[Merged by Bors] - feat(linear_algebra/*): Use alternating maps for wedge and determinant #5124
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| /-- The product of `n` terms of the form `ι R m` is an alternating map. | ||
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| This is a special case of `multilinear_map.mk_pi_algebra_fin` -/ | ||
| def ι_multi (n : ℕ) : |
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I'm not really sure what a good name for this is. wedge is another option, but it's weird that we use * as the symbol for a product that we call a wedge product.
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The name looks fine to me right now, since it's the (alternating) multilinear version of ι.
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src/linear_algebra/matrix.lean
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| map_eq_zero_of_eq' := begin | ||
| intros v i j h hij, | ||
| rw [←function.update_eq_self i v, h, ←det_transpose, he.to_matrix_update, ←update_row_transpose], | ||
| have : (he.to_matrix v)ᵀ j = he.repr (v j) := funext (λ _, rfl), |
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Zulip thread about whether this lemma should have a name.
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Seeing in this context, I believe it should have a name, but probably not be a simp lemma.
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src/linear_algebra/matrix.lean
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| map_eq_zero_of_eq' := begin | ||
| intros v i j h hij, | ||
| rw [←function.update_eq_self i v, h, ←det_transpose, he.to_matrix_update, ←update_row_transpose], | ||
| have : (he.to_matrix v)ᵀ j = he.repr (v j) := funext (λ _, rfl), |
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Seeing in this context, I believe it should have a name, but probably not be a simp lemma.
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…nt (#5124) This : * Adds `exterior_algebra.ι_multi`, where `ι_multi ![a, b ,c]` = `ι a * ι b * ι c` * Makes `det_row_multilinear` an `alternating_map` * Makes `is_basis.det` an `alternating_map`
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This :
exterior_algebra.ι_multi, whereι_multi ![a, b ,c]=ι a * ι b * ι cdet_row_multilinearanalternating_mapis_basis.detanalternating_mapI think
ι_multiis was what @zhangir-azerbayev was preparing to prove after they wrapped up #3770