PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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Orientations and positive basis classes agree in positive dimension

Statement

For $\dim V = n > 0$ , determinant-line rays are in bijection with positive-basis equivalence classes. For $n = 0$ , the unique empty basis sees only the positive ray.

Facts & Assumptions

Given: A finite-dimensional real vector space $V$ of dimension $n$ .

[L1]

An orientation of $V$ is a positive ray in $\det V = Λ^{n} V$ , including either ray of $R = Λ^{0} V$ when $n = 0$ (Determinant-line orientations of finite-dimensional real vector spaces).

Proof

technique · direct

1.1

The wedge of an ordered basis is nonzero in $\det V$ , and changing basis multiplies it by the determinant of the change-of-basis matrix.

givenL1algebra

2.1

Thus two bases determine the same ray exactly when their change determinant is positive. When $n = 0$ , the unique empty wedge is $+ 1$ , so it represents only the positive one of the two rays in [L1].

L1step 1.1∎

Depends on

Determinant-line orientations of finite-dimensional real vector spaces

Used by

Positive oriented atlases characterize orientations except for one-manifolds with boundary Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)