PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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The basic wedge map $(v_{1}, \dots, v_{k}) \mapsto v_{1} \land \dots \land v_{k}$ is multilinear and alternating

Statement

For every $k \geq 0$ , the map

$(v_{1}, \dots, v_{k}) ⟼ v_{1} \land \dots \land v_{k}$

from $V^{k}$ to $Λ^{k} V$ is $k$ -linear and alternating, in the sense of Alternating $k$ -linear maps.

Facts & Assumptions

Given: A vector space $V$ over a field $F$ and $k \geq 0$ .

[L1]

The exterior power is the quotient $Λ^{k} V = V^{\otimes k} / W_{k}$ , and the basic wedge map is the universal multilinear map composed with the quotient projection (The $k$ th exterior power as the tensor-power quotient by repeated-vector relations).

[L2]

The wedge map represents alternating $k$ -linear maps and is itself $k$ -linear and alternating (Exterior powers represent alternating multilinear maps and are unique up to unique isomorphism).

Proof

technique · direct

1.1

By [L1], the wedge of Decomposable $k$ -vectors and the basic wedge product is obtained from the universal multilinear map by a linear quotient map, so it is linear in each of its $k$ arguments.

1.2

By [L1], the quotient projection sends every pure tensor with a repeated pair to zero, so the wedge vanishes whenever two arguments are equal.

2.1

Steps 1.1 and 1.2 are exactly multilinearity and alternation; they match the structure map described in [L2], whose universal property supplies the same statements.

step 1.1step 1.2L2∎

Depends on

Used by

Exterior multiplication is well defined, graded, associative, unital, and graded-commutative Theorem

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

Keith Conrad, Exterior Powers (standard reference, not scraped)

The basic wedge map (v1,…,vk)↦v1∧⋯∧vk(v1​,…,vk​)↦v1​∧⋯∧vk​ is multilinear and alternating