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.
The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and on the zero space
Definition
Let be a linear operator on a finite-dimensional vector space over a field (Finite-dimensional vector space, and its dimension ; infinite-dimensional means having no finite basis).
If , choose an ordered basis and define
where is the coordinate matrix of (Coordinate columns and matrices of linear maps relative to ordered bases) and the determinant on the right is the matrix determinant (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix). If , define .
The positive-dimensional definition is justified by The determinant of a linear operator is independent of the chosen ordered basis ↗, which proves that the value does not depend on the chosen ordered basis.