Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03
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.

Kernel and image of a linear map

Definition

For a linear map T:V→WT:V\to W, its kernel and image are respectively

ker⁡T:={v∈V:T(v)=0W},im⁡T:={T(v):v∈V}.\ker T:=\{v\in V:T(v)=0_W\},\qquad \operatorname{im}T:=\{T(v):v\in V\}.

That both sets are linear subspaces, and that a trivial kernel characterises injectivity, is proved in The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial ↗.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources