Linear algebra MOC

Oriented vector space

An oriented vector space is a finite-dimensional vector space with a fixed choice of which ordered bases are positively oriented. linalg

In terms of oriented bases

Two ordered bases and of a vector space have the same orientation iff the unique linear automorphism has a positive determinant. This divides the possible bases of into two equivalence classes. An orientation of is thus a choice of which of these equivalence classes is positive and which is negative.

In terms of a top form

Tip

There is also a characterization in terms of the exterior algebra.

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