Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

Identity maps and composites of linear maps are linear

Statement

The identity map of a vector space is linear. If T:U→VT:U\to V and S:V→WS:V\to W are linear maps over the same field, then S∘T:U→WS\circ T:U\to W is linear.

Facts & Assumptions

Given: Vector spaces U,V,WU,V,W over a field FF, scalars a,b∈Fa,b\in F, vectors u,v∈Uu,v\in U, and linear maps T:U→VT:U\to V and S:V→WS:V\to W.

[L1]

A map RR is linear when R(au+bv)=aR(u)+bR(v)R(au+bv)=aR(u)+bR(v) for all scalars and vectors (Linear map between vector spaces over the same field).

Proof

technique · direct
1.1

The identity satisfies id⁡V(av+bw)=av+bw=aid⁡V(v)+bid⁡V(w)\operatorname{id}_V(av+bw)=av+bw=a\operatorname{id}_V(v)+b\operatorname{id}_V(w), so it is linear.

givenL1
2.1

Linearity of TT and then of SS gives S(T(au+bv))=S(aT(u)+bT(v))=aS(T(u))+bS(T(v))S(T(au+bv))=S(aT(u)+bT(v))=aS(T(u))+bS(T(v)).

step 1.1L1
3.1

The equality in step 2.1 is the defining linearity equation for S∘TS\circ T, so the composite is linear.

step 2.1L1∎

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