PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The top exterior power is one-dimensional

Statement

If $V$ is a finite-dimensional real vector space with $\dim V = n$ , then $⋀^{n} V$ and $⋀^{n} V^{*}$ are one-dimensional.

Facts & Assumptions

Given: A finite-dimensional real vector space $V$ with $\dim V = n$ .

[L1]

The dimension formula gives $\dim {Alt}^{n} (V) = (\binom{n}{n}) = 1$ (Dimension of the $k$ th exterior power is binomial).

[L2]

The pairing between $⋀^{n} V^{*}$ and $⋀^{n} V$ is nondegenerate (Exterior-power duality pairing).

Proof

technique · direct

1.1L1given

By [L1], the space $⋀^{n} V^{*} = {Alt}^{n} (V)$ is one-dimensional.

2.1L2step 1.1algebra

Since $⋀^{n} V$ is the dual of $⋀^{n} V^{*}$ by definition, or equivalently because [L2] identifies it with the dual through a nondegenerate pairing, it has the same dimension. Hence $\dim ⋀^{n} V = 1$ .

3.1step 1.1step 2.1∎

Therefore both top exterior powers are one-dimensional.

Depends on

Used by

A signed one-density on a real vector space Definition
Determinant-line orientations of finite-dimensional real vector spaces Definition
Determinant as the pairing of top exterior powers Example

Dependency tree · two levels

8 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

Will J. Merry, Differential Geometry (standard reference, not scraped)