DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

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The finite-dimensional exterior power of vectors

Definition

Let $V$ be a finite-dimensional real vector space and let $k \geq 0$ . The $k$ th exterior power of $V$ is the dual vector space

\overset{k}{⋀} V : = {Alt}^{k} (V)^{*} .

For $v_{1}, \dots, v_{k} \in V$ , the decomposable $k$ -vector $v_{1} \land \dots \land v_{k} \in ⋀^{k} V$ is the functional

ω \mapsto ω (v_{1}, \dots, v_{k}) .

Depends on

Alternating $k$-covectors

Used by

Exterior-power duality pairing Proposition
Universal property of the finite-dimensional exterior power Theorem

Dependency tree · two levels

2 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)