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.

Affine subspaces as translates $x + U$ of linear subspaces

Definition

Let $V$ be a vector space over $F$ , let $U \subseteq V$ be a linear subspace, and let $x \in V$ . The translate $x + U : = {x + u : u \in U}$ is an affine subspace of $V$ . The same affine subspace can have different representatives: if $y \in x + U$ , then $y + U = x + U$ . It is a linear subspace exactly when it contains $0$ , equivalently when $x \in U$ .

Depends on

Dependency tree · 2 levels

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §2.7

Affine subspaces as translates x+Ux+U of linear subspaces

Definition

Depends on

Dependency tree · 2 levels

Sources

Affine subspaces as translates $x + U$ of linear subspaces