DefinitionDefinition: Literature-sourcedProof: Not applicable

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.

Invertible matrices and the general linear group ${GL}_{n} (F)$

Definition

A matrix $A \in M_{n} (F)$ is invertible when there is a matrix $B \in M_{n} (F)$ such that

$A B = I_{n} = B A .$

Such a $B$ is unique and is denoted $A^{- 1}$ . The general linear set is

${GL}_{n} (F) : = {A \in M_{n} (F) : A is invertible} .$

It is the set of units of the ring $M_{n} (F)$ . Under matrix multiplication, ${GL}_{n} (F)$ is a group, including the trivial group ${GL}_{0} (F)$ . For $n = 0$ , the unique empty matrix is $I_{0}$ and is its own inverse.

Definition references

${GL}_{n} (F)$ is a group under matrix multiplication, including the trivial group ${GL}_{0} (F)$ ↗

Invertible matrices and the general linear group ${GL}_{n} (F)$

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

Invertible matrices and the general linear group GL⁡n(F)GLn​(F)

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

Invertible matrices and the general linear group ${GL}_{n} (F)$