LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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 $T : V \to W$ is linear, then $T (0_{V}) = 0_{W}$ , $T (- v) = - T (v)$ , and $T (u - v) = T (u) - T (v)$ for all $u, v \in V$ .

Facts & Assumptions

Given: Vector spaces $V, W$ over a field $F$ and a linear map $T : V \to W$ .

[L1]

A linear map satisfies $T (a u + b v) = a T (u) + b T (v)$ (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 $u - v = u + (- v)$ ; also $0_{F} v = 0_{V}$ and $(- 1_{F}) v = - v$ (Vector space over a field, In any vector space $0_{F} v = 0_{V}$ , $λ 0_{V} = 0_{V}$ , $(- λ) v = - (λ v)$ , $(- 1_{F}) v = - v$ , and $λ v = 0_{V}$ forces $λ = 0_{F}$ or $v = 0_{V}$ ).

Proof

technique · direct

1.1

Linearity gives $T (0_{V}) = T (0_{F} 0_{V} + 0_{F} 0_{V}) = T (0_{V}) + T (0_{V})$ ; cancellation in $W$ gives $T (0_{V}) = 0_{W}$ .

L1L2given

2.1

Since $0_{W} = T (0_{V}) = T (v + (- v)) = T (v) + T (- v)$ , the inverse law gives $T (- v) = - T (v)$ .

step 1.1L1L2given

3.1

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

step 2.1L1L2given∎

Depends on

Used by

The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial Theorem

Dependency tree · two levels

12 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

Axler, Linear Algebra Done Right, Chapter 3 (standard reference, not scraped)