Linear maps involving submodules of a module

In this file we define a number of linear maps involving submodules of a module.

Main declarations

• Submodule.subtype: Embedding of a submodule p to the ambient space M as a Submodule.
• LinearMap.domRestrict: The restriction of a semilinear map f : M → M₂ to a submodule p ⊆ M as a semilinear map p → M₂.
• LinearMap.restrict: The restriction of a linear map f : M → M₁ to a submodule p ⊆ M and q ⊆ M₁ (if q contains the codomain).
• Submodule.inclusion: the inclusion p ⊆ p' of submodules p and p' as a linear map.

Tags

submodule, subspace, linear map

def SMulMemClass.subtype {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {A : Type u_1} [SetLike A M] [AddSubmonoidClass A M] [SMulMemClass A R M] (S' : A) : ↥S' →ₗ[R] M

The natural R-linear map from a submodule of an R-module M to M.

Equations

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

@[simp]

theorem SMulMemClass.coeSubtype {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {A : Type u_1} [SetLike A M] [AddSubmonoidClass A M] [SMulMemClass A R M] (S' : A) : ⇑(SMulMemClass.subtype S') = Subtype.val

def Submodule.subtype {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : ↥p →ₗ[R] M

Embedding of a submodule p to the ambient space M.

Equations

• Submodule.subtype p = { toAddHom := { toFun := Subtype.val, map_add' := (_ : ∀ (a a_1 : ↥p), ↑a + ↑a_1 = ↑a + ↑a_1) }, map_smul' := (_ : ∀ (a : R) (a_1 : ↥p), a • ↑a_1 = a • ↑a_1) }

theorem Submodule.subtype_apply {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) (x : ↥p) : (Submodule.subtype p) x = ↑x

@[simp]

theorem Submodule.coeSubtype {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : ⇑(Submodule.subtype p) = Subtype.val

theorem Submodule.injective_subtype {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Function.Injective ⇑(Submodule.subtype p)

theorem Submodule.coe_sum {R : Type u} {M : Type v} {ι : Type w} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) (x : ι → ↥p) (s : Finset ι) : ↑(Finset.sum s fun (i : ι) => x i) = Finset.sum s fun (i : ι) => ↑(x i)

Note the AddSubmonoid version of this lemma is called AddSubmonoid.coe_finset_sum.

instance Submodule.instAddActionSubtypeMemSubmoduleInstMembershipSetLikeToAddMonoidToAddMonoidToAddSubmonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {α : Type u_1} [AddAction M α] : AddAction (↥p) α

The action by a submodule is the action by the underlying module.

Equations

• Submodule.instAddActionSubtypeMemSubmoduleInstMembershipSetLikeToAddMonoidToAddMonoidToAddSubmonoid p = AddAction.compHom α (LinearMap.toAddMonoidHom (Submodule.subtype p))

@[simp]

theorem Submodule.restrictScalarsEquiv_symm_apply (S : Type u') (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) : ∀ (a : ↥p), (LinearEquiv.symm (Submodule.restrictScalarsEquiv S R M p)) a = a

@[simp]

theorem Submodule.restrictScalarsEquiv_apply (S : Type u') (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) : ∀ (a : ↥p), (Submodule.restrictScalarsEquiv S R M p) a = a

def Submodule.restrictScalarsEquiv (S : Type u') (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) : ↥(Submodule.restrictScalars S p) ≃ₗ[R] ↥p

Turning p : Submodule R M into an S-submodule gives the same module structure as turning it into a type and adding a module structure.

Equations

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

def LinearMap.domRestrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) : ↥p →ₛₗ[σ₁₂] M₂

The restriction of a linear map f : M → M₂ to a submodule p ⊆ M gives a linear map p → M₂.

Equations

• LinearMap.domRestrict f p = LinearMap.comp f (Submodule.subtype p)

@[simp]

theorem LinearMap.domRestrict_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) (x : ↥p) : (LinearMap.domRestrict f p) x = f ↑x

def LinearMap.codRestrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (p : Submodule R₂ M₂) (f : M →ₛₗ[σ₁₂] M₂) (h : ∀ (c : M), f c ∈ p) : M →ₛₗ[σ₁₂] ↥p

A linear map f : M₂ → M whose values lie in a submodule p ⊆ M can be restricted to a linear map M₂ → p.

Equations

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

@[simp]

theorem LinearMap.codRestrict_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (p : Submodule R₂ M₂) (f : M →ₛₗ[σ₁₂] M₂) {h : ∀ (c : M), f c ∈ p} (x : M) : ↑((LinearMap.codRestrict p f h) x) = f x

@[simp]

theorem LinearMap.comp_codRestrict {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_5} {M₂ : Type u_7} {M₃ : Type u_8} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R₃ M₃) (h : ∀ (b : M₂), g b ∈ p) : LinearMap.comp (LinearMap.codRestrict p g h) f = LinearMap.codRestrict p (LinearMap.comp g f) (_ : ∀ (x : M), g (f x) ∈ p)

@[simp]

theorem LinearMap.subtype_comp_codRestrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (b : M), f b ∈ p) : LinearMap.comp (Submodule.subtype p) (LinearMap.codRestrict p f h) = f

def LinearMap.restrict {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] (f : M →ₗ[R] M₁) {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ x ∈ p, f x ∈ q) : ↥p →ₗ[R] ↥q

Restrict domain and codomain of a linear map.

Equations

• LinearMap.restrict f hf = LinearMap.codRestrict q (LinearMap.domRestrict f p) (_ : ∀ (x : ↥p), (LinearMap.domRestrict f p) x ∈ q)

@[simp]

theorem LinearMap.restrict_coe_apply {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] (f : M →ₗ[R] M₁) {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ x ∈ p, f x ∈ q) (x : ↥p) : ↑((LinearMap.restrict f hf) x) = f ↑x

theorem LinearMap.restrict_apply {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] {f : M →ₗ[R] M₁} {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ x ∈ p, f x ∈ q) (x : ↥p) : (LinearMap.restrict f hf) x = { val := f ↑x, property := (_ : f ↑x ∈ q) }

theorem LinearMap.restrict_comp {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {M₂ : Type u_10} {M₃ : Type u_11} [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {p : Submodule R M} {p₂ : Submodule R M₂} {p₃ : Submodule R M₃} {f : M →ₗ[R] M₂} {g : M₂ →ₗ[R] M₃} (hf : Set.MapsTo ⇑f ↑p ↑p₂) (hg : Set.MapsTo ⇑g ↑p₂ ↑p₃) (hfg : optParam (Set.MapsTo ⇑(g ∘ₗ f) ↑p ↑p₃) (_ : Set.MapsTo (⇑g ∘ fun (x : M) => f x) ↑p ↑p₃)) : LinearMap.restrict (g ∘ₗ f) hfg = LinearMap.restrict g hg ∘ₗ LinearMap.restrict f hf

theorem LinearMap.restrict_commute {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f : M →ₗ[R] M} {g : M →ₗ[R] M} (h : Commute f g) {p : Submodule R M} (hf : Set.MapsTo ⇑f ↑p ↑p) (hg : Set.MapsTo ⇑g ↑p ↑p) : Commute (LinearMap.restrict f hf) (LinearMap.restrict g hg)

theorem LinearMap.subtype_comp_restrict {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] {f : M →ₗ[R] M₁} {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ x ∈ p, f x ∈ q) : Submodule.subtype q ∘ₗ LinearMap.restrict f hf = LinearMap.domRestrict f p

theorem LinearMap.restrict_eq_codRestrict_domRestrict {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] {f : M →ₗ[R] M₁} {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ x ∈ p, f x ∈ q) : LinearMap.restrict f hf = LinearMap.codRestrict q (LinearMap.domRestrict f p) (_ : ∀ (x : ↥p), f ↑x ∈ q)

theorem LinearMap.restrict_eq_domRestrict_codRestrict {R : Type u_1} {M : Type u_5} {M₁ : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] {f : M →ₗ[R] M₁} {p : Submodule R M} {q : Submodule R M₁} (hf : ∀ (x : M), f x ∈ q) : LinearMap.restrict f (_ : ∀ x ∈ p, f x ∈ q) = LinearMap.domRestrict (LinearMap.codRestrict q f hf) p

instance LinearMap.uniqueOfLeft {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Subsingleton M] : Unique (M →ₛₗ[σ₁₂] M₂)

Equations

• LinearMap.uniqueOfLeft = let src := inferInstanceAs (Inhabited (M →ₛₗ[σ₁₂] M₂)); { toInhabited := { default := default }, uniq := (_ : ∀ (f : M →ₛₗ[σ₁₂] M₂), f = default) }

instance LinearMap.uniqueOfRight {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Subsingleton M₂] : Unique (M →ₛₗ[σ₁₂] M₂)

Equations

• LinearMap.uniqueOfRight = Function.Injective.unique (_ : Function.Injective FunLike.coe)

@[simp]

theorem LinearMap.evalAddMonoidHom_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (a : M) (f : M →ₛₗ[σ₁₂] M₂) : (LinearMap.evalAddMonoidHom a) f = f a

def LinearMap.evalAddMonoidHom {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (a : M) : (M →ₛₗ[σ₁₂] M₂) →+ M₂

Evaluation of a σ₁₂-linear map at a fixed a, as an AddMonoidHom.

Equations

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

@[simp]

theorem LinearMap.toAddMonoidHom'_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) : LinearMap.toAddMonoidHom' f = LinearMap.toAddMonoidHom f

def LinearMap.toAddMonoidHom' {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} : (M →ₛₗ[σ₁₂] M₂) →+ M →+ M₂

LinearMap.toAddMonoidHom promoted to an AddMonoidHom.

Equations

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

theorem LinearMap.sum_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} {ι : Type u_9} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (t : Finset ι) (f : ι → M →ₛₗ[σ₁₂] M₂) (b : M) : (Finset.sum t fun (d : ι) => f d) b = Finset.sum t fun (d : ι) => (f d) b

instance LinearMap.instNontrivialEnd {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial M] : Nontrivial (Module.End R M)

Equations

• (_ : Nontrivial (Module.End R M)) = (_ : Nontrivial (Module.End R M))

@[simp]

theorem LinearMap.coeFn_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {ι : Type u_10} (t : Finset ι) (f : ι → M →ₛₗ[σ₁₂] M₂) : ⇑(Finset.sum t fun (i : ι) => f i) = Finset.sum t fun (i : ι) => ⇑(f i)

theorem LinearMap.submodule_pow_eq_zero_of_pow_eq_zero {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {g : Module.End R ↥N} {G : Module.End R M} (h : G ∘ₗ Submodule.subtype N = Submodule.subtype N ∘ₗ g) {k : ℕ} (hG : G ^ k = 0) : g ^ k = 0

theorem LinearMap.pow_apply_mem_of_forall_mem {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} {p : Submodule R M} (n : ℕ) (h : ∀ x ∈ p, f' x ∈ p) (x : M) (hx : x ∈ p) : (f' ^ n) x ∈ p

theorem LinearMap.pow_restrict {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {f' : M →ₗ[R] M} {p : Submodule R M} (n : ℕ) (h : ∀ x ∈ p, f' x ∈ p) (h' : optParam (∀ x ∈ p, (f' ^ n) x ∈ p) (_ : ∀ x ∈ p, (f' ^ n) x ∈ p)) : LinearMap.restrict f' h ^ n = LinearMap.restrict (f' ^ n) h'

def LinearMap.domRestrict' {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) : (M →ₗ[R] M₂) →ₗ[R] ↥p →ₗ[R] M₂

Alternative version of domRestrict as a linear map.

Equations

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

@[simp]

theorem LinearMap.domRestrict'_apply {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M →ₗ[R] M₂) (p : Submodule R M) (x : ↥p) : ((LinearMap.domRestrict' p) f) x = f ↑x

def Submodule.inclusion {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {p' : Submodule R M} (h : p ≤ p') : ↥p →ₗ[R] ↥p'

If two submodules p and p' satisfy p ⊆ p', then inclusion p p' is the linear map version of this inclusion.

Equations

• Submodule.inclusion h = LinearMap.codRestrict p' (Submodule.subtype p) (_ : ∀ (x : ↥p), (Submodule.subtype p) x ∈ p')

@[simp]

theorem Submodule.coe_inclusion {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {p' : Submodule R M} (h : p ≤ p') (x : ↥p) : ↑((Submodule.inclusion h) x) = ↑x

theorem Submodule.inclusion_apply {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {p' : Submodule R M} (h : p ≤ p') (x : ↥p) : (Submodule.inclusion h) x = { val := ↑x, property := (_ : ↑x ∈ p') }

theorem Submodule.inclusion_injective {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {p' : Submodule R M} (h : p ≤ p') : Function.Injective ⇑(Submodule.inclusion h)

theorem Submodule.subtype_comp_inclusion {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (q : Submodule R M) (h : p ≤ q) : Submodule.subtype q ∘ₗ Submodule.inclusion h = Submodule.subtype p