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ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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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.

The quarter-turn (x,y)↦(−y,x)(x,y)\mapsto(-y,x) on R2\mathbb R^2 has matrix (0−110)\begin{pmatrix}0&-1\\1&0\end{pmatrix} and square −I2-I_2

Example

The linear map Q:R2→R2Q:\mathbb R^2\to\mathbb R^2 given by Q(x,y)=(−y,x)Q(x,y)=(-y,x) is a quarter-turn. In the standard basis,

[Q]=(0−110),[Q]2=−I2.[Q]=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad [Q]^2=-I_2.

Thus Q2(x,y)=(−x,−y)Q^2(x,y)=(-x,-y) is the half-turn.

Facts & Assumptions

Given: The real vector space R2\mathbb R^2 with its standard ordered basis and the displayed map QQ.

[L1]

The columns of a linear map's matrix are the coordinate columns of its basis-vector images (Coordinate columns [v]B[v]_{\mathcal B} and matrices [T]BC[T]_{\mathcal B}^{\mathcal C} of linear maps relative to ordered bases).

[L2]

The matrix of a composite is the product of the matrices ([S∘T]BD=[S]CD[T]BC[S\circ T]_{\mathcal B}^{\mathcal D}=[S]_{\mathcal C}^{\mathcal D}[T]_{\mathcal B}^{\mathcal C}).

Verification

technique · direct
1.1

Since Q(1,0)=(0,1)Q(1,0)=(0,1) and Q(0,1)=(−1,0)Q(0,1)=(-1,0), [L1] gives [Q]=(0−110)[Q]=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. Multiplying this matrix by itself gives (−100−1)=−I2\begin{pmatrix}-1&0\\0&-1\end{pmatrix}=-I_2.

givenL1
2.1

By [L2], the matrix square is [Q2][Q^2], and direct substitution gives Q2(x,y)=Q(−y,x)=(−x,−y)Q^2(x,y)=Q(-y,x)=(-x,-y), the half-turn.

step 1.1L1L2∎

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