ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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.

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The quotient of $F^{3}$ by a coordinate line and its canonical projection

Example

Let $W = F (1, 0, 0) \subseteq F^{3}$ . Every coset in $F^{3} / W$ has a unique representative $(0, y, z)$ , the cosets $(0, 1, 0) + W, (0, 0, 1) + W$ form a basis of the quotient, and the canonical projection is $π (x, y, z) = (x, y, z) + W = (0, y, z) + W .$ Thus $\dim (F^{3} / W) = 2$ and $\ker π = W$ .

Facts & Assumptions

Given: The coordinate line $W = F (1, 0, 0)$ in $F^{3}$ .

[L1]

$v + W = v^{'} + W$ exactly when $v - v^{'} \in W$ ; the quotient operations are independent of representatives and make $V / W$ a vector space; and the canonical projection $π : V \to V / W$ is a surjective linear map with $\ker π = W$ (Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ ).

[L2]

Representatives of a quotient basis, placed after a basis of $W$ , form a basis of the original space (A quotient basis lifts to a basis adapted to $W$ ).

Verification

technique · computation

1.1

Subtracting $(x, 0, 0) \in W$ shows $(x, y, z) + W = (0, y, z) + W$ ; if two such representatives agree, their difference $(0, y - y^{'}, z - z^{'})$ lies in $W$ , forcing $y = y^{'}$ and $z = z^{'}$ .

L1algebra

1.2

The standard list $((1, 0, 0), (0, 1, 0), (0, 0, 1))$ is a basis of $F^{3}$ , so [L2] makes the last two cosets a quotient basis.

L2algebra

2.1

The displayed formula for $π$ , its dimension, and $\ker π = W$ now follow from steps 1.1-1.2 and [L1].

step 1.1step 1.2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 11 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.

The quotient of F3F3 by a coordinate line and its canonical projection