Oriented vector space

An oriented vector space $(𝑉, 𝕂, s g n)$ is a finite-dimensional vector space $(𝑉, 𝕂)$ with a fixed choice of which ordered bases are positively oriented. linalg

In terms of oriented bases

Two ordered bases $B = (⃗ 𝐞 𝑖) 𝑛 𝑖 = 1$ and $B' = (⃗ 𝐞' 𝑖) 𝑛 𝑖 = 1$ of a vector space $𝑉$ have the same orientation iff the unique linear automorphism $𝐴 : B \mapsto B'$ has a positive determinant. This divides the possible bases of $𝑉$ into two equivalence classes. An orientation of $𝑉$ 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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Oriented vector space

In terms of oriented bases

In terms of a top form

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