Alphabeta MathHow 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.
Affine subspaces as translates of linear subspaces
Definition
Let be a vector space over , let be a linear subspace, and let . The translate is an affine subspace of . The same affine subspace can have different representatives: if , then . It is a linear subspace exactly when it contains , equivalently when .
Depends on
Used by
- Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice Definition
- The n hyperplanes xᵢ=1 cover {0,1}ⁿ except the origin, so the Alon–Füredi bound is tight Example
- Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet Lemma
- Point stabilizers, vertex residues, and rank-two boundary words Lemma
- Alcove transitivity, the affine Coxeter presentation, and the length function Theorem
- If Ax=b has one solution xₚ, then its full solution set is the affine subspace xₚ+N(A) Theorem
Dependency tree · two levels
11 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
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §2.7 (standard reference, not scraped)