Orientation (vector space)

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.

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.

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 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 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.

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.

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.

[2] For instance, in R3 a rotation around the Z Cartesian axis by an angle α is orientation-preserving:

The concept of orientation degenerates in the zero-dimensional case.

Consequently, the only basis of a zero-dimensional vector space 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.

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.

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.

In real coordinate space, an oriented line is also known as an axis.

A semi-infinite oriented line is called a ray.

For any n-dimensional real vector space V we can form the kth-exterior power of V, denoted ΛkV.

The vector space ΛnV (called the top exterior power) therefore has dimension 1.

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).

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.

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.

Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive.

The various objects of geometric algebra are charged with three attributes or features: attitude, orientation, and magnitude.

[5] 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 [6]), 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.