DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

How 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 $A x = b$ , its solution set, consistency, homogeneous systems and the augmented matrix $[A ∣ b]$

Definition

Let $A \in M_{m \times n} (F)$ and let $b \in F^{m}$ , viewed as an $m \times 1$ column. The matrix equation $A x = b$ asks for columns $x \in F^{n}$ satisfying the displayed equality. Equivalently it is the linear system $\sum_{j < n} a_{i j} x_{j} = b_{i} (i < m) .$

Its solution set is $S (A, b) : = {x \in F^{n} : A x = b}$ . The system is consistent when $S (A, b) \neq \emptyset$ and inconsistent otherwise. It is homogeneous when $b = 0$ .

The augmented matrix is $[A ∣ b] \in M_{m \times (n + 1)} (F)$ , obtained by adjoining $b$ as the last column. Elementary row operations act on every entry of a row, including its augmented entry.

Depends on

Dependency tree · 2 levels

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§1.2–1.3

Matrix equation Ax=bAx=b, its solution set, consistency, homogeneous systems and the augmented matrix [A∣b][A∣b]

Definition

Depends on

Dependency tree · 2 levels

Sources

Matrix equation $A x = b$ , its solution set, consistency, homogeneous systems and the augmented matrix $[A ∣ b]$