mathlib3 documentation

linear_algebra.finsupp_vector_space

Linear structures on function with finite support ι →₀ M #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains results on the R-module structure on functions of finite support from a type ι to an R-module M, in particular in the case that R is a field.

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theorem finsupp.linear_independent_single {R : Type u_1} {M : Type u_2} {ι : Type u_3} [ring R] [add_comm_group M] [module R M] {φ : ι → Type u_4} {f : Π (ι : ι), φ ι → M} (hf : ∀ (i : ι), linear_independent R (f i)) :
linear_independent R (λ (ix : Σ (i : ι), φ i), finsupp.single ix.fst (f ix.fst ix.snd))
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@[protected]
noncomputable def finsupp.basis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [semiring R] [add_comm_monoid M] [module R M] {φ : ι → Type u_4} (b : Π (i : ι), basis (φ i) R M) :
basis (Σ (i : ι), φ i) R (ι →₀ M)

The basis on ι →₀ M with basis vectors λ ⟨i, x⟩, single i (b i x).

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@[simp]
theorem finsupp.basis_repr {R : Type u_1} {M : Type u_2} {ι : Type u_3} [semiring R] [add_comm_monoid M] [module R M] {φ : ι → Type u_4} (b : Π (i : ι), basis (φ i) R M) (g : ι →₀ M) (ix : Σ (i : ι), φ i) :
⇑(⇑((finsupp.basis b).repr) g) ix = ⇑(⇑((b ix.fst).repr) (⇑g ix.fst)) ix.snd
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@[simp]
theorem finsupp.coe_basis {R : Type u_1} {M : Type u_2} {ι : Type u_3} [semiring R] [add_comm_monoid M] [module R M] {φ : ι → Type u_4} (b : Π (i : ι), basis (φ i) R M) :
⇑(finsupp.basis b) = λ (ix : Σ (i : ι), φ i), finsupp.single ix.fst (⇑(b ix.fst) ix.snd)
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@[protected]
noncomputable def finsupp.basis_single_one {R : Type u_1} {ι : Type u_3} [semiring R] :
basis ι R (ι →₀ R)

The basis on ι →₀ M with basis vectors λ i, single i 1.

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@[simp]
theorem finsupp.basis_single_one_repr {R : Type u_1} {ι : Type u_3} [semiring R] :
finsupp.basis_single_one.repr = linear_equiv.refl R (ι →₀ R)
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@[simp]
theorem finsupp.coe_basis_single_one {R : Type u_1} {ι : Type u_3} [semiring R] :
⇑finsupp.basis_single_one = λ (i : ι), finsupp.single i 1

TODO: move this section to an earlier file.

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theorem finset.sum_single_ite {R : Type u_1} {n : Type u_3} [decidable_eq n] [fintype n] [semiring R] (a : R) (i : n) :
finset.univ.sum (λ (x : n), finsupp.single x (ite (i = x) a 0)) = finsupp.single i a
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@[simp]
theorem basis.equiv_fun_symm_std_basis {R : Type u_1} {M : Type u_2} {n : Type u_3} [decidable_eq n] [fintype n] [semiring R] [add_comm_monoid M] [module R M] (b : basis n R M) (i : n) :
⇑(b.equiv_fun.symm) (⇑(linear_map.std_basis R (λ (_x : n), R) i) 1) = ⇑b i