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In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented (or right-handed) and which are "negatively" oriented (or left-handed). History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
Let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. There are precisely two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class and −1 to the other. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...
Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on Rn gives rise to a standard orientation on Rn. Any choice of a linear isomorphism between V and Rn will then give rise to an orientation on V in an obvious manner. In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
Note that the ordering of elements in a basis is crucial. Two basis with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ...
In mathematics, the permutations of a finite set (i. ...
In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ...
Alternate viewpoints We present two alternate (and more abstract) ways of understanding orientations:
1. For any real vector space V we can form the kth-exterior power of V, denoted ΛkV. This is a real vector space of dimension n-choose-k. The vector space ΛnV (called the top exterior power) therefore has dimension 1. That is, ΛnV is just a real line. A priori there is no choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero element ω of ΛnV determines an orientation of V by declaring ω to be in the positive direction. To connect with the basis point of view we say that the positively oriented bases are those on which ω evaluates to a positive number (since ω is a n-form we can evaluate it on a ordered set of n vectors, giving an element of R). The form ω is called an orientation form. If {ei} is a privileged basis for V then the orientation form giving the standard orientation is e1∧e2∧…∧en. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
In mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here m! denotes the factorial of m). ...
2. Let B be the set of all ordered bases for V. Then the general linear group GL(V) acts freely and transitively on B. (In fancy language, B is a GL(V)-torsor). This means that as a manifold, B is (noncanonically) homeomorphic to GL(V). Note that the group GL(V) is not connected, but rather has two connected components according to whether the determinant of the transformation is positive or negative. The identity component of GL(V) is denoted GL+(V) and consists of those transformations with positive determinant. The action of GL+(V) on B is not transitive: there are two orbits which correspond to the connected components of B. These orbits are precisely the equivalence classes referred to above. Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL(V) which does have a privileged component: the component of the identity. A specific choice of homeomorphism between B and GL(V) is equivalent to a choice of a privileged basis and therefore determines an orientation. In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
In mathematics, groups are often used to describe symmetries of objects. ...
In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...
In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
This word should not be confused with homomorphism. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
Connected components come up in topology and in graph theory, two related branches of mathematics. ...
In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ...
Orientation on manifolds One can also discuss orientation on manifolds. Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. One can assign to each of these vector spaces an orientation. However, one would like to know whether or not one can choose the orientations so that they "vary smoothly" from point to point. One may not be able do this, there are certain topological restrictions. A manifold which admits a smooth choice of orientations for its tangents spaces is said to be orientable. See the article on orientability for more on orientations of manifolds. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
This article or section should be merged with Orientable manifold. ...
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