Finite dimensional vector spaces

Definition and basic properties of finite dimensional vector spaces, of their dimensions, and of linear maps on such spaces.

Main definitions

Assume V is a vector space over a division ring K. There are (at least) three equivalent definitions of finite-dimensionality of V:

• it admits a finite basis.
• it is finitely generated.
• it is noetherian, i.e., every subspace is finitely generated.

We introduce a typeclass FiniteDimensional K V capturing this property. For ease of transfer of proof, it is defined using the second point of view, i.e., as Module.Finite. However, we prove that all these points of view are equivalent, with the following lemmas (in the namespace FiniteDimensional):

• fintypeBasisIndex states that a finite-dimensional vector space has a finite basis
• Module.finBasis and Module.finBasisOfFinrankEq are bases for finite dimensional vector spaces, where the index type is Fin (in Mathlib.LinearAlgebra.Dimension.Free)
• of_fintype_basis states that the existence of a basis indexed by a finite type implies finite-dimensionality
• of_finite_basis states that the existence of a basis indexed by a finite set implies finite-dimensionality
• IsNoetherian.iff_fg states that the space is finite-dimensional if and only if it is noetherian (in Mathlib.FieldTheory.Finiteness)

We make use of finrank, the dimension of a finite dimensional space, returning a Nat, as opposed to Module.rank, which returns a Cardinal. When the space has infinite dimension, its finrank is by convention set to 0. finrank is not defined using FiniteDimensional. For basic results that do not need the FiniteDimensional class, import Mathlib.LinearAlgebra.Finrank.

Preservation of finite-dimensionality and formulas for the dimension are given for

• submodules (FiniteDimensional.finiteDimensional_submodule)
• quotients (for the dimension of a quotient, see Submodule.finrank_quotient_add_finrank in Mathlib.LinearAlgebra.FiniteDimensional)
• linear equivs, in LinearEquiv.finiteDimensional
• image under a linear map (the rank-nullity formula is in LinearMap.finrank_range_add_finrank_ker in Mathlib.LinearAlgebra.FiniteDimensional)

Basic properties of linear maps of a finite-dimensional vector space are given. Notably, the equivalence of injectivity and surjectivity is proved in LinearMap.injective_iff_surjective, and the equivalence between left-inverse and right-inverse in LinearMap.mul_eq_one_comm and LinearMap.comp_eq_id_comm.

Implementation notes

Most results are deduced from the corresponding results for the general dimension (as a cardinal), in Mathlib.LinearAlgebra.Dimension. Not all results have been ported yet.

You should not assume that there has been any effort to state lemmas as generally as possible.

Plenty of the results hold for general fg modules or notherian modules, and they can be found in Mathlib.LinearAlgebra.FreeModule.Finite.Rank and Mathlib.RingTheory.Noetherian.

@[reducible, inline]

abbrev FiniteDimensional (K : Type u_1) (V : Type u_2) [DivisionRing K] [AddCommGroup V] [Module K V] : Prop

FiniteDimensional vector spaces are defined to be finite modules. Use FiniteDimensional.of_fintype_basis to prove finite dimension from another definition.

Equations

• FiniteDimensional K V = Module.Finite K V

theorem FiniteDimensional.of_injective {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] (f : V →ₗ[K] V₂) (w : Function.Injective ⇑f) [FiniteDimensional K V₂] : FiniteDimensional K V

If the codomain of an injective linear map is finite dimensional, the domain must be as well.

theorem FiniteDimensional.of_surjective {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] (f : V →ₗ[K] V₂) (w : Function.Surjective ⇑f) [FiniteDimensional K V] : FiniteDimensional K V₂

If the domain of a surjective linear map is finite dimensional, the codomain must be as well.

instance FiniteDimensional.finiteDimensional_pi (K : Type u) [DivisionRing K] {ι : Type u_1} [Finite ι] : FiniteDimensional K (ι → K)

instance FiniteDimensional.finiteDimensional_pi' (K : Type u) [DivisionRing K] {ι : Type u_1} [Finite ι] (M : ι → Type u_2) [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module K (M i)] [∀ (i : ι), FiniteDimensional K (M i)] : FiniteDimensional K ((i : ι) → M i)

theorem FiniteDimensional.of_fintype_basis {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type w} [Finite ι] (h : Basis ι K V) : FiniteDimensional K V

If a vector space has a finite basis, then it is finite-dimensional.

noncomputable def FiniteDimensional.fintypeBasisIndex {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_1} [FiniteDimensional K V] (b : Basis ι K V) : Fintype ι

If a vector space is FiniteDimensional, all bases are indexed by a finite type

Equations

• FiniteDimensional.fintypeBasisIndex b = Fintype.ofFinite ι

noncomputable instance FiniteDimensional.instFintypeElemOfVectorSpaceIndex {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] : Fintype ↑(Basis.ofVectorSpaceIndex K V)

If a vector space is FiniteDimensional, Basis.ofVectorSpace is indexed by a finite type.

Equations

• FiniteDimensional.instFintypeElemOfVectorSpaceIndex = inferInstance

theorem FiniteDimensional.of_finite_basis {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type w} {s : Set ι} (h : Basis (↑s) K V) (hs : s.Finite) : FiniteDimensional K V

If a vector space has a basis indexed by elements of a finite set, then it is finite-dimensional.

instance FiniteDimensional.finiteDimensional_submodule {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (S : Submodule K V) : FiniteDimensional K ↥S

A subspace of a finite-dimensional space is also finite-dimensional.

instance FiniteDimensional.finiteDimensional_quotient {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (S : Submodule K V) : FiniteDimensional K (V ⧸ S)

A quotient of a finite-dimensional space is also finite-dimensional.

theorem FiniteDimensional.of_finrank_pos {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : 0 < Module.finrank K V) : FiniteDimensional K V

theorem FiniteDimensional.of_finrank_eq_succ {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : ℕ} (hn : Module.finrank K V = n.succ) : FiniteDimensional K V

theorem FiniteDimensional.of_fact_finrank_eq_succ {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (n : ℕ) [hn : Fact (Module.finrank K V = n + 1)] : FiniteDimensional K V

We can infer FiniteDimensional K V in the presence of [Fact (finrank K V = n + 1)]. Declare this as a local instance where needed.

theorem Module.finrank_eq_rank' (K : Type u) (V : Type v) [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] : ↑(finrank K V) = Module.rank K V

In a finite-dimensional space, its dimension (seen as a cardinal) coincides with its finrank. This is a copy of finrank_eq_rank _ _ which creates easier typeclass searches.

theorem Module.finrank_of_infinite_dimensional {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : ¬FiniteDimensional K V) : finrank K V = 0

theorem Module.finiteDimensional_iff_of_rank_eq_nsmul {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {W : Type v} [AddCommGroup W] [Module K W] {n : ℕ} (hn : n ≠ 0) (hVW : Module.rank K V = n • Module.rank K W) : FiniteDimensional K V ↔ FiniteDimensional K W

theorem Module.finrank_eq_card_basis' {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {ι : Type w} (h : Basis ι K V) : ↑(finrank K V) = Cardinal.mk ι

If a vector space is finite-dimensional, then the cardinality of any basis is equal to its finrank.

theorem LinearIndependent.lt_aleph0_of_finiteDimensional {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type w} [FiniteDimensional K V] {v : ι → V} (h : LinearIndependent K v) : Cardinal.mk ι < Cardinal.aleph0

theorem Submodule.eq_top_of_finrank_eq {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {S : Submodule K V} (h : Module.finrank K ↥S = Module.finrank K V) : S = ⊤

If a submodule has maximal dimension in a finite dimensional space, then it is equal to the whole space.

instance FiniteDimensional.finiteDimensional_self (K : Type u) [DivisionRing K] : FiniteDimensional K K

theorem FiniteDimensional.span_of_finite (K : Type u) {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {A : Set V} (hA : A.Finite) : FiniteDimensional K ↥(Submodule.span K A)

The submodule generated by a finite set is finite-dimensional.

instance FiniteDimensional.span_singleton (K : Type u) {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (x : V) : FiniteDimensional K ↥(Submodule.span K {x})

The submodule generated by a single element is finite-dimensional.

instance FiniteDimensional.span_finset (K : Type u) {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Finset V) : FiniteDimensional K ↥(Submodule.span K ↑s)

The submodule generated by a finset is finite-dimensional.

instance FiniteDimensional.instSubtypeMemSubmoduleMapLinearMapId (K : Type u) {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] (f : V →ₗ[K] V₂) (p : Submodule K V) [FiniteDimensional K ↥p] : FiniteDimensional K ↥(Submodule.map f p)

Pushforwards of finite-dimensional submodules are finite-dimensional.

theorem FiniteDimensional.exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card {L : Type u_1} [LinearOrderedField L] {W : Type v} [AddCommGroup W] [Module L W] [FiniteDimensional L W] {t : Finset W} (h : Module.finrank L W + 1 < t.card) : ∃ (f : W → L), ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x

A slight strengthening of exists_nontrivial_relation_sum_zero_of_rank_succ_lt_card available when working over an ordered field: we can ensure a positive coefficient, not just a nonzero coefficient.

noncomputable def FiniteDimensional.basisSingleton {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (ι : Type u_1) [Unique ι] (h : Module.finrank K V = 1) (v : V) (hv : v ≠ 0) : Basis ι K V

In a vector space with dimension 1, each set {v} is a basis for v ≠ 0.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem FiniteDimensional.basisSingleton_repr_apply {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (ι : Type u_1) [Unique ι] (h : Module.finrank K V = 1) (v : V) (hv : v ≠ 0) (w : V) : (basisSingleton ι h v hv).repr w = Finsupp.single default (((Module.basisUnique ι h).repr w) default / ((Module.basisUnique ι h).repr v) default)

@[simp]

theorem FiniteDimensional.basisSingleton_apply {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (ι : Type u_1) [Unique ι] (h : Module.finrank K V = 1) (v : V) (hv : v ≠ 0) (i : ι) : (basisSingleton ι h v hv) i = v

@[simp]

theorem FiniteDimensional.range_basisSingleton {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (ι : Type u_1) [Unique ι] (h : Module.finrank K V = 1) (v : V) (hv : v ≠ 0) : Set.range ⇑(basisSingleton ι h v hv) = {v}

theorem FiniteDimensional.trans (F : Type u_1) (K : Type u_2) (A : Type u_3) [DivisionRing F] [DivisionRing K] [AddCommGroup A] [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [FiniteDimensional F K] [FiniteDimensional K A] : FiniteDimensional F A

theorem FiniteDimensional.of_rank_eq_nat {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : ℕ} (h : Module.rank K V = ↑n) : FiniteDimensional K V

theorem FiniteDimensional.of_rank_eq_zero {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : Module.rank K V = 0) : FiniteDimensional K V

theorem FiniteDimensional.of_rank_eq_one {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : Module.rank K V = 1) : FiniteDimensional K V

instance finiteDimensional_bot (K : Type u) (V : Type v) [DivisionRing K] [AddCommGroup V] [Module K V] : FiniteDimensional K ↥⊥

theorem Submodule.fg_iff_finiteDimensional {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Submodule K V) : s.FG ↔ FiniteDimensional K ↥s

A submodule is finitely generated if and only if it is finite-dimensional

theorem Submodule.finiteDimensional_of_le {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {S₁ S₂ : Submodule K V} [FiniteDimensional K ↥S₂] (h : S₁ ≤ S₂) : FiniteDimensional K ↥S₁

A submodule contained in a finite-dimensional submodule is finite-dimensional.

instance Submodule.finiteDimensional_inf_left {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (S₁ S₂ : Submodule K V) [FiniteDimensional K ↥S₁] : FiniteDimensional K ↥(S₁ ⊓ S₂)

The inf of two submodules, the first finite-dimensional, is finite-dimensional.

instance Submodule.finiteDimensional_inf_right {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (S₁ S₂ : Submodule K V) [FiniteDimensional K ↥S₂] : FiniteDimensional K ↥(S₁ ⊓ S₂)

The inf of two submodules, the second finite-dimensional, is finite-dimensional.

instance Submodule.finiteDimensional_sup {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (S₁ S₂ : Submodule K V) [h₁ : FiniteDimensional K ↥S₁] [h₂ : FiniteDimensional K ↥S₂] : FiniteDimensional K ↥(S₁ ⊔ S₂)

The sup of two finite-dimensional submodules is finite-dimensional.

instance Submodule.finiteDimensional_finset_sup {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_1} (s : Finset ι) (S : ι → Submodule K V) [∀ (i : ι), FiniteDimensional K ↥(S i)] : FiniteDimensional K ↥(s.sup S)

The submodule generated by a finite supremum of finite dimensional submodules is finite-dimensional.

Note that strictly this only needs ∀ i ∈ s, FiniteDimensional K (S i), but that doesn't work well with typeclass search.

instance Submodule.finiteDimensional_iSup {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Sort u_1} [Finite ι] (S : ι → Submodule K V) [∀ (i : ι), FiniteDimensional K ↥(S i)] : FiniteDimensional K ↥(⨆ (i : ι), S i)

The submodule generated by a supremum of finite dimensional submodules, indexed by a finite sort is finite-dimensional.

theorem LinearEquiv.finiteDimensional {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] (f : V ≃ₗ[K] V₂) [FiniteDimensional K V] : FiniteDimensional K V₂

Finite dimensionality is preserved under linear equivalence.

instance finiteDimensional_finsupp {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_1} [Finite ι] [FiniteDimensional K V] : FiniteDimensional K (ι →₀ V)

theorem Submodule.eq_of_le_of_finrank_le {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {S₁ S₂ : Submodule K V} [FiniteDimensional K ↥S₂] (hle : S₁ ≤ S₂) (hd : Module.finrank K ↥S₂ ≤ Module.finrank K ↥S₁) : S₁ = S₂

If a submodule is contained in a finite-dimensional submodule with the same or smaller dimension, they are equal.

theorem Submodule.eq_of_le_of_finrank_eq {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {S₁ S₂ : Submodule K V} [FiniteDimensional K ↥S₂] (hle : S₁ ≤ S₂) (hd : Module.finrank K ↥S₁ = Module.finrank K ↥S₂) : S₁ = S₂

If a submodule is contained in a finite-dimensional submodule with the same dimension, they are equal.

theorem Subalgebra.eq_of_le_of_finrank_le {K : Type u_1} {L : Type u_2} [Field K] [Ring L] [Algebra K L] {F E : Subalgebra K L} [hfin : FiniteDimensional K ↥E] (h_le : F ≤ E) (h_finrank : Module.finrank K ↥E ≤ Module.finrank K ↥F) : F = E

If a subalgebra is contained in a finite-dimensional subalgebra with the same or smaller dimension, they are equal.

theorem Subalgebra.eq_of_le_of_finrank_eq {K : Type u_1} {L : Type u_2} [Field K] [Ring L] [Algebra K L] {F E : Subalgebra K L} [hfin : FiniteDimensional K ↥E] (h_le : F ≤ E) (h_finrank : Module.finrank K ↥F = Module.finrank K ↥E) : F = E

If a subalgebra is contained in a finite-dimensional subalgebra with the same dimension, they are equal.

theorem LinearMap.surjective_of_injective {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f : V →ₗ[K] V} (hinj : Function.Injective ⇑f) : Function.Surjective ⇑f

On a finite-dimensional space, an injective linear map is surjective.

theorem LinearMap.finiteDimensional_of_surjective {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] [FiniteDimensional K V] (f : V →ₗ[K] V₂) (hf : range f = ⊤) : FiniteDimensional K V₂

The image under an onto linear map of a finite-dimensional space is also finite-dimensional.

instance LinearMap.finiteDimensional_range {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂] [Module K V₂] [FiniteDimensional K V] (f : V →ₗ[K] V₂) : FiniteDimensional K ↥(range f)

The range of a linear map defined on a finite-dimensional space is also finite-dimensional.

theorem LinearMap.injective_iff_surjective {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f : V →ₗ[K] V} : Function.Injective ⇑f ↔ Function.Surjective ⇑f

On a finite-dimensional space, a linear map is injective if and only if it is surjective.

theorem LinearMap.injOn_iff_surjOn {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {p : Submodule K V} [FiniteDimensional K ↥p] {f : V →ₗ[K] V} (h : ∀ x ∈ p, f x ∈ p) : Set.InjOn ⇑f ↑p ↔ Set.SurjOn ⇑f ↑p ↑p

theorem LinearMap.ker_eq_bot_iff_range_eq_top {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f : V →ₗ[K] V} : ker f = ⊥ ↔ range f = ⊤

theorem LinearMap.mul_eq_one_of_mul_eq_one {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f g : V →ₗ[K] V} (hfg : f * g = 1) : g * f = 1

In a finite-dimensional space, if linear maps are inverse to each other on one side then they are also inverse to each other on the other side.

theorem LinearMap.mul_eq_one_comm {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f g : V →ₗ[K] V} : f * g = 1 ↔ g * f = 1

In a finite-dimensional space, linear maps are inverse to each other on one side if and only if they are inverse to each other on the other side.

theorem LinearMap.comp_eq_id_comm {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f g : V →ₗ[K] V} : f ∘ₗ g = id ↔ g ∘ₗ f = id

In a finite-dimensional space, linear maps are inverse to each other on one side if and only if they are inverse to each other on the other side.

theorem LinearMap.comap_eq_sup_ker_of_disjoint {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {p : Submodule K V} [FiniteDimensional K ↥p] {f : V →ₗ[K] V} (h : ∀ x ∈ p, f x ∈ p) (h' : Disjoint p (ker f)) : Submodule.comap f p = p ⊔ ker f

theorem LinearMap.ker_comp_eq_of_commute_of_disjoint_ker {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f g : V →ₗ[K] V} (h : Commute f g) (h' : Disjoint (ker f) (ker g)) : ker (f ∘ₗ g) = ker f ⊔ ker g

theorem LinearMap.ker_noncommProd_eq_of_supIndep_ker {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {ι : Type u_1} {f : ι → V →ₗ[K] V} (s : Finset ι) (comm : (↑s).Pairwise (Function.onFun Commute f)) (h : s.SupIndep fun (i : ι) => ker (f i)) : ker (s.noncommProd f comm) = ⨆ i ∈ s, ker (f i)

noncomputable def LinearEquiv.ofInjectiveEndo {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) (h_inj : Function.Injective ⇑f) : V ≃ₗ[K] V

The linear equivalence corresponding to an injective endomorphism.

Equations

• LinearEquiv.ofInjectiveEndo f h_inj = LinearEquiv.ofBijective f ⋯

@[simp]

theorem LinearEquiv.coe_ofInjectiveEndo {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) (h_inj : Function.Injective ⇑f) : ⇑(ofInjectiveEndo f h_inj) = ⇑f

@[simp]

theorem LinearEquiv.ofInjectiveEndo_right_inv {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) (h_inj : Function.Injective ⇑f) : f * ↑(ofInjectiveEndo f h_inj).symm = 1

@[simp]

theorem LinearEquiv.ofInjectiveEndo_left_inv {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) (h_inj : Function.Injective ⇑f) : ↑(ofInjectiveEndo f h_inj).symm * f = 1

theorem LinearMap.isUnit_iff_ker_eq_bot {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) : IsUnit f ↔ ker f = ⊥

theorem LinearMap.isUnit_iff_range_eq_top {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (f : V →ₗ[K] V) : IsUnit f ↔ range f = ⊤

theorem finrank_zero_iff_forall_zero {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] : Module.finrank K V = 0 ↔ ∀ (x : V), x = 0

noncomputable def basisOfFinrankZero {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {ι : Type u_1} [IsEmpty ι] (hV : Module.finrank K V = 0) : Basis ι K V

If ι is an empty type and V is zero-dimensional, there is a unique ι-indexed basis.

Equations

• basisOfFinrankZero hV = Basis.empty V

theorem FiniteDimensional.exists_mul_eq_one (F : Type u_1) {K : Type u_2} [Field F] [Ring K] [IsDomain K] [Algebra F K] [FiniteDimensional F K] {x : K} (H : x ≠ 0) : ∃ (y : K), x * y = 1

noncomputable def divisionRingOfFiniteDimensional (F : Type u_1) (K : Type u_2) [Field F] [Ring K] [IsDomain K] [Algebra F K] [FiniteDimensional F K] : DivisionRing K

A domain that is module-finite as an algebra over a field is a division ring.

Equations

• divisionRingOfFiniteDimensional F K = DivisionRing.mk ⋯ zpowRec ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (fun (x : ℚ≥0) => HMul.hMul ↑x) ⋯ ⋯ (fun (x : ℚ) => HMul.hMul ↑x) ⋯

theorem FiniteDimensional.isUnit (F : Type u_1) {K : Type u_2} [Field F] [Ring K] [IsDomain K] [Algebra F K] [FiniteDimensional F K] {x : K} (H : x ≠ 0) : IsUnit x

noncomputable def fieldOfFiniteDimensional (F : Type u_1) (K : Type u_2) [Field F] [h : CommRing K] [IsDomain K] [Algebra F K] [FiniteDimensional F K] : Field K

An integral domain that is module-finite as an algebra over a field is a field.

Equations

• fieldOfFiniteDimensional F K = Field.mk ⋯ DivisionRing.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionRing.nnqsmul ⋯ ⋯ DivisionRing.qsmul ⋯

theorem finrank_span_singleton {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {v : V} (hv : v ≠ 0) : Module.finrank K ↥(Submodule.span K {v}) = 1

theorem exists_smul_eq_of_finrank_eq_one {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : Module.finrank K V = 1) {x : V} (hx : x ≠ 0) (y : V) : ∃ (c : K), c • x = y

In a one-dimensional space, any vector is a multiple of any nonzero vector

theorem Set.finrank_mono {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {s t : Set V} (h : s ⊆ t) : Set.finrank K s ≤ Set.finrank K t

We now give characterisations of finrank K V = 1 and finrank K V ≤ 1.

theorem finrank_eq_one_iff_of_nonzero {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (v : V) (nz : v ≠ 0) : Module.finrank K V = 1 ↔ Submodule.span K {v} = ⊤

A vector space with a nonzero vector v has dimension 1 iff v spans.

theorem finrank_eq_one_iff_of_nonzero' {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (v : V) (nz : v ≠ 0) : Module.finrank K V = 1 ↔ ∀ (w : V), ∃ (c : K), c • v = w

A module with a nonzero vector v has dimension 1 iff every vector is a multiple of v.

theorem surjective_of_nonzero_of_finrank_eq_one {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {W : Type u_1} {A : Type u_2} [Semiring A] [Module A V] [AddCommGroup W] [Module K W] [Module A W] [LinearMap.CompatibleSMul V W K A] (h : Module.finrank K W = 1) {f : V →ₗ[A] W} (w : f ≠ 0) : Function.Surjective ⇑f

theorem Subalgebra.finiteDimensional_toSubmodule {F : Type u_1} {E : Type u_2} [Field F] [Ring E] [Algebra F E] {S : Subalgebra F E} : FiniteDimensional F ↥(toSubmodule S) ↔ FiniteDimensional F ↥S

A Subalgebra is FiniteDimensional iff it is FiniteDimensional as a submodule.

theorem FiniteDimensional.of_subalgebra_toSubmodule {F : Type u_1} {E : Type u_2} [Field F] [Ring E] [Algebra F E] {S : Subalgebra F E} : FiniteDimensional F ↥(Subalgebra.toSubmodule S) → FiniteDimensional F ↥S

Alias of the forward direction of Subalgebra.finiteDimensional_toSubmodule.

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A Subalgebra is FiniteDimensional iff it is FiniteDimensional as a submodule.

theorem FiniteDimensional.subalgebra_toSubmodule {F : Type u_1} {E : Type u_2} [Field F] [Ring E] [Algebra F E] {S : Subalgebra F E} : FiniteDimensional F ↥S → FiniteDimensional F ↥(Subalgebra.toSubmodule S)

Alias of the reverse direction of Subalgebra.finiteDimensional_toSubmodule.

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A Subalgebra is FiniteDimensional iff it is FiniteDimensional as a submodule.

instance FiniteDimensional.finiteDimensional_subalgebra {F : Type u_1} {E : Type u_2} [Field F] [Ring E] [Algebra F E] [FiniteDimensional F E] (S : Subalgebra F E) : FiniteDimensional F ↥S

theorem Module.End.ker_pow_constant {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {f : End K V} {k : ℕ} (h : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)) (m : ℕ) : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ (k + m))