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Affine group, the Glossary

In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself.^[1]

42 relations: Affine plane, Affine space, Affine transformation, Block matrix, Cambridge University Press, Coordinate system, Coxeter group, Eigenvalues and eigenvectors, Euclidean plane, Euclidean space, Exact sequence, Fixed point (mathematics), General linear group, Group (mathematics), Group extension, Group representation, Holomorph (mathematics), Homography, Hyperplane, Hyperplane at infinity, Isometry, Lattice (group), Lie group, Lorentz group, Mathematics, Matrix similarity, Non-abelian group, Orthogonal group, Perpendicular, Poincaré group, Projective geometry, Real number, Scaling (geometry), Semidirect product, Shear mapping, Similarity (geometry), Special linear group, Subgroup, The American Mathematical Monthly, Theory of relativity, Translation (geometry), Volume form.
Affine geometry

Affine plane

In geometry, an affine plane is a two-dimensional affine space.

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In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces 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 segments. Affine group and affine space are affine geometry.

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Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. Affine group and affine transformation are affine geometry.

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Block matrix

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.

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Cambridge University Press

Cambridge University Press is the university press of the University of Cambridge.

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Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation.

See Affine group and Eigenvalues and eigenvectors

Euclidean plane

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted \textbf^2 or \mathbb^2.

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Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

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Exact sequence

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.

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Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation.

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General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication. Affine group and general linear group are lie groups.

See Affine group and General linear group

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element. Affine group and group (mathematics) are group theory.

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Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. Affine group and group extension are group theory.

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Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Affine group and group representation are group theory.

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Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group G, denoted \operatorname(G), is a group that simultaneously contains (copies of) G and its automorphism group \operatorname(G). Affine group and holomorph (mathematics) are group theory.

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Homography

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.

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Hyperplane

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Affine group and hyperplane are affine geometry.

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Hyperplane at infinity

In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity.

See Affine group and Hyperplane at infinity

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

See Affine group and Isometry

Lattice (group)

In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Affine group and lattice (group) are lie groups.

See Affine group and Lattice (group)

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. Affine group and Lie group are lie groups.

See Affine group and Lie group

Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. Affine group and Lorentz group are group theory and lie groups.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Matrix similarity

In linear algebra, two n-by-n matrices and are called similar if there exists an invertible n-by-n matrix such that B.

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Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. Affine group and orthogonal group are lie groups.

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Perpendicular

In geometry, two geometric objects are perpendicular if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot.

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Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. Affine group and Poincaré group are lie groups.

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Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.

See Affine group and Projective geometry

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Scaling (geometry)

In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

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Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

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Shear mapping

In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction.

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Similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other.

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Special linear group

In mathematics, the special linear group of degree n over a commutative ring R is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. Affine group and special linear group are lie groups.

See Affine group and Special linear group

References

[1] https://en.wikipedia.org/wiki/Affine_group

Also known as Affine general linear group, Affine symmetry, General affine group, Special affine group.

Affine group, the Glossary

Table of Contents

See also

Affine geometry

References