Orientation (mathematics)

:See also Orientation (rigid body).

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In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented (or right-handed) and which are "negatively" oriented (or left-handed).

Related Topics:
Mathematics - Real vector space - Bases

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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. 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.

Related Topics:
Linear algebra - Linear transformation - Determinant - Equivalence relation - Equivalence class

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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 gives rise to a standard orientation on Rn. Any choice of a linear isomorphism between V and Rn will then give rise to an orientation on V in an obvious manner.

Related Topics:
Standard basis - Isomorphism

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Note that the ordering of elements in a basis is crucial. Two basis 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.

Related Topics:
Permutation - Signature - Permutation matrix

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