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
- Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent Corollary
- Determinant is a group homomorphism GL(V)→ F^×, and det(T⁻¹)=det(T)⁻¹ Corollary
- Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension Corollary
- A finite-dimensional representation ρ:G→ GL(V) over a field, and its degree Definition
- Conjugation and the adjoint representation of a Lie group Definition
- The exponential map of a flat torus is not injective Example
- For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible Proposition
- 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]_B^C is a vector-space isomorphism L(V,W)≅ M_m× n(F) Theorem
Dependency tree · two levels
6 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
- 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)