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, 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 **unoriented**.

- Definition
- Zero-dimensional case
- On a line
- Alternate viewpoints
- Multilinear algebra
- Lie group theory
- Geometric algebra
- Orientation on manifolds
- See also
- References
- External links

In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. 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, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral).

Let *V* be a finite-dimensional real vector space and let *b*_{1} and *b*_{2} 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*_{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; 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 classes determined by this relation. An **orientation** on *V* is an assignment of +1 to one equivalence class and −1 to the other.^{ [1] }

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**^{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 on which it is built). Any choice of a linear 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. 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.

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.^{ [2] } For instance, in **R**^{3} a rotation around the *Z* Cartesian axis by an angle *α* is orientation-preserving:

while a reflection by the *XY* Cartesian plane is not orientation-preserving:

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 . 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

It is therefore possible to orient a point in two different ways, positive and negative.

Because there is only a single ordered basis , a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing or 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 [*a*, *b*] is a one-dimensional manifold with boundary, and its boundary is the set {*a*, *b*}. In order to get the correct statement of the fundamental theorem of calculus, the point *b* should be oriented positively, while the point *a* should be oriented negatively.

The one-dimensional case deals with a line which may be traversed in one of two directions. There are two orientations to a 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 directed line segments. An orientable surface sometimes has the selected orientation indicated by the orientation of a line perpendicular to the surface.

For any *n*-dimensional real vector space *V* we can form the *k*th-exterior power of *V*, denoted Λ^{k}*V*. This is a real vector space of dimension . 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 *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 {*e*_{i}} is a privileged basis for *V* and {*e*_{i}^{∗}} is the dual basis, then the orientation form giving the standard orientation is *e*_{1}^{∗} ∧ *e*_{2}^{∗} ∧ … ∧ *e*_{n}^{∗}.

The connection of this with the determinant point of view is: the determinant of an endomorphism can be interpreted as the induced action on the top exterior power.

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 (except for GL_{0}, 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^{+}(*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: , and the Stiefel manifold of *n*-frames in is a -torsor, so is a torsor over , i.e., its 2 points, and a choice of one of them is an orientation.

The various objects of geometric algebra are charged with three attributes or *features*: attitude, orientation, and magnitude.^{ [4] } For example, a 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 planes associated with it (possibly specified by the normal line common to these planes ^{ [5] }), 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.^{ [6] }

Each point *p* on an *n*-dimensional differentiable manifold has a tangent space *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 restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be * orientable *.

In mathematics, the **determinant** is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants . The determinant of a matrix *A* is denoted det(*A*), det *A*, or |*A*|.

In mathematics, a set B of vectors in a vector space *V* is called a **basis** if every element of *V* may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as **components** or **coordinates** of the vector with respect to B. The elements of a basis are called **basis vectors**.

**Linear algebra** is the branch of mathematics concerning linear equations such as:

In linear algebra, the **rank** of a matrix A is the dimension of the vector space generated by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.

In mathematics, the **dimension** of a vector space *V* is the cardinality of a basis of *V* over its base field. It is sometimes called **Hamel dimension** or **algebraic dimension** to distinguish it from other types of dimension.

In linear algebra, the **trace** of a square matrix **A**, denoted tr(**A**), is defined to be the sum of elements on the main diagonal of **A**.

In algebra, the **kernel** of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the *null space*, is the kernel of the linear map defined by the matrix.

In mathematics, the **general linear group** of degree *n* is the set of *n*×*n* invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

In mathematics, the **special linear group**SL(*n*, *F*) of degree *n* over a field *F* is the set of *n* × *n* matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, a **symplectic matrix** is a matrix with real entries that satisfies the condition

In mathematics, **orientability** is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is **orientable** if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an **orientation** of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is **non-orientable** if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed in its own mirror image . A Möbius strip is an example of a non-orientable space.

In mathematics, the **exterior product** or **wedge product** of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by , is called a bivector and lives in a space called the *exterior square*, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and , which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and , but, unlike the cross product, the exterior product is associative.

In mathematics, a **volume form** or **top-dimensional form** is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension , a volume form is an -form. It is a section of the line bundle A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

In mathematics, and specifically differential geometry, a **density** is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the **density bundle**. An element of the density bundle at *x* is a function that assigns a volume for the parallelotope spanned by the *n* given tangent vectors at *x*.

In projective geometry and linear algebra, the **projective orthogonal group** PO is the induced action of the orthogonal group of a quadratic space *V* = (*V*,*Q*) on the associated projective space P(*V*). Explicitly, the projective orthogonal group is the quotient group

In mathematics, the **sign** of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative, or it may be considered both positive and negative. Whenever not specifically mentioned, this article adheres to the first convention.

In universal algebra, a **basis** is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner. It also represents the endomorphisms of an algebra by certain indexings of algebra elements, which can correspond to the usual matrices when the free algebra is a vector space.

An **oriented matroid** is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary matroid abstracts the dependence properties that are common both to graphs, which are not necessarily *directed*, and to arrangements of vectors over fields, which are not necessarily *ordered*.

This is a **glossary of representation theory** in mathematics.

- ↑ W., Weisstein, Eric. "Vector Space Orientation".
*mathworld.wolfram.com*. Retrieved 2017-12-08. - ↑ W., Weisstein, Eric. "Orientation-Preserving".
*mathworld.wolfram.com*. Retrieved 2017-12-08. - ↑ Leo Dorst; Daniel Fontijne; Stephen Mann (2009).
*Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry*(2nd ed.). Morgan Kaufmann. p. 32. ISBN 978-0-12-374942-0.The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.

- ↑ B Jancewicz (1996). "Tables 28.1 & 28.2 in section 28.3:
*Forms and pseudoforms*". In William Eric Baylis (ed.).*Clifford (geometric) algebras with applications to physics, mathematics, and engineering*. Springer. p. 397. ISBN 0-8176-3868-7. - ↑ William Anthony Granville (1904). "§178 Normal line to a surface".
*Elements of the differential and integral calculus*. Ginn & Company. p. 275. - ↑ David Hestenes (1999).
*New foundations for classical mechanics: Fundamental Theories of Physics*(2nd ed.). Springer. p. 21. ISBN 0-7923-5302-1.

- "Orientation",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

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