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Affine Space
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. 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 vectors'', also called ''translation'' 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 may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space R3
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Origin
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat (geometry)
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In a -dimensional space, there are flats of every dimension from 0 to ; flats of dimension are called ''hyperplanes''. Flats are the affine subspaces of Euclidean spaces, which means that they are similar to linear subspaces, except that they need not pass through the origin. Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations. A flat is a manifold and an algebraic variety, and is sometimes called a ''linear manifold'' or ''linear variety'' to distinguish it from other manifolds or varieties. Descriptions By equations A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Homogeneous Space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-empty set ''X'' on which ''G'' acts freely and transitively (meaning that, for any ''x'', ''y'' in ''X'', there exists a unique ''g'' in ''G'' such that , where · denotes the (right) action of ''G'' on ''X''). An analogous definition holds in other categories, where, for example, *''G'' is a topological group, ''X'' is a topological space and the action is continuous, *''G'' is a Lie group, ''X'' is a smooth manifold and the action is smooth, *''G'' is an algebraic group, ''X'' is an algebraic variety and the action is regular. Definition If ''G'' is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). 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. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the ''additive group'' of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...s. References {{DEFAULTSORT:Additive group Algebraic structures Group theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Combination
In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ are elements of . The elements can also be points of a Euclidean space, and, more generally, of an affine space over a field . In this case the \alpha_ are elements of (or \mathbb R for a Euclidean space), and the affine combination is also a point. See for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinations commute with any affine transformation in the sense that : T\sum_^ = \sum_^. In particular, any affine combination of the fixed points of a given affine transformatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
