[Merged by Bors] - feat: port LinearAlgebra.BilinearMap #2434
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jcommelin
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Feb 22, 2023
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| ! if you have ported upstream changes. | ||
| -/ | ||
| import Mathlib.LinearAlgebra.Basic | ||
| import Mathlib.LinearAlgebra.Basis |
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This import doesn't exist in mathlib3 master
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| variable (b₁ : Basis ι₁ R M) (b₂ : Basis ι₂ S N) (b₁' : Basis ι₁ R Mₗ) (b₂' : Basis ι₂ R Nₗ) | ||
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| /-- Two bilinear maps are equal when they are equal on all basis vectors. -/ | ||
| theorem ext_basis {B B' : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (h : ∀ i j, B (b₁ i) (b₂ j) = B' (b₁ i) (b₂ j)) : | ||
| B = B' := | ||
| b₁.ext fun i => b₂.ext fun j => h i j | ||
| #align linear_map.ext_basis LinearMap.ext_basis | ||
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| /-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis. | ||
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| Version for semi-bilinear maps, see `sum_repr_mul_repr_mul` for the bilinear version. -/ | ||
| theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) : | ||
| ((b₁.repr x).Sum fun i xi => (b₂.repr y).Sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) = | ||
| B x y := by | ||
| conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y] | ||
| simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂, | ||
| LinearMap.map_smulₛₗ] | ||
| #align linear_map.sum_repr_mul_repr_mulₛₗ LinearMap.sum_repr_mul_repr_mulₛₗ | ||
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| /-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis. | ||
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| Version for bilinear maps, see `sum_repr_mul_repr_mulₛₗ` for the semi-bilinear version. -/ | ||
| theorem sum_repr_mul_repr_mul {B : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ} (x y) : | ||
| ((b₁'.repr x).Sum fun i xi => (b₂'.repr y).Sum fun j yj => xi • yj • B (b₁' i) (b₂' j)) = | ||
| B x y := by | ||
| conv_rhs => rw [← b₁'.total_repr x, ← b₂'.total_repr y] | ||
| simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smul₂, | ||
| LinearMap.map_smul] | ||
| #align linear_map.sum_repr_mul_repr_mul LinearMap.sum_repr_mul_repr_mul |
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These lemmas don't exist in mathlib3 master
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I think mathport has rerun, so this is ready to re-port |
Mathbin -> Mathlib fix certain import statements move "by" to end of line add import to Mathlib.lean
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I re-ported this file. @jcommelin |
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Co-authored-by: casavaca <96765450+casavaca@users.noreply.github.com>
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