Bases in a vector space

This file provides results for bases of a vector space.

Some of these results should be merged with the results on free modules. We state these results in a separate file to the results on modules to avoid an import cycle.

Main statements

• Basis.ofVectorSpace states that every vector space has a basis.
• Module.Free.of_divisionRing states that every vector space is a free module.

Tags

basis, bases

noncomputable def Basis.extend {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndependent (ι := { x : V // x ∈ s }) K Subtype.val) : Basis (↑(LinearIndependent.extend hs (_ : s ⊆ Set.univ))) K V

If s is a linear independent set of vectors, we can extend it to a basis.

Equations

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

theorem Basis.extend_apply_self {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndependent (ι := { x : V // x ∈ s }) K Subtype.val) (x : ↑(LinearIndependent.extend hs (_ : s ⊆ Set.univ))) : (Basis.extend hs) x = ↑x

@[simp]

theorem Basis.coe_extend {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndependent (ι := { x : V // x ∈ s }) K Subtype.val) : ⇑(Basis.extend hs) = Subtype.val

theorem Basis.range_extend {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndependent (ι := { x : V // x ∈ s }) K Subtype.val) : Set.range ⇑(Basis.extend hs) = LinearIndependent.extend hs (_ : s ⊆ Set.univ)

def Basis.sumExtendIndex {ι : Type u_1} {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} (hs : LinearIndependent K v) : Set V

Auxiliary definition: the index for the new basis vectors in Basis.sumExtend.

The specific value of this definition should be considered an implementation detail.

Equations

• Basis.sumExtendIndex hs = LinearIndependent.extend (_ : LinearIndependent (ι := { x : V // x ∈ Set.range v }) K Subtype.val) (_ : Set.range v ⊆ Set.univ) \ Set.range v

noncomputable def Basis.sumExtend {ι : Type u_1} {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} (hs : LinearIndependent K v) : Basis (ι ⊕ ↑(Basis.sumExtendIndex hs)) K V

If v is a linear independent family of vectors, extend it to a basis indexed by a sum type.

Equations

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

theorem Basis.subset_extend {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndependent (ι := { x : V // x ∈ s }) K Subtype.val) : s ⊆ LinearIndependent.extend hs (_ : s ⊆ Set.univ)

noncomputable def Basis.ofVectorSpaceIndex (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : Set V

A set used to index Basis.ofVectorSpace.

Equations

• Basis.ofVectorSpaceIndex K V = LinearIndependent.extend (_ : LinearIndependent K fun (x : ↑∅) => ↑x) (_ : ∅ ⊆ Set.univ)

noncomputable def Basis.ofVectorSpace (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V

Each vector space has a basis.

Equations

• Basis.ofVectorSpace K V = Basis.extend (_ : LinearIndependent K fun (x : ↑∅) => ↑x)

instance Module.Free.of_divisionRing (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : Module.Free K V

Equations

• (_ : Module.Free K V) = (_ : Module.Free K V)

theorem Basis.ofVectorSpace_apply_self (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] (x : ↑(Basis.ofVectorSpaceIndex K V)) : (Basis.ofVectorSpace K V) x = ↑x

@[simp]

theorem Basis.coe_ofVectorSpace (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : ⇑(Basis.ofVectorSpace K V) = Subtype.val

theorem Basis.ofVectorSpaceIndex.linearIndependent (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : LinearIndependent (ι := { x : V // x ∈ Basis.ofVectorSpaceIndex K V }) K Subtype.val

theorem Basis.range_ofVectorSpace (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : Set.range ⇑(Basis.ofVectorSpace K V) = Basis.ofVectorSpaceIndex K V

theorem Basis.exists_basis (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] : ∃ (s : Set V), Nonempty (Basis (↑s) K V)

theorem VectorSpace.card_fintype (K : Type u_3) (V : Type u_4) [DivisionRing K] [AddCommGroup V] [Module K V] [Fintype K] [Fintype V] : ∃ (n : ℕ), Fintype.card V = Fintype.card K ^ n

theorem nonzero_span_atom {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] (v : V) (hv : v ≠ 0) : IsAtom (Submodule.span K {v})

For a module over a division ring, the span of a nonzero element is an atom of the lattice of submodules.

theorem atom_iff_nonzero_span {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] (W : Submodule K V) : IsAtom W ↔ ∃ (v : V), v ≠ 0 ∧ W = Submodule.span K {v}

The atoms of the lattice of submodules of a module over a division ring are the submodules equal to the span of a nonzero element of the module.

instance instIsAtomisticSubmoduleToSemiringToDivisionSemiringToAddCommMonoidCompleteLattice {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] : IsAtomistic (Submodule K V)

The lattice of submodules of a module over a division ring is atomistic.

Equations

• (_ : IsAtomistic (Submodule K V)) = (_ : IsAtomistic (Submodule K V))

theorem LinearMap.exists_leftInverse_of_injective {K : Type u_3} {V : Type u_4} {V' : Type u_5} [DivisionRing K] [AddCommGroup V] [AddCommGroup V'] [Module K V] [Module K V'] (f : V →ₗ[K] V') (hf_inj : LinearMap.ker f = ⊥) : ∃ (g : V' →ₗ[K] V), g ∘ₗ f = LinearMap.id

theorem Submodule.exists_isCompl {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] (p : Submodule K V) : ∃ (q : Submodule K V), IsCompl p q

instance Module.Submodule.complementedLattice {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] : ComplementedLattice (Submodule K V)

Equations

• (_ : ComplementedLattice (Submodule K V)) = (_ : ComplementedLattice (Submodule K V))

theorem LinearMap.exists_rightInverse_of_surjective {K : Type u_3} {V : Type u_4} {V' : Type u_5} [DivisionRing K] [AddCommGroup V] [AddCommGroup V'] [Module K V] [Module K V'] (f : V →ₗ[K] V') (hf_surj : LinearMap.range f = ⊤) : ∃ (g : V' →ₗ[K] V), f ∘ₗ g = LinearMap.id

theorem LinearMap.exists_extend {K : Type u_3} {V : Type u_4} {V' : Type u_5} [DivisionRing K] [AddCommGroup V] [AddCommGroup V'] [Module K V] [Module K V'] {p : Submodule K V} (f : ↥p →ₗ[K] V') : ∃ (g : V →ₗ[K] V'), g ∘ₗ Submodule.subtype p = f

Any linear map f : p →ₗ[K] V' defined on a subspace p can be extended to the whole space.

theorem Submodule.exists_le_ker_of_lt_top {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] (p : Submodule K V) (hp : p < ⊤) : ∃ (f : V →ₗ[K] K), f ≠ 0 ∧ p ≤ LinearMap.ker f

If p < ⊤ is a subspace of a vector space V, then there exists a nonzero linear map f : V →ₗ[K] K such that p ≤ ker f.

theorem quotient_prod_linearEquiv {K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] (p : Submodule K V) : Nonempty (((V ⧸ p) × ↥p) ≃ₗ[K] V)