[Merged by Bors] - feat(LinearAlgebra/Transvection): the determinant of a transvection is equal to 1.#32757
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Is it easier to use Matrix.det_of_upperTriangular?
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Over a field, that's essentially what is done. But that wouldn't suffice over a ring. |
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Oh, I see, it's because |
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Exactly! |
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Co-authored-by: Eric Wieser <[email protected]>
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Thanks for this lovely work and also for your extraordinary patience!
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…s equal to 1. (#32757) Prove that the determinant of a transvection is equal to 1 The proof goes by showing that the determinant of `LinearMap.transvection f v` is `1 + f v` (even if `f v` is nonzero). I first treat the case of a field, distinguishing whether `f v = 0` (so that we get a transvection) or not (so that we have a dilation). Then, by base change to the field of fractions, I can handle the case of a domain. Finally, the general case is treated by base change from the “universal case” where the base ring is the ring of polynomials in $$2n$$ indeterminates corresponding to the coefficients of $$f$$ and $$v$$. Co-authored-by: pre-commit-ci-lite[bot] <117423508+pre-commit-ci-lite[bot]@users.noreply.github.com>
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…s equal to 1. (leanprover-community#32757) Prove that the determinant of a transvection is equal to 1 The proof goes by showing that the determinant of `LinearMap.transvection f v` is `1 + f v` (even if `f v` is nonzero). I first treat the case of a field, distinguishing whether `f v = 0` (so that we get a transvection) or not (so that we have a dilation). Then, by base change to the field of fractions, I can handle the case of a domain. Finally, the general case is treated by base change from the “universal case” where the base ring is the ring of polynomials in $$2n$$ indeterminates corresponding to the coefficients of $$f$$ and $$v$$. Co-authored-by: pre-commit-ci-lite[bot] <117423508+pre-commit-ci-lite[bot]@users.noreply.github.com>
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…s equal to 1. (leanprover-community#32757) Prove that the determinant of a transvection is equal to 1 The proof goes by showing that the determinant of `LinearMap.transvection f v` is `1 + f v` (even if `f v` is nonzero). I first treat the case of a field, distinguishing whether `f v = 0` (so that we get a transvection) or not (so that we have a dilation). Then, by base change to the field of fractions, I can handle the case of a domain. Finally, the general case is treated by base change from the “universal case” where the base ring is the ring of polynomials in $$2n$$ indeterminates corresponding to the coefficients of $$f$$ and $$v$$. Co-authored-by: pre-commit-ci-lite[bot] <117423508+pre-commit-ci-lite[bot]@users.noreply.github.com>
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Prove that the determinant of a transvection is equal to 1
The proof goes by showing that the determinant of
LinearMap.transvection f vis1 + f v(even iff vis nonzero).I first treat the case of a field, distinguishing whether
f v = 0(so that we geta transvection) or not (so that we have a dilation).
Then, by base change to the field of fractions, I can handle the case of a domain. Finally, the general case is treated by base change from the “universal case” where the base ring is the ring of polynomials in$$2n$$ indeterminates corresponding to the coefficients of $$f$$ and $$v$$ .