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Mathlib.LinearAlgebra.Dimension.DivisionRing

Dimension of vector spaces #

In this file we provide results about Module.rank and FiniteDimensional.finrank of vector spaces over division rings.

Main statements #

For vector spaces (i.e. modules over a field), we have

If a vector space has a finite dimension, the index set of Basis.ofVectorSpace is finite.

theorem rank_quotient_add_rank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (p : Submodule K V) :
theorem rank_range_add_rank_ker {K : Type u} {V : Type v} {V₁ : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₁] [Module K V₁] (f : V →ₗ[K] V₁) :

The rank-nullity theorem

theorem rank_eq_of_surjective {K : Type u} {V : Type v} {V₁ : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₁] [Module K V₁] (f : V →ₗ[K] V₁) (h : Function.Surjective f) :
theorem exists_linearIndependent_cons_of_lt_rank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : } {v : Fin nV} (hv : LinearIndependent K v) (h : n < Module.rank K V) :
∃ (x : V), LinearIndependent K (Fin.cons x v)

Given a family of n linearly independent vectors in a space of dimension > n, one may extend the family by another vector while retaining linear independence.

theorem exists_linearIndependent_snoc_of_lt_rank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : } {v : Fin nV} (hv : LinearIndependent K v) (h : n < Module.rank K V) :
∃ (x : V), LinearIndependent K (Fin.snoc v x)

Given a family of n linearly independent vectors in a space of dimension > n, one may extend the family by another vector while retaining linear independence.

theorem exists_linearIndependent_pair_of_one_lt_rank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : 1 < Module.rank K V) {x : V} (hx : x 0) :
∃ (y : V), LinearIndependent K ![x, y]

Given a nonzero vector in a space of dimension > 1, one may find another vector linearly independent of the first one.

theorem rank_add_rank_split {K : Type u} {V : Type v} {V₁ : Type v} {V₂ : Type v} {V₃ : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] [AddCommGroup V₃] [Module K V₃] (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂) (ce : V₁ →ₗ[K] V₃) (hde : LinearMap.range db LinearMap.range eb) (hgd : LinearMap.ker cd = ) (eq : db ∘ₗ cd = eb ∘ₗ ce) (eq₂ : ∀ (d : V₂) (e : V₃), db d = eb e∃ (c : V₁), cd c = d ce c = e) :

This is mostly an auxiliary lemma for Submodule.rank_sup_add_rank_inf_eq.

theorem Submodule.rank_sup_add_rank_inf_eq {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Submodule K V) (t : Submodule K V) :
Module.rank K (s t) + Module.rank K (s t) = Module.rank K s + Module.rank K t
theorem Submodule.rank_add_le_rank_add_rank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Submodule K V) (t : Submodule K V) :
Module.rank K (s t) Module.rank K s + Module.rank K t
def Basis.ofRankEqZero {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [IsEmpty ι] (hV : Module.rank K V = 0) :
Basis ι K V

The ι indexed basis on V, where ι is an empty type and V is zero-dimensional.

See also FiniteDimensional.finBasis.

Equations
Instances For
    @[simp]
    theorem Basis.ofRankEqZero_apply {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [IsEmpty ι] (hV : Module.rank K V = 0) (i : ι) :
    theorem le_rank_iff_exists_linearIndependent {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {c : Cardinal.{v}} :
    c Module.rank K V ∃ (s : Set V), Cardinal.mk s = c LinearIndependent (ι := { x : V // x s }) K Subtype.val
    theorem le_rank_iff_exists_linearIndependent_finset {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : } :
    n Module.rank K V ∃ (s : Finset V), s.card = n LinearIndependent (ι := { x : V // x s }) K Subtype.val
    theorem rank_le_one_iff {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] :
    Module.rank K V 1 ∃ (v₀ : V), ∀ (v : V), ∃ (r : K), r v₀ = v

    A vector space has dimension at most 1 if and only if there is a single vector of which all vectors are multiples.

    theorem rank_submodule_le_one_iff {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Submodule K V) :
    Module.rank K s 1 ∃ v₀ ∈ s, s Submodule.span K {v₀}

    A submodule has dimension at most 1 if and only if there is a single vector in the submodule such that the submodule is contained in its span.

    theorem rank_submodule_le_one_iff' {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (s : Submodule K V) :
    Module.rank K s 1 ∃ (v₀ : V), s Submodule.span K {v₀}

    A submodule has dimension at most 1 if and only if there is a single vector, not necessarily in the submodule, such that the submodule is contained in its span.

    theorem exists_linearIndependent_snoc_of_lt_finrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : } {v : Fin nV} (hv : LinearIndependent K v) (h : n < FiniteDimensional.finrank K V) :
    ∃ (x : V), LinearIndependent K (Fin.snoc v x)

    Given a family of n linearly independent vectors in a finite-dimensional space of dimension > n, one may extend the family by another vector while retaining linear independence.

    theorem exists_linearIndependent_cons_of_lt_finrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {n : } {v : Fin nV} (hv : LinearIndependent K v) (h : n < FiniteDimensional.finrank K V) :
    ∃ (x : V), LinearIndependent K (Fin.cons x v)

    Given a family of n linearly independent vectors in a finite-dimensional space of dimension > n, one may extend the family by another vector while retaining linear independence.

    theorem exists_linearIndependent_pair_of_one_lt_finrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] (h : 1 < FiniteDimensional.finrank K V) {x : V} (hx : x 0) :
    ∃ (y : V), LinearIndependent K ![x, y]

    Given a nonzero vector in a finite-dimensional space of dimension > 1, one may find another vector linearly independent of the first one.

    theorem linearIndependent_iff_card_eq_finrank_span {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] {b : ιV} :

    A finite family of vectors is linearly independent if and only if its cardinality equals the dimension of its span.

    noncomputable def basisOfTopLeSpanOfCardEqFinrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] (b : ιV) (le_span : Submodule.span K (Set.range b)) (card_eq : Fintype.card ι = FiniteDimensional.finrank K V) :
    Basis ι K V

    A family of finrank K V vectors forms a basis if they span the whole space.

    Equations
    Instances For
      @[simp]
      theorem coe_basisOfTopLeSpanOfCardEqFinrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {ι : Type u_2} [Fintype ι] (b : ιV) (le_span : Submodule.span K (Set.range b)) (card_eq : Fintype.card ι = FiniteDimensional.finrank K V) :
      (basisOfTopLeSpanOfCardEqFinrank b le_span card_eq) = b
      @[simp]
      theorem finsetBasisOfTopLeSpanOfCardEqFinrank_repr_apply {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Finset V} (le_span : Submodule.span K s) (card_eq : s.card = FiniteDimensional.finrank K V) :
      ∀ (a : V), (finsetBasisOfTopLeSpanOfCardEqFinrank le_span card_eq).repr a = (LinearIndependent.repr (_ : LinearIndependent K Subtype.val)) ((LinearMap.codRestrict (Submodule.span K (Set.range Subtype.val)) LinearMap.id (_ : ∀ (x : V), LinearMap.id x Submodule.span K (Set.range Subtype.val))) a)
      noncomputable def finsetBasisOfTopLeSpanOfCardEqFinrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Finset V} (le_span : Submodule.span K s) (card_eq : s.card = FiniteDimensional.finrank K V) :
      Basis { x : V // x s } K V

      A finset of finrank K V vectors forms a basis if they span the whole space.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[simp]
        theorem setBasisOfTopLeSpanOfCardEqFinrank_repr_apply {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} [Fintype s] (le_span : Submodule.span K s) (card_eq : (Set.toFinset s).card = FiniteDimensional.finrank K V) :
        ∀ (a : V), (setBasisOfTopLeSpanOfCardEqFinrank le_span card_eq).repr a = (LinearIndependent.repr (_ : LinearIndependent (ι := s) K Subtype.val)) ((LinearMap.codRestrict (Submodule.span K (Set.range Subtype.val)) LinearMap.id (_ : ∀ (x : V), LinearMap.id x Submodule.span K (Set.range Subtype.val))) a)
        noncomputable def setBasisOfTopLeSpanOfCardEqFinrank {K : Type u} {V : Type v} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} [Fintype s] (le_span : Submodule.span K s) (card_eq : (Set.toFinset s).card = FiniteDimensional.finrank K V) :
        Basis (s) K V

        A set of finrank K V vectors forms a basis if they span the whole space.

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

          Key lemma towards the Erdős-Kaplansky theorem from https://mathoverflow.net/a/168624

          theorem rank_fun_infinite {K : Type u} [DivisionRing K] {ι : Type v} [hι : Infinite ι] :
          Module.rank K (ιK) = Cardinal.mk (ιK)

          The Erdős-Kaplansky Theorem: the dual of an infinite-dimensional vector space over a division ring has dimension equal to its cardinality.

          The Erdős-Kaplansky Theorem over a field.