DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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

Used by

If Ax=b has one solution xₚ, then its full solution set is the affine subspace xₚ+N(A) Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §2.7 (standard reference, not scraped)

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