Orientation (vector space)

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).

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: $${\displaystyle {\mathbf{A}}_1=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)}$$ while a reflection by the XY Cartesian plane is not orientation-preserving: $${\displaystyle {\mathbf{A}}_2=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right)}$$

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

Because there is only a single ordered basis ${\displaystyle \mathrm{\varnothing }}$, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing ${\displaystyle \{\mathrm{\varnothing }\}\mapsto +1}$ or ${\displaystyle \{\mathrm{\varnothing }\}\mapsto -1}$ 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.

On a line

The one-dimensional case deals with an oriented line (or directed line), which may be traversed in one of two directions. In real coordinate space, an oriented line is also known as an axis. There are two orientations to a line (forward and backward) just as there are two orientations to an oriented circle (clockwise and anti-clockwise). A semi-infinite oriented line is called a ray (sometimes also a half-axis or an oriented half-line). In the case of a line segment (a connected subset of a line), the two possible orientations result in directed line segments.

On a surface

An orientable surface sometimes has the selected orientation indicated by the orientation of a surface normal. An oriented plane can be defined by a pseudovector.