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The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called . In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
orientability is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
is a broader notion that, in two dimensions, allows one to say when a
cycle Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
over the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
impossible to replicate by means of a simple
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a displacement alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, the two possible basis orientations are called
right-handed In human biology, handedness is the better, faster, or more precise performance or individual preference for use of a hand, known as the dominant hand. The incapable, less capable or less preferred hand is called the non-dominant hand. Right-ha ...

right-handed
and left-handed (or right-chiral and left-chiral).


Definition

Let ''V'' be a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
real vector space and let ''b''1 and ''b''2 be two ordered bases for ''V''. It is a standard result in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
that there exists a unique
linear transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
''A'' : ''V'' → ''V'' that takes ''b''1 to ''b''2. The bases ''b''1 and ''b''2 are said to have the ''same orientation'' (or be consistently oriented) if ''A'' has positive
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

determinant
; otherwise they have ''opposite orientations''. The property of having the same orientation defines an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on the set of all ordered bases for ''V''. If ''V'' is non-zero, there are precisely two
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es determined by this relation. An orientation on ''V'' is an assignment of +1 to one equivalence class and −1 to the other. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on R''n'' provides a standard orientation on R''n'' (in turn, the orientation of the standard basis depends on the orientation of the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
on which it is built). Any choice of a linear
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
between ''V'' and R''n'' will then provide an orientation on ''V''. The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
. They will have the same/opposite orientations according to whether the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...
of this permutation is ±1. This is because the determinant of a
permutation matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

permutation matrix
is equal to the signature of the associated permutation. Similarly, let ''A'' be a nonsingular linear mapping of vector space R''n'' to R''n''. This mapping is orientation-preserving if its determinant is positive. For instance, in R3 a rotation around the ''Z'' Cartesian axis by an angle ''α'' is orientation-preserving: \mathbf _1 = \begin \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end while a reflection by the ''XY'' Cartesian plane is not orientation-preserving: \mathbf _2 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end


Zero-dimensional case

The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set \emptyset. Therefore, there is a single equivalence class of ordered bases, namely, the class \ whose sole member is the empty set. This means that an orientation of a zero-dimensional space is a function \ \to \. It is therefore possible to orient a point in two different ways, positive and negative. Because there is only a single ordered basis \emptyset, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing \ \mapsto +1 or \ \mapsto -1 therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
as an instance of
Stokes' theorem Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
. A closed interval is a one-dimensional
manifold with boundary The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
, and its boundary is the set . In order to get the correct statement of the fundamental theorem of calculus, the point should be oriented positively, while the point should be oriented negatively.


On a line

The one-dimensional case deals with a line which may be traversed in one of two directions. There are two orientations to a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
just as there are two orientations to a circle. In the case of a
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

line segment
(a connected subset of a line), the two possible orientations result in directed line segments. An orientable surface sometimes has the selected orientation indicated by the orientation of a line perpendicular to the surface.


Alternate viewpoints


Multilinear algebra

For any ''n''-dimensional real vector space ''V'' we can form the ''k''th-
exterior power In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
of ''V'', denoted Λ''k''''V''. This is a real vector space of dimension \tbinom. The vector space Λ''n''''V'' (called the ''top exterior power'') therefore has dimension 1. That is, Λ''n''''V'' is just a real line. There is no ''a priori'' choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero
linear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear form
''ω'' on Λ''n''''V'' determines an orientation of ''V'' by declaring that ''x'' is in the positive direction when ''ω''(''x'') > 0. 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 an ''n''-form we can evaluate it on an ordered set of ''n'' vectors, giving an element of R). The form ''ω'' is called an orientation form. If is a privileged basis for ''V'' and is the
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
, then the orientation form giving the standard orientation is . The connection of this with the determinant point of view is: the determinant of an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
T : V \to V can be interpreted as the induced action on the top exterior power.


Lie group theory

Let ''B'' be the set of all ordered bases for ''V''. Then the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
GL(''V'')
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament The New Te ...
freely and transitively on ''B''. (In fancy language, ''B'' is a GL(''V'')-
torsor In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), 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 gr ...
). This means that as a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
, ''B'' is (noncanonically)
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
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 (except for GL0, which is the trivial group and thus has a single connected component; this corresponds to the canonical orientation on a zero-dimensional vector space). The
identity component In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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. More formally: \pi_0(\operatorname(V)) = (\operatorname(V)/\operatorname^+(V) = \, and the
Stiefel manifoldIn mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal k-frame, ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vector (mathematics), vectors in \R^n. It is named after Swiss mathematician Edu ...
of ''n''-frames in V is a \operatorname(V)-
torsor In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), 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 gr ...
, so V_n(V)/\operatorname^+(V) is a
torsor In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), 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 gr ...
over \, i.e., its 2 points, and a choice of one of them is an orientation.


Geometric algebra

The various objects of
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
are charged with three attributes or ''features'': attitude, orientation, and magnitude. For example, a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a
bivector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

bivector
in three dimensions has an attitude given by the family of
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
s associated with it (possibly specified by the normal line common to these planes ), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its ''circulation''), and a magnitude given by the area of the parallelogram defined by its two vectors.


Orientation on manifolds

Each point ''p'' on an ''n''-dimensional differentiable
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
has a
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
''T''''p''''M'' which is an ''n''-dimensional real vector space. Each of these vector spaces can be assigned an orientation. Some orientations "vary smoothly" from point to point. Due to certain
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topological
restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be ''
orientable is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
''.


See also

*
Sign conventionIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...
*
Rotation formalisms in three dimensions In geometry, various formalisms exist to express a rotation (mathematics), rotation in three dimension (vector space), dimensions as a mathematical transformation (geometry), transformation. In physics, this concept is applied to classical mechanic ...
*
Chirality (mathematics) In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
*
Right-hand rule In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

Right-hand rule
* Even and odd permutations *
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
*
Pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

Pseudovector
*
Orientation of a vector bundleIn mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: ''E'' →''B'', an orientation of ''E'' means: for each fiber ''E'x'', there is an orientation of ...


References


External links

* {{DEFAULTSORT:Orientation (vector space) Linear algebra Analytic geometry Orientation (geometry)