ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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Determinant as the pairing of top exterior powers

Example

On $R^{n}$ with standard basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ , the pairing

$⟨ e^{1} \land \dots \land e^{n}, v_{1} \land \dots \land v_{n} ⟩$

is exactly $\det [v_{1} \dots v_{n}]$ .

Facts & Assumptions

Given: Vectors $v_{1}, \dots, v_{n} \in R^{n}$ .

[L1]

The exterior-power pairing on decomposable elements is the determinant of the evaluation matrix (Exterior-power duality pairing).

[L2]

The top exterior power is one-dimensional (The top exterior power is one-dimensional).

Verification

technique · direct

1.1given

The matrix with entries $e^{i} (v_{j})$ is exactly the coordinate matrix $[v_{1} \dots v_{n}]$ of the ordered $n$ -tuple $(v_{1}, \dots, v_{n})$ .

2.1L1step 1.1algebra

Applying [L1] to $α_{i} = e^{i}$ gives $⟨ e^{1} \land \dots \land e^{n}, v_{1} \land \dots \land v_{n} ⟩ = \det (e^{i} (v_{j})) = \det [v_{1} \dots v_{n}] .$

3.1L2step 2.1

By [L2], this scalar determines the decomposable top wedge relative to the standard volume form.

4.1step 2.1step 3.1∎

Thus the determinant is the top-degree pairing against $e^{1} \land \dots \land e^{n}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)