ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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 quarter-turn $(x, y) \mapsto (- y, x)$ on $R^{2}$ has matrix $(\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ and square $- I_{2}$

Example

The linear map $Q : R^{2} \to R^{2}$ given by $Q (x, y) = (- y, x)$ is a quarter-turn. In the standard basis,

$[Q] = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array}), [Q]^{2} = - I_{2} .$

Thus $Q^{2} (x, y) = (- x, - y)$ is the half-turn.

Facts & Assumptions

Given: The real vector space $R^{2}$ with its standard ordered basis and the displayed map $Q$ .

[L1]

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

[L2]

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

Verification

technique · direct

1.1

Since $Q (1, 0) = (0, 1)$ and $Q (0, 1) = (- 1, 0)$ , [L1] gives $[Q] = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ . Multiplying this matrix by itself gives $(\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array}) = - I_{2}$ .

givenL1

2.1

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

step 1.1L1L2∎

Depends on

Used by

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Dependency tree · two levels

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The quarter-turn (x,y)↦(−y,x)(x,y)↦(−y,x) on R2R2 has matrix (0−110)(01​−10​) and square −I2−I2​