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# Vector Space Orientation

An ordered vector basis for a finite-dimensional vector space defines an orientation. Another basis gives the same orientation if the matrix has a positive determinant, in which case the basis is called oriented.

Any vector space has two possible orientations since the determinant of an nonsingular matrix is either positive or negative. For example, in , is one orientation and is the other orientation. In three dimensions, the cross product uses the right-hand rule by convention, reflecting the use of the canonical orientation as .

An orientation can be given by a nonzero element in the top exterior power of , i.e., . For example, gives the canonical orientation on and gives the other orientation.

Some special vector space structures imply an orientation. For example, if is a symplectic form on , of dimension , then gives an orientation. Also, if is a complex vector space, then as a real vector space of dimension , the complex structure gives an orientation.

Bundle Orientation, Manifold Orientation

This entry contributed by Todd Rowland

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Rowland, Todd. "Vector Space Orientation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VectorSpaceOrientation.html