Orientation (vector space)
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered
In
Definition
Let V be a
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
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:
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 . 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
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
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
Alternate viewpoints
Multilinear algebra
For any n-dimensional real vector space V we can form the kth-
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.
Lie group theory
Let B be the set of all ordered bases for V. Then the
More formally: , and the Stiefel manifold of n-frames in is a -
Geometric algebra
The various objects of
Orientation on manifolds
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.
See also
- Sign convention
- Rotation formalisms in three dimensions
- Chirality (mathematics)
- Right-hand rule
- Even and odd permutations
- Cartesian coordinate system
- Pseudovector
- Orientation of a vector bundle
References
- ^ W., Weisstein, Eric. "Vector Space Orientation". mathworld.wolfram.com. Retrieved 2017-12-08.
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: CS1 maint: multiple names: authors list (link) - ^ W., Weisstein, Eric. "Orientation-Preserving". mathworld.wolfram.com. Retrieved 2017-12-08.
{{cite web}}
: CS1 maint: multiple names: authors list (link) - ^
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 978-0-12-374942-0.
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 (ed.). 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.
- ^
ISBN 0-7923-5302-1.
External links
- "Orientation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]