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).

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.

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.

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.

Alternate viewpoints

We present two alternate (and more abstract) ways of understanding orientations:

1. For any real vector space V we can form the k^th-exterior power of V, denoted Λ^kV. 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 {e_i} is a privileged basis for V then the orientation form giving the standard orientation is e₁∧e₂∧…∧e_n.

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.

Orientation on manifolds

One can also discuss orientation on manifolds. Each point p on an n-dimensional differentiable manifold has a tangent space T_{pM 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.}

See also

• Orientability
• Chirality (mathematics)
• Even and odd permutations
• Handedness
• Cartesian coordinate system