Orientation (vector space)

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.^[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 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.^[2] 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}}

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 $\operatorname {GL}_{n}\to \pm 1$ ). 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 R⁰, and thus is oriented.
The 0th exterior power of a vector space is the ground field $K$ , which here is R¹, which has an orientation (given by the standard basis).

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 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.

Alternate viewpoints

Multilinear algebra

For any n-dimensional real vector space V we can form the kth-exterior power of V, denoted Λ^kV. This is a real vector space of dimension ${\tbinom {n}{k}}$ . 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: $\pi _{0}(\operatorname {GL}(V))=(\operatorname {GL}(V)/\operatorname {GL}^{+}(V)=\{\pm 1\}$ , and the Stiefel manifold of n-frames in $V$ is a $\operatorname {GL} (V)$ -torsor, so $V_{n}(V)/\operatorname {GL}^{+}(V)$ is a torsor over $\{\pm 1\}$ , i.e., its 2 points, and a choice of one of them is an orientation.

Geometric algebra

Parallel plane segments with the same attitude, magnitude and orientation, all corresponding to the same bivector a ∧ b.^[3]

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]

Orientation on manifolds

Main article: Orientability

The orientation of a volume may be determined by the orientation on its boundary, indicated by the circulating arrows.

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.

References

↑ Rowland, Todd. "Vector Space Orientation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/VectorSpaceOrientation.html
↑ Weisstein, Eric W. "Orientation-Preserving." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Orientation-Preserving.html
↑ 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 0-12-374942-5. 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. 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.

External links

Hazewinkel, Michiel, ed. (2001), "Orientation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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