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.
Matrix equation , its solution set, consistency, homogeneous systems and the augmented matrix
Definition
Let and let , viewed as an column. The matrix equation asks for columns satisfying the displayed equality. Equivalently it is the linear system
Its solution set is . The system is consistent when and inconsistent otherwise. It is homogeneous when .
The augmented matrix is , obtained by adjoining as the last column. Elementary row operations act on every entry of a row, including its augmented entry.
Depends on
Used by
- Row reducing [A∣ I] yields [I∣ A⁻¹] exactly when A is invertible Corollary
- Over F₂, the equation x+y=0 has exactly two solutions, so the infinite-field hypothesis is necessary Counterexample
- Row space, column space, nullspace, row rank, column rank and matrix rank Definition
- Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system Theorem
- If Ax=b has one solution xₚ, then its full solution set is the affine subspace xₚ+N(A) Theorem
- Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables Theorem
Dependency tree · two levels
14 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
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§1.2–1.3 (standard reference, not scraped)