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.
Linear map between vector spaces over the same field
Definition
Let and be vector spaces over the same field . A function is linear when
for all and . Such a function is a linear map, or linear transformation.
Depends on
Used by
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms Definition
- Coordinate columns [v]_mathcal B and matrices [T]_mathcal B^mathcal C of linear maps relative to ordered bases Definition
- Invertible linear maps, linear isomorphisms, and inverse linear maps Definition
- Kernel and image of a linear map Definition
- The space mathcal L(V,W) of linear maps with pointwise addition and scalar multiplication Definition
- The subspace Γ of directions along which a series converges absolutely, and its orthogonal complement Γ^⊥ Definition
- The forward shift on F^ℕ is injective but not surjective Example
- A linear map preserves zero, negatives, and subtraction Lemma
- Extending a basis of the kernel to a basis of the domain gives a basis of the image Lemma
- Identity maps and composites of linear maps are linear Lemma
- mathcal L(V,W) is a vector space over the common scalar field Proposition
- Vector spaces over a fixed field and linear maps form the large locally small category Vect_F Proposition
- Conventions of this page, the standing n ≥ 1 hypothesis, and what is taken up elsewhere in the reading order Remark
- [T(v)]_mathcal C=[T]_mathcal B^mathcal C[v]_mathcal B Theorem
- The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 10 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
- Axler, Linear Algebra Done Right, Chapter 3 (standard reference, not scraped)