Векторлық кеңістіктің бағдарлануы және оның айналасы

Choice of reference for distinguishing an object and its mirror image

The orientation of a real vector space or simply orientation of a 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 selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. 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).

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 Therefore, there is a single equivalence class of ordered bases, namely, the class whose sole member 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. Because there is only a single ordered basis , a zero dimensional vector space is the same as a zero dimensional vector space with ordered basis. Choosing or 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.

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.

For any n dimensional real vector space V we can form the kth exterior power of V, denoted ΛkV. This is a real vector space of dimension 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 {ei} is a privileged basis for V and {ei∗} is the dual basis, then the orientation form giving the standard orientation is e1∗ ∧ e2∗ ∧ ∧ en∗. The connection of this with the determinant point of view is: the determinant of an endomorphism can be interpreted as the induced action on the top exterior power.

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

Each point p on an n dimensional differentiable manifold has a tangent space TpM which is an n dimensional real vector space. Each of these vector spaces can be assigned an orientation. Some orientations "vary smoothly" from point to point. Due to certain topological restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be orientable.