Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The wedge product is alternating and bilinear

Statement

If α∈Alt⁡k(V) and β∈Alt⁡ℓ(V), then α∧β is alternating of degree k+ℓ, and the wedge product is bilinear in (α,β).

Facts & Assumptions

Given: Alternating covectors α,α′∈Alt⁡k(V), β,β′∈Alt⁡ℓ(V), and scalars a,b.

[F1]

The wedge product is the normalized alternation of the tensor product, equivalently the signed shuffle sum (The wedge product of alternating covectors).

Proof

technique · direct
1.1F1given

By [F1], α∧β is obtained by applying the alternation operator to α⊗β. Alternation produces an alternating multilinear form, so α∧β∈Alt⁡k+ℓ(V).

1.2F1givenalgebra

Both tensor product and alternation are linear in each argument, so [F1] gives (aα+bα′)∧β=a(α∧β)+b(α′∧β), and similarly in the second slot.

2.1step 1.1step 1.2∎

Therefore the wedge product is alternating and bilinear.

Depends on

Used by

Dependency tree · two levels

4 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