Alphabeta MathHow 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.
Invertible linear maps, linear isomorphisms, and inverse linear maps
Definition
Let be linear. It is invertible when there is a linear map such that
Such an is the inverse linear map of , denoted . The two inverse equations make bijective and determine uniquely. An invertible linear map is also called a linear isomorphism, and and are linearly isomorphic, written , when such a map exists.
Depends on
Used by
- Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension Corollary
- A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms Theorem
- Invertible matrix theorem: invertibility, full pivot rank, RREF I, trivial nullspace and unique solvability are equivalent Theorem
- T↦[T]_mathcal B^mathcal C is a vector-space isomorphism mathcal L(V,W)≅ M_m× n(F) Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 19 results over 11 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
- S. Schiavone, MIT 18.700 Day 9, Definition 21 (standard reference, not scraped)
- S. Axler, Linear Algebra Done Right, 4th ed., §3D (standard reference, not scraped)