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 underdetermined consistent system has infinitely many exact solutions but a unique minimum-norm one

Example

Let

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

Then the exact solutions are

$x (t) = [\begin{array}{c} 1 - t \\ 1 - t \\ t \end{array}] (t \in F),$

and the minimum-norm one is $A^{+} b = (1 / 3, 1 / 3, 2 / 3)^{T}$ .

Facts & Assumptions

Given: The displayed matrix $A$ and right-hand side $b$ .

[L1]

If $A$ has full row rank, then $A^{+} = A^{*} (A A^{*})^{- 1}$ (If $A$ has full row rank, then $A^{+} = A^{*} (A A^{*})^{- 1}$ ).

[L2]

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

Verification

technique · computation

1.1

Solving $A x = b$ gives the one-parameter family $x (t) = (1 - t, 1 - t, t)^{T}$ . Thus the system is consistent and has infinitely many exact solutions.

algebra

1.2

Here $A A^{*} = [\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}], (A A^{*})^{- 1} = \frac{1}{3} [\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array}],$ so [L1] gives $A^{+} b = A^{*} (A A^{*})^{- 1} b = [\begin{array}{c} \frac{1}{3} \\ \frac{1}{3} \\ \frac{2}{3} \end{array}] .$

L1algebra

2.1

The vector of step 1.2 equals $x (2 / 3)$ from step 1.1, so it is an exact solution. By [L2], it is the unique exact solution of minimum Euclidean norm.

L2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.