DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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.

A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates

Definition

Let $m,n\ge1$ . A map $L:\mathbb R^m\to\mathbb R^n$ is linear when

$L(\alpha u+\beta v)=\alpha L(u)+\beta L(v)$

for all $u,v\in\mathbb R^m$ and all $\alpha,\beta\in\mathbb R$ . Both spaces carry their Euclidean vector-space operations and Euclidean norms from The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ .

Remarks

This is the concrete Euclidean notion required for total differentiation. It makes no assertion about linear maps between arbitrary vector spaces.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 79 results over 17 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

J. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)

A linear map L:Rm→RnL:\mathbb{R}^m\to\mathbb{R}^nL:Rm→Rn in Euclidean coordinates