View source for Orientation (vector space)

{{short description|Choice of reference for distinguishing an object and its mirror image}}
{{see also|Orientation (geometry)|Orientability}}
[[File:Cartesian coordinate system handedness.svg|thumb|The left-handed orientation is shown on the left, and the right-handed on the right.]]

In [[mathematics]], '''orientation''' is a geometric notion that in two dimensions allows one to say when a [[Cycle (graph)|cycle]] goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.  In [[linear algebra]], the notion of orientation makes sense in arbitrary finite dimension.  In this setting, the orientation of an [[ordered basis]] is a kind of asymmetry that makes a [[reflection (mathematics)|reflection]] impossible to replicate by means of a simple [[rotation (mathematics)|rotation]].  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 rotation alone, but it is possible to do so by reflecting the figure in a mirror.  As a result, in the three-dimensional [[Euclidean space]], the two possible basis orientations are called [[right hand rule|right-handed]] and left-handed (or right-chiral and left-chiral).

The orientation on a [[real number|real]] [[vector space]] is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional [[Euclidean space]], 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 with an orientation selected is called an '''oriented''' vector space, while one not having an orientation selected, is called '''{{visible anchor|unoriented}}'''.

==Definition==
Let ''V'' be a [[finite-dimensional]] real vector space and let ''b''<sub>1</sub> and ''b''<sub>2</sub> 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 ''b''<sub>1</sub> to ''b''<sub>2</sub>. The bases ''b''<sub>1</sub> and ''b''<sub>2</sub> 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''. If ''V'' is non-zero, there are precisely two [[equivalence class]]es determined by this relation. An '''orientation''' on ''V'' is an assignment of +1 to one equivalence class and &minus;1 to the other.<ref>{{Cite web|url=http://mathworld.wolfram.com/VectorSpaceOrientation.html|title=Vector Space Orientation|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2017-12-08}}</ref>

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 '''R'''<sup>''n''</sup> provides a '''standard orientation''' on '''R'''<sup>''n''</sup> (in turn, the orientation of the standard basis depends on the orientation of the [[Cartesian coordinate system]] on which it is built). Any choice of a linear [[isomorphism]] between ''V'' and '''R'''<sup>''n''</sup> 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]]. They will have the same/opposite orientations according to whether the [[signature (permutation)|signature]] of this permutation is ±1. This is because the determinant of a [[permutation matrix]] is equal to the signature of the associated permutation.

Similarly, let ''A'' be a nonsingular linear mapping of vector space '''R'''<sup>''n''</sup> to '''R'''<sup>''n''</sup>. This mapping is '''orientation-preserving''' if its determinant is positive.<ref>{{Cite web|url=http://mathworld.wolfram.com/Orientation-Preserving.html|title=Orientation-Preserving|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2017-12-08}}</ref> For instance, in '''R'''<sup>3</sup> a rotation around the ''Z'' Cartesian axis by an angle ''α'' is orientation-preserving:
::<math>
\mathbf {A}_1 = \begin{pmatrix}
  \cos \alpha & -\sin \alpha & 0 \\
  \sin \alpha & \cos \alpha & 0 \\
  0 & 0 & 1  
\end{pmatrix}
</math>

while a reflection by the ''XY'' Cartesian plane is not orientation-preserving:
::<math>
\mathbf {A}_2 = \begin{pmatrix}
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  0 & 0 & -1  
\end{pmatrix}
</math>

===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 <math>\emptyset</math>. Therefore, there is a single equivalence class of ordered bases, namely, the class <math>\{\emptyset\}</math> whose sole member is the empty set. This means that an orientation of a zero-dimensional space is a function
:<math>\{\{\emptyset\}\} \to \{\pm 1\}.</math>
It is therefore possible to orient a point in two different ways, positive and negative.

Because there is only a single ordered basis <math>\emptyset</math>, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing <math>\{\emptyset\} \mapsto +1</math> or <math>\{\emptyset\} \mapsto -1</math> 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]] as an instance of [[Stokes' theorem]].  A closed interval {{math|[''a'', ''b'']}} is a one-dimensional [[manifold with boundary]], and its boundary is the set {{math|{''a'', ''b''}}}.  In order to get the correct statement of the fundamental theorem of calculus, the point {{math|''b''}} should be oriented positively, while the point {{math|''a''}} 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 (geometry)|line]] just as there are two orientations to a circle. In the case of a [[line segment]] (a connected subset of a line), the two possible orientations result in [[line segment#Directed line segment|directed line segments]]. An [[orientability|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]] of ''V'', denoted Λ<sup>''k''</sup>''V''. This is a real vector space of dimension [[binomial coefficient|<math>\tbinom{n}{k}</math>]]. The vector space Λ<sup>''n''</sup>''V'' (called the ''top exterior power'') therefore has dimension 1. That is, Λ<sup>''n''</sup>''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]] ω on Λ<sup>''n''</sup>''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 {''e''<sub>''i''</sub>} is a privileged basis for ''V'' and {''e''<sub>''i''</sub><sup>∗</sup>} is the [[dual basis]], then the orientation form giving the standard orientation is {{nowrap|''e''<sub>1</sub><sup>∗</sup> ∧ ''e''<sub>2</sub><sup>∗</sup> ∧ … ∧ ''e''<sub>''n''</sub><sup>∗</sup>}}.

The connection of this with the determinant point of view is: the determinant of an [[endomorphism]] <math>T\colon V \to V</math> 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]] GL(''V'') [[Group action (mathematics)|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 space|connected]], but rather has two [[connected space|connected component]]s according to whether the determinant of the transformation is positive or negative (except for GL<sub>0</sub>, 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]] of GL(''V'') is denoted GL<sup>+</sup>(''V'') and consists of those transformations with positive determinant. The action of GL<sup>+</sup>(''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: <math>\pi_0(\operatorname{GL}(V)) = (\operatorname{GL}(V)/\operatorname{GL}^+(V) = \{\pm 1\}</math>,
and the [[Stiefel manifold]] of ''n''-frames in <math>V</math> is a <math>\operatorname{GL}(V)</math>-[[torsor]], so <math>V_n(V)/\operatorname{GL}^+(V)</math> is a [[torsor]] over <math>\{\pm 1\}</math>, i.e., its 2 points, and a choice of one of them is an orientation.

===Geometric algebra===
[[File:Wedge product.JPG|thumb|left|150px|Parallel plane segments with the same attitude, magnitude and orientation, all corresponding to the same bivector {{nowrap|'''a''' ∧ '''b'''}}.<ref name=Dorst>
{{cite book |author1=Leo Dorst |author2=Daniel Fontijne |author3=Stephen Mann |title=Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry |url=https://books.google.com/books?id=-1-zRTeCXwgC&pg=PA32#v=onepage&q=&f=false |page=32 |isbn=0-12-374942-5 |publisher=Morgan Kaufmann |year=2009 |edition=2nd| quote=The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.}}</ref>]]

The various objects of [[geometric algebra]] are charged with three attributes or ''features'': attitude, orientation, and magnitude.<ref name=Jancewicz1>

{{cite book |author=B Jancewicz |url=https://books.google.com/books?id=0Nji78YQKfQC&pg=PA403&dq=attitude+%22Table+28.1%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=5#v=onepage&q=attitude%20%22Table%2028.1%22&f=false |page=397 |editor=William Eric Baylis |chapter=Tables 28.1 & 28.2 in section 28.3: '' Forms and pseudoforms'' |isbn=0-8176-3868-7 |year=1996 |publisher=Springer |title=Clifford (geometric) algebras with applications to physics, mathematics, and engineering}}

</ref> For example, a [[Euclidean vector|vector]] 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 three dimensions has an attitude given by the family of [[Plane (geometry)|plane]]s associated with it (possibly specified by the [[Tangential and normal components|normal line]] common to these planes <ref name=Granville>
{{cite book |author=William Anthony Granville |title =Elements of the differential and integral calculus |page=275 |url=https://books.google.com/books?id=0jcAAAAAYAAJ&pg=PA275 |chapter=§178 Normal line to a surface |publisher=Ginn & Company |year=1904}}

</ref>), 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.<ref name=Hestenes>
{{cite book |title=New foundations for classical mechanics: Fundamental Theories of Physics |author=David Hestenes |url=https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA21 |page=21 |isbn=0-7923-5302-1 |edition=2nd |year=1999 |publisher=Springer| authorlink = David Hestenes}}
</ref>

==Orientation on manifolds==
{{Main|Orientability}}
[[File:Surface orientation.pdf|thumb|The orientation of a volume may be determined by the orientation on its boundary, indicated by the circulating arrows.]]

Each point ''p'' on an ''n''-dimensional differentiable [[manifold]] has a [[tangent space]] ''T''<sub>''p''</sub>''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 [[topology|topological]] restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be ''[[Orientability|orientable]]''.

==See also==
*[[Sign convention]]
*[[Rotation formalisms in three dimensions]]
*[[Chirality (mathematics)]]
*[[Right-hand rule]]
*[[Even and odd permutations]]
*[[Cartesian coordinate system]]
*[[Pseudovector]] &mdash; Pseudovectors are a consequence of oriented spaces.
*[[Orientability]] &mdash; Discussion about the possibility of having orientations in a space.
*[[Orientation of a vector bundle]]

==References==
{{Reflist}}

==External links==
* {{springer|title=Orientation|id=p/o070200}}

{{DEFAULTSORT:Orientation (vector space)}}
[[Category:Linear algebra]]
[[Category:Analytic geometry]]
[[Category:Orientation (geometry)]]