DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $L : R^{m} \to R^{n}$ in Euclidean coordinates

Definition

Let $m, n \geq 1$ . A map $L : R^{m} \to R^{n}$ is linear when

$L (α u + β v) = α L (u) + β L (v)$

for all $u, v \in R^{m}$ and all $α, β \in R$ . Both spaces carry their Euclidean vector-space operations and Euclidean norms from The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

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 · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

A linear map L:Rm→RnL:Rm→Rn in Euclidean coordinates