Affine space - Knowino (2024)

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The 2-dimensional plane, well-known from elementary Euclidean geometry, is an example of an affine space. Remember that in elementary geometry none of the points in the plane is special—there is no origin. A real n-dimensional affine space is distinguished from the vector space Affine space - Knowino (2) by having no special point, no fixed origin.[1]

From elementary geometry we know that any two points in a plane (a collection of infinitely many points) can be connected by a line segment. If the points P and Q in a plane are ordered with P before Q, the line segment connecting the two becomes an arrow pointing from P to Q. This arrow can be mapped onto a vector, the difference vector, denoted by Affine space - Knowino (3).[2] If all arrows in a plane can be mapped onto vectors of a 2-dimensional vector space V2, called the difference space, the plane is an affine space of dimension 2, denoted by A2.Arrows that are mapped onto the same vector in the difference space are said to be parallel, they differ from each other by translation.

In elementary analytic geometry, the map of arrows onto vectors is almost always defined by the choice of an origin O, which is a point somewhere in the plane. Clearly, an arbitrary point P is the head of an arrow with tail in the origin and corresponding with the unique difference vector Affine space - Knowino (4). All arrows with tail in O are mapped one-to-one onto a 2-dimensional difference space V2, with the vector addition in V2 in one-to-one correspondence with the parallelogram rule for the addition of arrows in the plane.

Usually one equips the difference space with an inner product, turning it into an inner product space. Its elements have well-defined length, namely, the square root of the inner product of the vector with itself. The distance between any two points P and Q may now be defined as the length of Affine space - Knowino (5) in V2. A two-dimensional affine space, with this distance defined between the points, is the Euclidean plane known from high-school geometry.

Upon formalizing and generalizing the definition of an affine space, we replace the dimension 2 by an arbitrary finite dimension n and replace arrows by ordered pairs of points ("head" and "tail") in a given point space A. Briefly, A is an affine space of dimension n if there exists a map of the Cartesian product, A × A onto a vector space of dimension n. This map must satisfy certain axioms that are treated in the next section. If the dimension needs to be exhibited, we may write An for the affine space of dimension n.


  • 1 Formal definition
  • 2 Parallelogram law
  • 3 Position vector
  • 4 Affine coordinate systems
  • 5 Affine maps
  • 6 Euclidean space
  • 7 Notes and references

[edit] Formal definition

We will restrict the definition to vector spaces over the field Affine space - Knowino (6) of real numbers.

Let V be an n-dimensional vector space and A a set of elements that we will call points. Assume that a relation between points and vectors is defined in the following way:

  1. To every ordered pair P, Q of A there is assigned a vector of V, called the difference vector, denoted by Affine space - Knowino (7).
  2. To every point P of A and every vector Affine space - Knowino (8) of V there exists exactly one point Q in A, such that Affine space - Knowino (9).
  3. If P, Q, and R are three arbitrary points in A, then
Affine space - Knowino (10)

If these three postulates hold, the set A is an n-dimensional affine space with difference space V.

Two immediate and important consequences are:

Affine space - Knowino (11)

Lemma 1:

Affine space - Knowino (12).

Proof: If the points coincide, Affine space - Knowino (13), we just saw that the difference vector is the zero vector. Conversely, assume that Affine space - Knowino (14) and Affine space - Knowino (15). Then for an arbitrary point Affine space - Knowino (16),

Affine space - Knowino (17)

which implies that the same vector in V connects Affine space - Knowino (18) in A with two different points, which by postulate 2 is forbidden.

[edit] Parallelogram law

The parallelogram law in the 2-dimensional Euclidean plane.

Consider four points in A: P1, P2, Q1, and Q2. Assume that the following difference vectors are equal,

Affine space - Knowino (21)

then we may exchange Q1 and P2,

Affine space - Knowino (22)

See the figure for a concrete example in which the four points form a parallelogram.

Proof: Subtract the following equations:

Affine space - Knowino (23)

This gives

Affine space - Knowino (24)

[edit] Position vector

Choose a fixed point O in the affine space A, an origin. Every point P is uniquely determined by the vector Affine space - Knowino (25). Indeed, suppose that there is another point Q such that Affine space - Knowino (26), it then follows from lemma 1 that P = Q.

The vector Affine space - Knowino (27) is the position vector of P with respect to O. After choosing O every point P in A is uniquely identified by its corresponding position vector Affine space - Knowino (28).

Choice of another origin O gives a translation of Affine space - Knowino (29) by Affine space - Knowino (30), for

Affine space - Knowino (31)

with Affine space - Knowino (32) the position vector of P with respect to O .

[edit] Affine coordinate systems

An affine coordinate system

Affine space - Knowino (33)

consists of an origin O in A and a basis Affine space - Knowino (34) of the difference space V. Then every point P in A determines a system of n real numbers xi (i = 1, ..., n) by

Affine space - Knowino (35)

The numbers xi (i = 1, ..., n) are the affine coordinates of P with respect to the given coordinate system. Note that O has the coordinates xi = 0.

Consider now two different affine coordinate systems,

Affine space - Knowino (36)


Affine space - Knowino (37)

The matrix (Ai j) transforms the one basis of V into the other, hence it is a square regular (invertible) matrix. The real numbers ti are the affine coordinates of O relative to Affine space - Knowino (38).

Express a fixed point P with respect to Affine space - Knowino (39) and Affine space - Knowino (40),

Affine space - Knowino (41)

Insert into the second equation

Affine space - Knowino (42)

and express Affine space - Knowino (43) in Affine space - Knowino (44) , then

Affine space - Knowino (45)

so that the transformation from the one affine coordinate system to the other is,

Affine space - Knowino (46)

where we introduced bold lowercase letters for real column-vectors (stacks of n real numbers) andthe boldface capital indicates an n × n matrix.Inversion of the non-singular (regular) matrix (Ai j) gives the inverse transformation,

Affine space - Knowino (47)

[edit] Affine maps

Let PP′ be a mapping of the affine space A into itself; if the map satisfies the following two conditions, it is an affine map.

(i) Conserve "parallelism". (In the Euclidean plane this condition implies that sets of mutually parallel arrows are mapped onto sets of mutually parallel arrows. Note, however, that in general the mapped arrows are not parallel to the original arrows):

Affine space - Knowino (48)

(ii) The map is linear in the difference space. That is, the map φ: VV defined by

Affine space - Knowino (49)

is linear.[3]A translation is an affine map with φ the identity operation,

Affine space - Knowino (50)

Here the origin and the image are parallel.

Fig. 1. Affine map. Here φ is linear, but not necessarily length preserving.

Given two points O and O′ and a linear map φ: VV. There exists exactly one affine map that sends O into O′ and induces φ on V. This is the map

Affine space - Knowino (53)

because (see Fig. 1),

Affine space - Knowino (54)

Note: since Affine space - Knowino (55) by the parallelogram law (see above), the map φ may also be defined as

Affine space - Knowino (56)

as was done in Eq. (1).

Choose a basis

Affine space - Knowino (57)


Affine space - Knowino (58)

from which follows the matrix-vector expression for an affine map,

Affine space - Knowino (59)

Often[4] one writes the last expression for an affine map with the aid of a square (n+1) × (n+1) matrix that contains F on the diagonal and that is augmented with the translation vector t and the number 1,

Affine space - Knowino (60)

[edit] Euclidean space

Let A be an affine space with difference space V on which a positive-definite inner product is defined. Then A is called a Euclidean space. The distance between two point P and Qis defined by the length ,

Affine space - Knowino (61)

where the expression between round brackets indicates the inner product of the vector with itself.It follows from the properties of the real inner product that the distance has the usual properties,

  1. ρ(P,Q) ≥ 0 and ρ(P,Q) = 0 if and only if P = Q
  2. ρ(P,Q) = ρ(Q,P)
  3. ρ(P,Q) ≤ ρ(P,R) + ρ(R,Q)

A rigid motion of a Euclidean space is an affine map which preserves distances. The linear map φ on V is then a rotation. Conversely, given a rotation φ and two points P and P′ then there exists exactly one rigid motion which sends P into P′ and induces φ on V.

[edit] Notes and references

  1. V. I . Arnold, Mathematical Methods of Classical Mechanics, translated from the Russian by K. Vogtmann and A. Weinstein. Springer, New York (1978).
  2. Sometimes it is stated: "the arrow is a vector", but in the present context it is necessary to carefully distinguish arrows from vectors.
  3. Recall that φ is linear when
    Affine space - Knowino (62)
  4. The computer language PostScript calls the augmented matrix (for n = 2) the "current transformation matrix" (CTM).
  • A. Lichnerowicz, Elements of Tensor Calculus, Translated from the French by J. W. Leech and D. J. Newman, Methuen (London) 1962.
  • W. H. Greub, Linear Algebra, 2nd edition, Springer (Berlin) 1963.

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I'm an enthusiast with demonstrable knowledge of the topic at hand, affine spaces and related concepts in mathematics. I'll provide information on all the concepts mentioned in the article you provided:

1. Affine Space:

  • An affine space is a mathematical concept used in geometry and linear algebra.
  • In an affine space, there is no fixed origin or special point, unlike a vector space.
  • It is a set of points where you can calculate differences between points, and these differences are represented by vectors.

2. Difference Vector:

  • The difference vector between two points in an affine space represents the translation from one point to another.
  • It is denoted as (P - Q), where P and Q are points in the affine space.
  • The difference vectors between all pairs of points in an affine space can be mapped to a vector space called the "difference space."

3. Difference Space:

  • The difference space is a vector space where difference vectors from an affine space are mapped.
  • In a 2-dimensional affine space, this difference space is referred to as V2.

4. Parallel Arrows:

  • Arrows in an affine space that are mapped onto the same vector in the difference space are said to be parallel.
  • They differ from each other only by translation.

5. Inner Product Space:

  • The difference space can be equipped with an inner product, making it an inner product space.
  • Inner product spaces have well-defined lengths for their elements, which is calculated as the square root of the inner product of the vector with itself.

6. Euclidean Plane:

  • A two-dimensional affine space with a defined distance between points, calculated from the inner product, is known as the Euclidean plane.
  • The Euclidean plane is the familiar geometric plane studied in high-school geometry.

7. Formal Definition of Affine Space:

  • The article provides a formal definition of an n-dimensional affine space, where n is an arbitrary finite dimension.
  • The definition involves a set of points and their relations with vectors, subject to certain axioms.

8. Parallelogram Law:

  • The article discusses the parallelogram law, which deals with four points in an affine space.
  • If certain difference vectors are equal, it allows for the exchange of points, forming a parallelogram.

9. Position Vector:

  • A position vector represents a point in an affine space with respect to a chosen origin.
  • It is denoted as P = O + v, where O is the origin and v is a vector.
  • The position vector uniquely identifies a point.

10. Affine Coordinate Systems:

  • Affine coordinate systems consist of an origin and a basis for the difference space.
  • These systems allow points to be represented by sets of real numbers (affine coordinates) relative to the given coordinate system.

11. Affine Maps:

  • Affine maps are mappings of an affine space into itself.
  • They must conserve "parallelism" and be linear in the difference space.
  • These maps can include translations, rotations, and other transformations.

12. Euclidean Space:

  • An affine space with a positive-definite inner product is called a Euclidean space.
  • In Euclidean spaces, distances between points are defined based on the inner product, and they follow typical distance properties.

These concepts are fundamental in various areas of mathematics, including geometry, linear algebra, and vector calculus. They provide a framework for understanding the relationships between points and vectors in different spaces.

Affine space - Knowino (2024)


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