DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Transverse linear subspaces

Definition

Let $V$ be a finite-dimensional real vector space, and let $A, B \leq V$ be linear subspaces. They are transverse, written $A ⋔ B$ , when

$A + B = V .$

Equivalently,

$\dim (A \cap B) = \dim A + \dim B - \dim V$

by The dimension formula: for finite-dimensional linear subspaces $U$ and $W$ of $V$ , the subspaces $U + W$ and $U \cap W$ are finite-dimensional and $\dim_{F} (U + W) + \dim_{F} (U \cap W) = \dim_{F} U + \dim_{F} W$ .

Depends on

Used by

The local oriented intersection sign Definition
Preimage orientation agrees with the local intersection sign Lemma

Dependency tree · two levels

33 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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)