TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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

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${rank}_{Q} W_{1, k} (n) = n$ for $1 \leq k \leq n - 1$

Statement

Let $1 \leq k \leq n - 1$ . Then

${rank}_{Q} W_{1, k} (n) = n .$

Facts & Assumptions

Given: a natural number $n$ and an index $k$ with $1 \leq k \leq n - 1$ .

[F1]

The row of $W_{1, k} (n)$ indexed by ${i}$ records membership of the point $i$ in each $k$ -set (The inclusion matrix $W_{t, k} (n)$ of $t$ -sets against $k$ -sets).

Proof

technique · direct

1.1

Suppose $\sum_{i < n} c_{i} R_{i} = 0$ , where $R_{i}$ is the row indexed by ${i}$ . Looking at the column indexed by a $k$ -set $K$ gives the equation $\sum_{i \in K} c_{i} = 0$ .

F1assume-contra

2.1

Let $i \neq j$ . Because $1 \leq k \leq n - 1$ , there is a $k$ -set containing $i$ but not $j$ ; replacing $i$ by $j$ gives another $k$ -set. Subtracting the two equations from step 1.1 yields $c_{i} = c_{j}$ .

F1step 1.1

3.1

All coefficients are therefore equal to some common value $c$ . Choosing any $k$ -set $K$ in step 1.1 gives $k c = 0$ , and since $k \geq 1$ in $Q$ this forces $c = 0$ . So the rows are linearly independent, and there are $n$ of them.

step 1.1step 2.1algebradischarge-contradiction∎

Remarks

The range is sharp. At $k = n$ the matrix has one column and rank $1$ , while at $k = 0$ there is no point row at all.

Depends on

Used by

Dependency tree · two levels

31 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

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §7.1 (standard reference, not scraped)

rank⁡QW1,k(n)=nrankQ​W1,k​(n)=n for 1≤k≤n−11≤k≤n−1