ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

An overdetermined inconsistent system has a unique minimum-norm least-squares solution given by $A^{+} b$

Example

Take

$A = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{array}], b = [\begin{array}{c} 1 \\ 1 \\ 0 \end{array}] .$

Then $b \notin im A$ , the least-squares problem is inconsistent, and

$A^{+} b = [\begin{array}{c} \frac{1}{3} \\ \frac{1}{3} \end{array}] .$

Facts & Assumptions

Given: The matrix $A$ and right-hand side $b$ in the statement.

[L1]

Reduced QR gives the pseudoinverse of a full-column-rank matrix without forming normal equations (Reduced QR gives the full-column and full-row-rank pseudoinverse formulas without forming normal equations).

[L2]

$A^{+} b$ is the unique least-squares solution of minimum norm (For every right-hand side $b$ , $A^{+} b$ is the unique least-squares solution of minimum Euclidean norm).

Verification

technique · computation

1.1L1algebra

The columns of $A$ are independent, so $A$ has full column rank. Directly, $A^{*} A = [\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}], (A^{*} A)^{- 1} = \frac{1}{3} [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}] .$ Hence $A^{+} b = (A^{*} A)^{- 1} A^{*} b = (1 / 3, 1 / 3)^{T}$ .

2.1step 1.1algebra

The residual is $r = b - A A^{+} b = [\begin{array}{c} \frac{2}{3} \\ \frac{2}{3} \\ - \frac{2}{3} \end{array}],$ so $r \neq 0$ and the system is inconsistent.

3.1L2step 1.1step 2.1∎

By [L2], the vector from step 1.1 is nevertheless the unique least-squares solution of minimum Euclidean norm.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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An overdetermined inconsistent system has a unique minimum-norm least-squares solution given by A+bA+b