allows easy characterization of the elements of the triangle that do not involve angles or distance: The vertices are the points of barycentric coordinates , and . The lines supporting the edges are the points that have a zero coordinate. The edges themselves are the points that have a zero coordinate and two nonnegative coordinates. The interior of the triangle are the points whose all coordinates are positive. The median (geometry), medians are the points that have two equal coordinates, and the centroid is the point of coordinates .Change of coordinates

Case of affine coordinates

Case of barycentric coordinates

Properties of affine homomorphisms

Matrix representation

Image and fibers

Let : $f\; \backslash colon\; E\; \backslash to\; F$ be an affine homomorphism, with : $\backslash overrightarrow\; \backslash colon\; \backslash overrightarrow\; \backslash to\; \backslash overrightarrow$ as associated linear map. The image of is the affine subspace of , which has $\backslash overrightarrow(\backslash overrightarrow)$ as associated vector space. As an affine space does not have a zero element, an affine homomorphism does not have a kernel (algebra), kernel. However, for any point of , the inverse image of is an affine subspace of , of direction $\backslash overrightarrow^(\backslash overrightarrow)$. This affine subspace is called the fiber (mathematics), fiber of .Projection

An important example is the projection parallel to some direction onto an affine subspace. The importance of this example lies in the fact thatEuclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensi ...

s are affine spaces, and that this kind of projections is fundamental in Euclidean geometry.
More precisely, given an affine space with associated vector space $\backslash overrightarrow$, let be an affine subspace of direction $\backslash overrightarrow$, and be a complementary subspace of $\backslash overrightarrow$ in $\backslash overrightarrow$ (this means that every vector of $\backslash overrightarrow$ may be decomposed in a unique way as the sum of an element of $\backslash overrightarrow$ and an element of ). For every point of , its projection to parallel to is the unique point in such that
: $p(x)\; -\; x\; \backslash in\; D.$
This is an affine homomorphism whose associated linear map $\backslash overrightarrow$ is defined by
: $\backslash overrightarrow(x\; -\; y)\; =\; p(x)\; -\; p(y),$
for and in .
The image of this projection is , and its fibers are the subspaces of direction .
Quotient space

Although kernels are not defined for affine spaces, quotient spaces are defined. This results from the fact that "belonging to the same fiber of an affine homomorphism" is an equivalence relation. Let be an affine space, and be alinear subspace
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the associated vector space $\backslash overrightarrow$. The quotient of by is the quotient by an equivalence relation, quotient of by the equivalence relation
: $x\; -\; y\; \backslash in\; D.$
This quotient is an affine space, which has $\backslash overrightarrow/D$ as associated vector space.
For every affine homomorphism $E\; \backslash to\; F$, the image is isomorphic to the quotient of by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces.
Affine transformation

Axioms

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space. axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line. Affine planes satisfy the following axioms : (in which two lines are called parallel if they are equal or disjoint): * Any two distinct points lie on a unique line. * Given a point and line there is a unique line which contains the point and is parallel to the line * There exist three non-collinear points. As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. gives axioms for higher-dimensional affine spaces.Relation to projective spaces

Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure (mathematics), closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines. Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant (mathematics)#Invariant set, invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.Affine algebraic geometry

In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called ''polynomial functions over the affine space''. For defining a ''polynomial function over the affine space'', one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates. The choice of a system of affine coordinates for an affine space $\backslash mathbb\_k^n$ of dimension over a field (mathematics), field induces an affine isomorphism between $\backslash mathbb\_k^n$ and the affine coordinate space . This explains why, for simplification, many textbooks write $\backslash mathbb\_k^n\; =\; k^n$, and introduce affine algebraic varieties as the common zeros of polynomial functions over . As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties.Ring of polynomial functions

By the definition above, the choice of an affine frame of an affine space $\backslash mathbb\_k^n$ allows one to identify the polynomial functions on $\backslash mathbb\_k^n$ with polynomials in variables, the ''i''th variable representing the function that maps a point to its th coordinate. It follows that the set of polynomial functions over $\backslash mathbb\_k^n$ is a algebra over a field, -algebra, denoted $k\backslash left[\backslash mathbb\_k^n\backslash right]$, which is isomorphic to the polynomial ring $k\backslash left[X\_1,\; \backslash dots,\; X\_n\backslash right]$. When one changes coordinates, the isomorphism between $k\backslash left[\backslash mathbb\_k^n\backslash right]$ and $k[X\_1,\; \backslash dots,\; X\_n]$ changes accordingly, and this induces an automorphism of $k\backslash left[X\_1,\; \backslash dots,\; X\_n\backslash right]$, which maps each indeterminate to a polynomial of degree one. It follows that the total degree defines a Filtration (mathematics), filtration of $k\backslash left[\backslash mathbb\; A\_k^n\backslash right]$, which is independent from the choice of coordinates. The total degree defines also a graded ring, graduation, but it depends on the choice of coordinates, as a change of affine coordinates may map indeterminates on non-homogeneous polynomials.Zariski topology

Affine spaces over topological fields, such as the real or the complex numbers, have a natural topology (structure), topology. The Zariski topology, which is defined for affine spaces over any field, allows use of topological methods in any case. Zariski topology is the unique topology on an affine space whose closed sets are affine algebraic sets (that is sets of the common zeros of polynomials functions over the affine set). As, over a topological field, polynomial functions are continuous, every Zariski closed set is closed for the usual topology, if any. In other words, over a topological field, Zariski topology is coarser topology, coarser than the natural topology. There is a natural injective function from an affine space into the set of prime ideals (that is the spectrum of a ring, spectrum) of its ring of polynomial functions. When affine coordinates have been chosen, this function maps the point of coordinates $\backslash left(a\_1,\; \backslash dots,\; a\_n\backslash right)$ to the maximal ideal $\backslash left\backslash langle\; X\_1\; -\; a\_1,\; \backslash dots,\; X\_n\; -\; a\_n\backslash right\backslash rangle$. This function is a homeomorphism (for the Zariski topology of the affine space and of the spectrum of the ring of polynomial functions) of the affine space onto the image of the function. The case of an algebraically closed field, algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz). This is the starting idea of scheme theory of Grothendieck, which consists, for studying algebraic varieties, of considering as "points", not only the points of the affine space, but also all the prime ideals of the spectrum. This allows gluing together algebraic varieties in a similar way as, for manifolds, chart (topology), charts are glued together for building a manifold.Cohomology

Like all affine varieties, local data on an affine space can always be patched together globally: the cohomology of affine space is trivial. More precisely, $H^i\backslash left(\backslash mathbb\_k^n,\backslash mathbf\backslash right)\; =\; 0$ for all coherent sheaves F, and integers $i\; >\; 0$. This property is also enjoyed by all other affine variety, affine varieties. But also all of the etale cohomology groups on affine space are trivial. In particular, every line bundle is trivial. More generally, the Quillen–Suslin theorem implies that ''every'' algebraic vector bundle over an affine space is trivial.See also

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References

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