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.
Row echelon form, reduced row echelon form, leading entries, pivots, pivot positions and pivot columns
Definition
In a nonzero row of a matrix over a field, its leading entry is its leftmost nonzero entry. A matrix is in row echelon form when:
- every zero row lies below every nonzero row;
- the leading entry of each nonzero row lies strictly to the right of the leading entry in the row above it;
- every entry below a leading entry is zero.
A leading entry in row echelon form is a pivot. Its location is a pivot position, and a column containing a pivot is a pivot column.
A row echelon matrix is in reduced row echelon form when every pivot equals and is the only nonzero entry in its column. Zero-row, zero-column and all-zero matrices satisfy these conditions vacuously.