ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

Gram–Schmidt on an explicit basis of $R^{3}$

Example

Applying Gram–Schmidt in standard $R^{3}$ to

$v_{0} = (1, 1, 1), v_{1} = (0, 1, 2), v_{2} = (1, 0, 2)$

produces

$e_{0} = \frac{(1, 1, 1)}{\sqrt{3}}, e_{1} = \frac{(- 1, 0, 1)}{\sqrt{2}}, e_{2} = \frac{(1, - 2, 1)}{\sqrt{6}} .$

Facts & Assumptions

Given: The displayed ordered basis in standard $R^{3}$ .

[L1]

The standard coordinate inner product is the sum of coordinatewise products (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[L2]

Gram–Schmidt subtracts earlier orthogonal projections, normalises the nonzero residual, and preserves each successive span (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).

Verification

technique · computation

1.1

By [L1], $∥ v_{0} ∥ = \sqrt{3}$ , giving $e_{0}$ . The second residual is $u_{1} = v_{1} - ⟨ v_{1}, e_{0} ⟩ e_{0} = (- 1, 0, 1)$ , whose norm is $\sqrt{2}$ , giving $e_{1}$ .

L1L2algebra

1.2

The third residual is $u_{2} = v_{2} - ⟨ v_{2}, e_{0} ⟩ e_{0} - ⟨ v_{2}, e_{1} ⟩ e_{1} = (1 / 2, - 1, 1 / 2)$ . Its norm is $\sqrt{6} / 2$ , so normalisation gives the displayed $e_{2}$ .

L1L2algebra

2.1

Direct dot products give $⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ . Moreover $e_{0}$ is a nonzero multiple of $v_{0}$ , $u_{1} = v_{1} - v_{0}$ , and $u_{2}$ differs from $v_{2}$ by a combination of the first two residual directions, verifying all three successive-span equalities from [L2].

step 1.1step 1.2L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 10 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

Sergei Treil, Linear Algebra Done Wrong, §5.3.2 (standard reference, not scraped)

Gram–Schmidt on an explicit basis of R3R3