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

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dimension of the $k$ th exterior power is binomial

Statement

If $V$ is a finite-dimensional real vector space with $\dim V = n$ and $k \in N$ , then

\dim {Alt}^{k} (V) = (\binom{n}{k}) .

In particular, ${Alt}^{k} (V) = 0$ for $k > n$ .

Facts & Assumptions

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

[L1]

The wedges $e^{i_{1}} \land \dots \land e^{i_{k}}$ with $i_{1} < \dots < i_{k}$ form a basis of ${Alt}^{k} (V)$ (Wedge monomials in a dual basis form a basis).

[F1]

The binomial coefficient $(\binom{n}{k})$ counts the $k$ -element subsets of an $n$ -element set (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

Proof

technique · direct

1.1

Choose a basis $e_{1}, \dots, e_{n}$ of $V$ . By [L1], ${Alt}^{k} (V)$ has one basis vector for each strictly increasing $k$ -tuple $(i_{1}, \dots, i_{k})$ from ${1, \dots, n}$ .

L1givenchoose

2.1

Such tuples are the same thing as $k$ -element subsets of an $n$ -element set, so [F1] counts them by $(\binom{n}{k})$ . Therefore $\dim {Alt}^{k} (V) = (\binom{n}{k})$ . If $k > n$ , there are no such tuples, so the basis is empty and ${Alt}^{k} (V) = 0$ .

F1step 1.1algebra

3.1

This is exactly the claimed dimension formula and vanishing statement.

step 2.1∎

Depends on

Used by

A k-form on an n-manifold must vanish when k>n False statement
The top exterior power is one-dimensional Proposition

Dependency tree · two levels

24 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)

Dimension of the kkth exterior power is binomial