DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

The Krylov subspace $K_{m} (A, b) = span {b, A b, \dots, A^{m - 1} b}$

Definition

Let $A$ be a square matrix over $R$ or $C$ and let $b$ be a vector of matching size. Define

$K_{0} (A, b) : = {0},$

and for $m \geq 1$ define the $m$ th Krylov subspace

$K_{m} (A, b) : = span {b, A b, \dots, A^{m - 1} b} .$

Thus $K_{m} (A, b) \subseteq K_{m + 1} (A, b)$ for every $m \geq 0$ .

Depends on

Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes

Used by

Dependency tree · two levels

3 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

Yousef Saad, CSCI 8314 Set 13: Krylov subspace methods; Introduction; Krylov subspaces; Gram-Schmidt process (review); The Arnoldi process; FOM and GMRES (standard reference, not scraped)

The Krylov subspace Km(A,b)=span⁡{b,Ab,…,Am−1b}Km​(A,b)=span{b,Ab,…,Am−1b}