Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

A linear map preserves zero, negatives, and subtraction

Statement

If T:V→WT:V\to W is linear, then T(0V)=0WT(0_V)=0_W, T(−v)=−T(v)T(-v)=-T(v), and T(u−v)=T(u)−T(v)T(u-v)=T(u)-T(v) for all u,v∈Vu,v\in V.

Facts & Assumptions

Given: Vector spaces V,WV,W over a field FF and a linear map T:V→WT:V\to W.

[L1]

A linear map satisfies T(au+bv)=aT(u)+bT(v)T(au+bv)=aT(u)+bT(v) (Linear map between vector spaces over the same field).

[L2]

The additive structure of a vector space is an abelian group, so it has identities, inverses, cancellation, and u−v=u+(−v)u-v=u+(-v); also 0Fv=0V0_Fv=0_V and (−1F)v=−v(-1_F)v=-v (Vector space over a field, In any vector space 0Fv=0V0_F v = 0_V, λ0V=0V\lambda 0_V = 0_V, (−λ)v=−(λv)(-\lambda)v = -(\lambda v), (−1F)v=−v(-1_F)v = -v, and λv=0V\lambda v = 0_V forces λ=0F\lambda = 0_F or v=0Vv = 0_V).

Proof

technique · direct
1.1

Linearity gives T(0V)=T(0F0V+0F0V)=T(0V)+T(0V)T(0_V)=T(0_F0_V+0_F0_V)=T(0_V)+T(0_V); cancellation in WW gives T(0V)=0WT(0_V)=0_W.

L1L2given
2.1

Since 0W=T(0V)=T(v+(−v))=T(v)+T(−v)0_W=T(0_V)=T(v+(-v))=T(v)+T(-v), the inverse law gives T(−v)=−T(v)T(-v)=-T(v).

step 1.1L1L2given
3.1

Using u−v=u+(−v)u-v=u+(-v) and step 2.1, T(u−v)=T(u)+T(−v)=T(u)−T(v)T(u-v)=T(u)+T(-v)=T(u)-T(v).

step 2.1L1L2given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 12 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