In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an
affine space is a geometric structure that generalizes the
affineIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
properties of
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin. The solution set of an inhomogeneous linear equation is either empty or an affine space.
Informal descriptions
The following
characterizationIn mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
may be easier to understand than the usual formal definition: an affine space is what is left of a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
after you've forgotten which point is the origin (or, in the words of the French mathematician
Marcel BergerMarcel Berger is a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques , France...
, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the true origin, and Bob believes that another point — call it
p — is the origin. Two vectors,
a and
b, are to be added. Bob draws an arrow from
p to
a and another arrow from
p to
b, and completes the parallelogram to find what Bob thinks is
a +
b, but Alice knows that it is actually
 p + (a − p) + (b − p).
Similarly, Alice and Bob may evaluate any
linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of
a and
b, 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
 λa + (1 − λ)b
then Bob can similarly travel to
 p + λ(a − p) + (1 − λ)(b − p) = λa + (1 − λ)b.
Then, for all coefficients λ + (1 − λ) = 1, Alice and Bob describe the same point with the same linear combination, starting from different origins.
Here is the punch line: Alice knows the "linear structure", but 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. An underlying set with an affine structure is an affine space.
Definition
An
affine space is a set
together with a
group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, considered as an
AbelianIn mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel: Group theory :*Abelian group, a group in which the binary operation is commutative...
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
acting on the left such that the only vector acting with a fixpoint is
and there is a single orbit (i.e., the action is free and transitive). It is an example of (left)
set, where
is the additive group
.
Explicitly, an affine space is a point set
together with a map
with the following properties:
 Left identity

 Associativity

 Uniqueness
 is a bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
.
The
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is said to underlie the
affine space and is also called the
difference space.
By choosing an origin,
, one can thus identify
with
, hence turn
into a vector space. Conversely, any vector space,
, is an affine space over itself. The
uniqueness property ensures that subtraction of any two elements of
is well defined, producing a vector of
.
If
,
, and
are points in
and
is a scalar, then
is independent of
. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.
By noting that one can define subtraction of points of an affine space as follows:
is the unique vector in
such that
,
one can equivalently define an affine space as a point set
, together with a vector space
, and a subtraction map
with the following properties :
 there is a unique point such that and
 .
These two properties are called Weyl's axioms.
Examples
 When children find the answers to sums such as 4+3 or 4−2 by counting right or left on a number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as speciallymarked points evenly spaced on the line...
, they are treating the number line as a onedimensional affine space.
 Any coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
of a subspace of a vector space is an affine space over .
 If is a 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 y 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 .
Affine subspaces
An
affine subspace (sometimes called a
linear manifold,
linear variety, or a
flat) of a vector space
is a subset closed under affine combinations of vectors in the space. For example, the set
is an affine space, where
is a family of vectors in
– this space is the
affine span of these points. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace
of
This affine subspace can be equivalently described as the coset of the
action
where
is any element of
, or equivalently as any
level setIn mathematics, a level set of a realvalued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
of the quotient map
A choice of
gives a base point of
and an identification of
with
but there is no natural choice, nor a natural identification of
with
A linear transformation is a function that preserves all
linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s; an affine transformation is a function that preserves all
affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.
For example, in
, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.
Affine combinations and affine dependence
An
affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are
affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three noncollinear points is the plane that contains all three. Vectors
 v_{1}, v_{2}, ..., v_{n}
are linearly dependent if there exist scalars
a_{1},
a_{2}, …,
a_{n}, not all zero, for which
Similarly they are
affinely dependent if the same is true and also
Equation is an
affine relation among the vectors
v_{1},
v_{2}, …,
v_{n}.
Axioms
Affine space is usually studied as
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
using coordinates, or equivalently vector spaces. It can also be studied as
synthetic geometrySynthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.Logical synthesis:The process...
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 geometryOrdered geometry is a form of geometry featuring the concept of intermediacy but, like projective geometry, omitting the basic notion of measurement...
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 noncollinear points.
As well as affine planes over fields (or
division ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with...
s), there are also many
nonDesarguesian planeIn mathematics, a nonDesarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...
s satisfying these axioms. gives axioms for higher dimensional affine spaces.
Relation to projective spaces
Affine spaces are
subspaceIn mathematics:* Euclidean subspace, in linear algebra, a set of vectors in ndimensional Euclidean space that is closed under addition and scalar multiplication...
s of
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
s: an affine plane can be obtained from any
projective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
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
closureIn mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
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 preserve the points at infinity as a set) yield transformations of affine space, and conversely any affine linear transformation extends uniquely to a projective linear transformations, so affine transformations are a subset of projective transforms. Most familiar is that
Möbius transformations (transformations of the projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.
However, one cannot take the projectivization of an affine space, so projective spaces are not naturally
quotientIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
s of affine spaces: one can only take the projectivization of a
vector space, since the projective space is lines
through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.
See also
 affine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
 affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
 affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
 equipollence (geometry)
In geometry, equipollence is a certain relationship between ordered pairs of points. A pair of points and another pair are equipollent precisely if the distance and direction from a to b are respectively the same as the distance and direction from c to d. In affine spaces over a field...
 interval measurement, an affine observable in statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
 heap (mathematics)
In abstract algebra, a heap is a mathematical generalisation of a group. Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space can be viewed as a vector space in which the 0 element has been "forgotten"...
 Space (mathematics)