Orientation (vector space)

See also: orientation (geometry)

The left-handed orientation is shown on the left, and the right-handed on the right.

In linear algebra, the orientation (also called handedness or chirality) of an ordered basis is a kind of asymmetry that makes a reflexion impossible to replicate by means of a simple rotation. All bases are asymmetric, and they have two possible orientations, as well as the right and left hand of the human body. In the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral).

The orientation on a 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 is called an oriented vector space, while one without a choice of orientation is called unoriented.

Definition

Let V be a finite-dimensional real vector space and 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. 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.

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 provides a standard orientation on Rⁿ (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 Rⁿ 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 Rⁿ to Rⁿ. 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:

${\mathbf {A}}_{1}={\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}$

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

${\mathbf {A}}_{2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}$

Non-numerical information

It is impossible to define the orientation on a vector space without referring to the orientation of a basis, and hence to the orientation of a coordinate system. In turn, it is impossible to unequivocally describe the orientation of a coordinate system only with numbers. For instance, the definition of a right-handed Cartesian coordinate system must be based on non-numerical information, such as the shape of the right hand of a human being, or the combined concepts of down-up, South-North, and West-East, or down-up, left-right, and backward-forward. Similarly, it is impossible to explain which is the right hand or side of a human being without either showing it, or referring to other non-numerical knowledge, such as the definition of a Cartesian coordinate system. This is also true for the definition of the clockwise and counterclockwise senses of rotation, which cannot be only based on numbers, as it is related to the definition of left-handed and right-handed coordinate systems (in a right-handed coordinate system xyz, counterclockwise is the sense of the shortest rotation from axis x to y, as seen from a point in the positive z semiaxis).

Zero-dimensional case

The concept of orientation defined above did not quite apply to zero-dimensional vector spaces (as the only empty matrix is the identity (with determinant 1), so there will be only one equivalence class). However, it is useful to be able to assign different orientations to a point (e.g. orienting the boundary of a 1-dimensional manifold). A more general definition of orientation that works regardless of dimension is the following: An orientation on V is a map from the set of ordered bases of V to the set $\{\pm 1\}$ that is invariant under base changes with positive determinant and changes sign under base changes with negative determinant (it is equivarient with respect to the homomorphism $GL_{n}\to \pm 1$). The set of ordered bases of the zero-dimensional vector space has one element (the empty set), and so there are two maps from this set to $\{\pm 1\}$.

A subtle point is that a zero-dimensional vector space is naturally (canonically) oriented, so we can talk about an orientation being positive (agreeing with the canonical orientation) or negative (disagreeing). An application is interpreting the Fundamental theorem of calculus as a special case of Stokes' theorem.

Two ways of seeing this are:

• A zero-dimensional vector space is a point, and there is a unique map from a point to a point, so every zero-dimensional vector space is naturally identified with $R^{0}$, and thus is oriented.
• The 0th exterior power of a vector space is the ground field $K$, which here is $R^{1}$, which has an orientation (given by the standard basis).

Alternate viewpoints

Multilinear algebra

For any n-dimensional 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. 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 ΛⁿV 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 e₁^*∧e₂^*∧…∧e_n^*.

The connection of this with the determinant point of view is: the determinant of an endomorphism $T\colon V\to V$ 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 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 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 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: $\pi _{0}(GL(V))=(GL(V)/GL^{+}(V)=\{\pm 1\}$, and the Stiefel manifold of n-frames in $V$ is a $GL(V)$-torsor, so $V_{n}(V)/GL^{+}(V)$ is a torsor over $\{\pm 1\}$, i.e., it's 2 points, and a choice of one of them is an orientation.

Orientation on manifolds

Main article: Orientability

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 it is possible to choose the orientations so that they "vary smoothly" from point to point. Due to certain topological restrictions, there are situations when this is impossible. 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

• Rotation representation (mathematics)
• Chirality (mathematics)
• Right-hand rule
• Even and odd permutations
• Cartesian coordinate system
• Attitude (geometry)