Documentation

Mathlib.Algebra.Module.LinearMap.Defs

(Semi)linear maps #

In this file we define

We then provide LinearMap with the following instances:

Implementation notes #

To ensure that composition works smoothly for semilinear maps, we use the typeclasses RingHomCompTriple, RingHomInvPair and RingHomSurjective from Mathlib.Algebra.Ring.CompTypeclasses.

Notation #

TODO #

Tags #

linear map

structure IsLinearMap (R : Type u) {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : MM₂) :

A map f between modules over a semiring is linear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = c • f x. The predicate IsLinearMap R f asserts this property. A bundled version is available with LinearMap, and should be favored over IsLinearMap most of the time.

  • map_add : ∀ (x y : M), f (x + y) = f x + f y

    A linear map preserves addition.

  • map_smul : ∀ (c : R) (x : M), f (c x) = c f x

    A linear map preserves scalar multiplication.

Instances For
    theorem IsLinearMap.map_add {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {f : MM₂} (self : IsLinearMap R f) (x : M) (y : M) :
    f (x + y) = f x + f y

    A linear map preserves addition.

    theorem IsLinearMap.map_smul {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {f : MM₂} (self : IsLinearMap R f) (c : R) (x : M) :
    f (c x) = c f x

    A linear map preserves scalar multiplication.

    structure LinearMap {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] (σ : R →+* S) (M : Type u_19) (M₂ : Type u_20) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends AddHom :
    Type (max u_19 u_20)

    A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x. Elements of LinearMap σ M M₂ (available under the notation M →ₛₗ[σ] M₂) are bundled versions of such maps. For plain linear maps (i.e. for which σ = RingHom.id R), the notation M →ₗ[R] M₂ is available. An unbundled version of plain linear maps is available with the predicate IsLinearMap, but it should be avoided most of the time.

    • toFun : MM₂
    • map_add' : ∀ (x y : M), self.toFun (x + y) = self.toFun x + self.toFun y
    • map_smul' : ∀ (m : R) (x : M), self.toFun (m x) = σ m self.toFun x

      The proposition that the function commutes with the actions.

    Instances For
      @[reducible]
      abbrev LinearMap.toMulActionHom {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] {σ : R →+* S} {M : Type u_19} {M₂ : Type u_20} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M →ₛₗ[σ] M₂) :
      M →ₑ[σ] M₂

      The MulActionHom underlying a LinearMap.

      Equations
      • self.toMulActionHom = { toFun := self.toFun, map_smul' := }
      Instances For

        M →ₛₗ[σ] N is the type of σ-semilinear maps from M to N.

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

          M →ₗ[R] N is the type of R-linear maps from M to N.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            class SemilinearMapClass (F : Type u_17) {R : outParam (Type u_18)} {S : outParam (Type u_19)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) (M : outParam (Type u_20)) (M₂ : outParam (Type u_21)) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] [FunLike F M M₂] extends AddHomClass , MulActionSemiHomClass :

            SemilinearMapClass F σ M M₂ asserts F is a type of bundled σ-semilinear maps M → M₂.

            See also LinearMapClass F R M M₂ for the case where σ is the identity map on R.

            A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

              Instances
                @[reducible, inline]
                abbrev LinearMapClass (F : Type u_17) (R : outParam (Type u_18)) (M : Type u_19) (M₂ : Type u_20) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [FunLike F M M₂] :

                LinearMapClass F R M M₂ asserts F is a type of bundled R-linear maps M → M₂.

                This is an abbreviation for SemilinearMapClass F (RingHom.id R) M M₂.

                Equations
                Instances For
                  theorem LinearMapClass.map_smul {R : outParam (Type u_17)} {M : outParam (Type u_18)} {M₂ : outParam (Type u_19)} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {F : Type u_20} [FunLike F M M₂] [LinearMapClass F R M M₂] (f : F) (r : R) (x : M) :
                  f (r x) = r f x
                  @[instance 100]
                  instance SemilinearMapClass.instAddMonoidHomClass {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} (F : Type u_17) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] :
                  Equations
                  • =
                  @[instance 100]
                  instance SemilinearMapClass.distribMulActionSemiHomClass {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} (F : Type u_17) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] :
                  Equations
                  • =
                  theorem SemilinearMapClass.map_smul_inv {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} {F : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : F) [FunLike F M M₃] [SemilinearMapClass F σ M M₃] {σ' : S →+* R} [RingHomInvPair σ σ'] (c : S) (x : M) :
                  c f x = f (σ' c x)
                  def SemilinearMapClass.semilinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} {F : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : F) [FunLike F M M₃] [SemilinearMapClass F σ M M₃] :
                  M →ₛₗ[σ] M₃

                  Reinterpret an element of a type of semilinear maps as a semilinear map.

                  Equations
                  • f = { toFun := f, map_add' := , map_smul' := }
                  Instances For
                    instance SemilinearMapClass.instCoeToSemilinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} {F : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] :
                    CoeHead F (M →ₛₗ[σ] M₃)

                    Reinterpret an element of a type of semilinear maps as a semilinear map.

                    Equations
                    • SemilinearMapClass.instCoeToSemilinearMap = { coe := fun (f : F) => f }
                    @[reducible, inline]
                    abbrev LinearMapClass.linearMap {R : Type u_1} {M₁ : Type u_10} {M₂ : Type u_11} {F : Type u_17} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : F) [FunLike F M₁ M₂] [LinearMapClass F R M₁ M₂] :
                    M₁ →ₗ[R] M₂

                    Reinterpret an element of a type of linear maps as a linear map.

                    Equations
                    Instances For
                      instance LinearMapClass.instCoeToLinearMap {R : Type u_1} {M₁ : Type u_10} {M₂ : Type u_11} {F : Type u_17} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] [FunLike F M₁ M₂] [LinearMapClass F R M₁ M₂] :
                      CoeHead F (M₁ →ₗ[R] M₂)

                      Reinterpret an element of a type of linear maps as a linear map.

                      Equations
                      • LinearMapClass.instCoeToLinearMap = { coe := fun (f : F) => f }
                      instance LinearMap.instFunLike {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} :
                      FunLike (M →ₛₗ[σ] M₃) M M₃
                      Equations
                      • LinearMap.instFunLike = { coe := fun (f : M →ₛₗ[σ] M₃) => f.toFun, coe_injective' := }
                      instance LinearMap.semilinearMapClass {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} :
                      SemilinearMapClass (M →ₛₗ[σ] M₃) σ M M₃
                      Equations
                      • =
                      @[simp]
                      theorem LinearMap.coe_coe {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {F : Type u_17} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] {f : F} :
                      f = f
                      def LinearMap.toDistribMulActionHom {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) :
                      M →ₑ+[σ] M₃

                      The DistribMulActionHom underlying a LinearMap.

                      Equations
                      • f.toDistribMulActionHom = { toFun := f.toFun, map_smul' := , map_zero' := , map_add' := }
                      Instances For
                        @[simp]
                        theorem LinearMap.coe_toAddHom {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) :
                        f.toAddHom = f
                        theorem LinearMap.toFun_eq_coe {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {f : M →ₛₗ[σ] M₃} :
                        f.toFun = f
                        theorem LinearMap.ext_iff {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {f : M →ₛₗ[σ] M₃} {g : M →ₛₗ[σ] M₃} :
                        f = g ∀ (x : M), f x = g x
                        theorem LinearMap.ext {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {f : M →ₛₗ[σ] M₃} {g : M →ₛₗ[σ] M₃} (h : ∀ (x : M), f x = g x) :
                        f = g
                        def LinearMap.copy {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (f' : MM₃) (h : f' = f) :
                        M →ₛₗ[σ] M₃

                        Copy of a LinearMap with a new toFun equal to the old one. Useful to fix definitional equalities.

                        Equations
                        • f.copy f' h = { toFun := f', map_add' := , map_smul' := }
                        Instances For
                          @[simp]
                          theorem LinearMap.coe_copy {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (f' : MM₃) (h : f' = f) :
                          (f.copy f' h) = f'
                          theorem LinearMap.copy_eq {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (f' : MM₃) (h : f' = f) :
                          f.copy f' h = f
                          @[simp]
                          theorem LinearMap.coe_mk {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : AddHom M M₃) (h : ∀ (m : R) (x : M), f.toFun (m x) = σ m f.toFun x) :
                          { toAddHom := f, map_smul' := h } = f
                          @[simp]
                          theorem LinearMap.coe_addHom_mk {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : AddHom M M₃) (h : ∀ (m : R) (x : M), f.toFun (m x) = σ m f.toFun x) :
                          { toAddHom := f, map_smul' := h } = f
                          theorem LinearMap.coe_semilinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {F : Type u_17} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] (f : F) :
                          f = f
                          theorem LinearMap.toLinearMap_injective {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {F : Type u_17} [FunLike F M M₃] [SemilinearMapClass F σ M M₃] {f : F} {g : F} (h : f = g) :
                          f = g
                          def LinearMap.id {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] :

                          Identity map as a LinearMap

                          Equations
                          • LinearMap.id = { toFun := id, map_add' := , map_smul' := }
                          Instances For
                            theorem LinearMap.id_apply {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) :
                            LinearMap.id x = x
                            @[simp]
                            theorem LinearMap.id_coe {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] :
                            LinearMap.id = id
                            @[simp]
                            theorem LinearMap.id'_apply {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] {σ : R →+* R} [RingHomId σ] (x : M) :
                            LinearMap.id' x = x
                            def LinearMap.id' {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] {σ : R →+* R} [RingHomId σ] :

                            A generalisation of LinearMap.id that constructs the identity function as a σ-semilinear map for any ring homomorphism σ which we know is the identity.

                            Equations
                            • LinearMap.id' = { toFun := fun (x : M) => x, map_add' := , map_smul' := }
                            Instances For
                              @[simp]
                              theorem LinearMap.id'_coe {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] {σ : R →+* R} [RingHomId σ] :
                              LinearMap.id' = id
                              theorem LinearMap.isLinear {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (fₗ : M →ₗ[R] M₂) :
                              IsLinearMap R fₗ
                              theorem LinearMap.coe_injective {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} :
                              Function.Injective DFunLike.coe
                              theorem LinearMap.congr_arg {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {f : M →ₛₗ[σ] M₃} {x : M} {x' : M} :
                              x = x'f x = f x'
                              theorem LinearMap.congr_fun {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} {f : M →ₛₗ[σ] M₃} {g : M →ₛₗ[σ] M₃} (h : f = g) (x : M) :
                              f x = g x

                              If two linear maps are equal, they are equal at each point.

                              @[simp]
                              theorem LinearMap.mk_coe {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (h : ∀ (m : R) (x : M), (↑f).toFun (m x) = σ m (↑f).toFun x) :
                              { toAddHom := f, map_smul' := h } = f
                              theorem LinearMap.map_add {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (x : M) (y : M) :
                              f (x + y) = f x + f y
                              theorem LinearMap.map_zero {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) :
                              f 0 = 0
                              @[simp]
                              theorem LinearMap.map_smulₛₗ {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (c : R) (x : M) :
                              f (c x) = σ c f x
                              theorem LinearMap.map_smul {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (fₗ : M →ₗ[R] M₂) (c : R) (x : M) :
                              fₗ (c x) = c fₗ x
                              theorem LinearMap.map_smul_inv {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) {σ' : S →+* R} [RingHomInvPair σ σ'] (c : S) (x : M) :
                              c f x = f (σ' c x)
                              @[simp]
                              theorem LinearMap.map_eq_zero_iff {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) (h : Function.Injective f) {x : M} :
                              f x = 0 x = 0
                              class LinearMap.CompatibleSMul (M : Type u_9) (M₂ : Type u_11) [AddCommMonoid M] [AddCommMonoid M₂] (R : Type u_17) (S : Type u_18) [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] :

                              A typeclass for SMul structures which can be moved through a LinearMap. This typeclass is generated automatically from an IsScalarTower instance, but exists so that we can also add an instance for AddCommGroup.toIntModule, allowing z • to be moved even if S does not support negation.

                              • map_smul : ∀ (fₗ : M →ₗ[S] M₂) (c : R) (x : M), fₗ (c x) = c fₗ x

                                Scalar multiplication by R of M can be moved through linear maps.

                              Instances
                                theorem LinearMap.CompatibleSMul.map_smul {M : Type u_9} {M₂ : Type u_11} :
                                ∀ {inst : AddCommMonoid M} {inst_1 : AddCommMonoid M₂} {R : Type u_17} {S : Type u_18} {inst_2 : Semiring S} {inst_3 : SMul R M} {inst_4 : Module S M} {inst_5 : SMul R M₂} {inst_6 : Module S M₂} [self : LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (c : R) (x : M), fₗ (c x) = c fₗ x

                                Scalar multiplication by R of M can be moved through linear maps.

                                @[instance 100]
                                instance LinearMap.IsScalarTower.compatibleSMul {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] {R : Type u_17} {S : Type u_18} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] [SMul R S] [IsScalarTower R S M] [IsScalarTower R S M₂] :
                                Equations
                                • =
                                instance LinearMap.IsScalarTower.compatibleSMul' {M : Type u_9} [AddCommMonoid M] {R : Type u_17} {S : Type u_18} [Semiring S] [SMul R M] [Module S M] [SMul R S] [IsScalarTower R S M] :
                                Equations
                                • =
                                @[simp]
                                theorem LinearMap.map_smul_of_tower {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] {R : Type u_17} {S : Type u_18} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (c : R) (x : M) :
                                fₗ (c x) = c fₗ x
                                theorem LinearMap.isScalarTower_of_injective {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] (R : Type u_17) {S : Type u_18} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] [SMul R S] [LinearMap.CompatibleSMul M M₂ R S] [IsScalarTower R S M₂] (f : M →ₗ[S] M₂) (hf : Function.Injective f) :
                                theorem LinearMap.isLinearMap_of_compatibleSMul (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M →ₗ[S] M₂) :
                                def LinearMap.toAddMonoidHom {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) :
                                M →+ M₃

                                convert a linear map to an additive map

                                Equations
                                • f.toAddMonoidHom = { toFun := f, map_zero' := , map_add' := }
                                Instances For
                                  @[simp]
                                  theorem LinearMap.toAddMonoidHom_coe {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} (f : M →ₛₗ[σ] M₃) :
                                  f.toAddMonoidHom = f
                                  def LinearMap.restrictScalars (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) :
                                  M →ₗ[R] M₂

                                  If M and M₂ are both R-modules and S-modules and R-module structures are defined by an action of R on S (formally, we have two scalar towers), then any S-linear map from M to M₂ is R-linear.

                                  See also LinearMap.map_smul_of_tower.

                                  Equations
                                  • R fₗ = { toFun := fₗ, map_add' := , map_smul' := }
                                  Instances For
                                    instance LinearMap.coeIsScalarTower (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] :
                                    CoeHTCT (M →ₗ[S] M₂) (M →ₗ[R] M₂)
                                    Equations
                                    @[simp]
                                    theorem LinearMap.coe_restrictScalars (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (f : M →ₗ[S] M₂) :
                                    (R f) = f
                                    theorem LinearMap.restrictScalars_apply (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (x : M) :
                                    (R fₗ) x = fₗ x
                                    theorem LinearMap.restrictScalars_injective (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] :
                                    @[simp]
                                    theorem LinearMap.restrictScalars_inj (R : Type u_1) {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] (fₗ : M →ₗ[S] M₂) (gₗ : M →ₗ[S] M₂) :
                                    R fₗ = R gₗ fₗ = gₗ
                                    theorem LinearMap.toAddMonoidHom_injective {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₃] [Module R M] [Module S M₃] {σ : R →+* S} :
                                    Function.Injective LinearMap.toAddMonoidHom
                                    theorem LinearMap.ext_ring_iff {R : Type u_1} {S : Type u_6} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M₃] [Module S M₃] {σ : R →+* S} {f : R →ₛₗ[σ] M₃} {g : R →ₛₗ[σ] M₃} :
                                    f = g f 1 = g 1
                                    theorem LinearMap.ext_ring {R : Type u_1} {S : Type u_6} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M₃] [Module S M₃] {σ : R →+* S} {f : R →ₛₗ[σ] M₃} {g : R →ₛₗ[σ] M₃} (h : f 1 = g 1) :
                                    f = g

                                    If two σ-linear maps from R are equal on 1, then they are equal.

                                    theorem LinearMap.ext_ring_op_iff {R : Type u_1} {S : Type u_6} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M₃] [Module S M₃] {σ : Rᵐᵒᵖ →+* S} {f : R →ₛₗ[σ] M₃} {g : R →ₛₗ[σ] M₃} :
                                    f = g f 1 = g 1
                                    theorem LinearMap.ext_ring_op {R : Type u_1} {S : Type u_6} {M₃ : Type u_12} [Semiring R] [Semiring S] [AddCommMonoid M₃] [Module S M₃] {σ : Rᵐᵒᵖ →+* S} {f : R →ₛₗ[σ] M₃} {g : R →ₛₗ[σ] M₃} (h : f 1 = g 1) :
                                    f = g
                                    @[simp]
                                    theorem RingHom.toSemilinearMap_apply {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R →+* S) :
                                    ∀ (a : R), f.toSemilinearMap a = (↑f).toFun a
                                    def RingHom.toSemilinearMap {R : Type u_1} {S : Type u_6} [Semiring R] [Semiring S] (f : R →+* S) :

                                    Interpret a RingHom f as an f-semilinear map.

                                    Equations
                                    • f.toSemilinearMap = { toFun := (↑f).toFun, map_add' := , map_smul' := }
                                    Instances For
                                      def LinearMap.comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₂] M₂) :
                                      M₁ →ₛₗ[σ₁₃] M₃

                                      Composition of two linear maps is a linear map

                                      Equations
                                      • f.comp g = { toFun := f g, map_add' := , map_smul' := }
                                      Instances For

                                        Pretty printer defined by notation3 command.

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

                                          ∘ₗ is notation for composition of two linear (not semilinear!) maps into a linear map. This is useful when Lean is struggling to infer the RingHomCompTriple instance.

                                          Equations
                                          Instances For
                                            theorem LinearMap.comp_apply {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₂] M₂) (x : M₁) :
                                            (f.comp g) x = f (g x)
                                            @[simp]
                                            theorem LinearMap.coe_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₂] M₂) :
                                            (f.comp g) = f g
                                            @[simp]
                                            theorem LinearMap.comp_id {R₂ : Type u_3} {R₃ : Type u_4} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₂] [Semiring R₃] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₂₃ : R₂ →+* R₃} (f : M₂ →ₛₗ[σ₂₃] M₃) :
                                            f.comp LinearMap.id = f
                                            @[simp]
                                            theorem LinearMap.id_comp {R₂ : Type u_3} {R₃ : Type u_4} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₂] [Semiring R₃] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₂₃ : R₂ →+* R₃} (f : M₂ →ₛₗ[σ₂₃] M₃) :
                                            LinearMap.id.comp f = f
                                            theorem LinearMap.comp_assoc {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_17} {M₄ : Type u_18} [Semiring R₄] [AddCommMonoid M₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (f : M₁ →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₃ →ₛₗ[σ₃₄] M₄) :
                                            (h.comp g).comp f = h.comp (g.comp f)
                                            theorem Function.Surjective.injective_linearMapComp_right {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {g : M₁ →ₛₗ[σ₁₂] M₂} (hg : Function.Surjective g) :
                                            Function.Injective fun (f : M₂ →ₛₗ[σ₂₃] M₃) => f.comp g

                                            The linear map version of Function.Surjective.injective_comp_right

                                            @[simp]
                                            theorem LinearMap.cancel_right {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {f : M₂ →ₛₗ[σ₂₃] M₃} {g : M₁ →ₛₗ[σ₁₂] M₂} {f' : M₂ →ₛₗ[σ₂₃] M₃} (hg : Function.Surjective g) :
                                            f.comp g = f'.comp g f = f'
                                            theorem Function.Injective.injective_linearMapComp_left {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {f : M₂ →ₛₗ[σ₂₃] M₃} (hf : Function.Injective f) :
                                            Function.Injective fun (g : M₁ →ₛₗ[σ₁₂] M₂) => f.comp g

                                            The linear map version of Function.Injective.comp_left

                                            @[simp]
                                            theorem LinearMap.cancel_left {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M₁ : Type u_10} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] {module_M₁ : Module R₁ M₁} {module_M₂ : Module R₂ M₂} {module_M₃ : Module R₃ M₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {f : M₂ →ₛₗ[σ₂₃] M₃} {g : M₁ →ₛₗ[σ₁₂] M₂} {g' : M₁ →ₛₗ[σ₁₂] M₂} (hf : Function.Injective f) :
                                            f.comp g = f.comp g' g = g'
                                            def LinearMap.inverse {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] (f : M →ₛₗ[σ] M₂) (g : M₂M) (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
                                            M₂ →ₛₗ[σ'] M

                                            If a function g is a left and right inverse of a linear map f, then g is linear itself.

                                            Equations
                                            • f.inverse g h₁ h₂ = { toFun := g, map_add' := , map_smul' := }
                                            Instances For
                                              theorem LinearMap.map_neg {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommGroup M] [AddCommGroup M₂] {module_M : Module R M} {module_M₂ : Module S M₂} {σ : R →+* S} (f : M →ₛₗ[σ] M₂) (x : M) :
                                              f (-x) = -f x
                                              theorem LinearMap.map_sub {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring S] [AddCommGroup M] [AddCommGroup M₂] {module_M : Module R M} {module_M₂ : Module S M₂} {σ : R →+* S} (f : M →ₛₗ[σ] M₂) (x : M) (y : M) :
                                              f (x - y) = f x - f y
                                              instance LinearMap.CompatibleSMul.intModule {M : Type u_9} {M₂ : Type u_11} [AddCommGroup M] [AddCommGroup M₂] {S : Type u_17} [Semiring S] [Module S M] [Module S M₂] :
                                              Equations
                                              • =
                                              instance LinearMap.CompatibleSMul.units {M : Type u_9} {M₂ : Type u_11} [AddCommGroup M] [AddCommGroup M₂] {R : Type u_17} {S : Type u_18} [Monoid R] [MulAction R M] [MulAction R M₂] [Semiring S] [Module S M] [Module S M₂] [LinearMap.CompatibleSMul M M₂ R S] :
                                              Equations
                                              • =
                                              @[simp]
                                              theorem Module.compHom.toLinearMap_apply {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] (g : R →+* S) (a : R) :
                                              def Module.compHom.toLinearMap {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] (g : R →+* S) :

                                              g : R →+* S is R-linear when the module structure on S is Module.compHom S g .

                                              Equations
                                              Instances For
                                                def DistribMulActionHom.toSemilinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] [Semiring R] [Module R M] [Semiring S] [Module S M₂] {σ : R →+* S} (fₗ : M →ₑ+[σ] M₂) :
                                                M →ₛₗ[σ] M₂

                                                A DistribMulActionHom between two modules is a linear map.

                                                Equations
                                                • fₗ = { toFun := fₗ.toFun, map_add' := , map_smul' := }
                                                Instances For
                                                  instance DistribMulActionHom.instSemilinearMapClass {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] [Semiring R] [Module R M] [Semiring S] [Module S M₂] {σ : R →+* S} :
                                                  SemilinearMapClass (M →ₑ+[σ] M₂) σ M M₂
                                                  Equations
                                                  • =
                                                  instance DistribMulActionHom.instCoeTCSemilinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] [Semiring R] [Module R M] [Semiring S] [Module S M₂] {σ : R →+* S} :
                                                  CoeTC (M →ₑ+[σ] M₂) (M →ₛₗ[σ] M₂)
                                                  Equations
                                                  • DistribMulActionHom.instCoeTCSemilinearMap = { coe := DistribMulActionHom.toSemilinearMap }
                                                  def DistribMulActionHom.toLinearMap {R : Type u_1} {M : Type u_9} {M₃ : Type u_12} [AddCommMonoid M] [AddCommMonoid M₃] [Semiring R] [Module R M] [Module R M₃] (fₗ : M →+[R] M₃) :
                                                  M →ₗ[R] M₃

                                                  A DistribMulActionHom between two modules is a linear map.

                                                  Equations
                                                  • fₗ.toLinearMap = { toFun := fₗ.toFun, map_add' := , map_smul' := }
                                                  Instances For
                                                    instance DistribMulActionHom.instCoeTCLinearMap {R : Type u_1} {M : Type u_9} {M₃ : Type u_12} [AddCommMonoid M] [AddCommMonoid M₃] [Semiring R] [Module R M] [Module R M₃] :
                                                    CoeTC (M →+[R] M₃) (M →ₗ[R] M₃)
                                                    Equations
                                                    • DistribMulActionHom.instCoeTCLinearMap = { coe := DistribMulActionHom.toLinearMap }
                                                    instance DistribMulActionHom.instLinearMapClassId {R : Type u_1} {M : Type u_9} {M₃ : Type u_12} [AddCommMonoid M] [AddCommMonoid M₃] [Semiring R] [Module R M] [Module R M₃] :
                                                    LinearMapClass (M →+[R] M₃) R M M₃

                                                    A DistribMulActionHom between two modules is a linear map.

                                                    Equations
                                                    • =
                                                    @[simp]
                                                    theorem DistribMulActionHom.coe_toLinearMap {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] [Semiring R] [Module R M] [Semiring S] [Module S M₂] {σ : R →+* S} (f : M →ₑ+[σ] M₂) :
                                                    f = f
                                                    theorem DistribMulActionHom.toLinearMap_injective {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] [Semiring R] [Module R M] [Semiring S] [Module S M₂] {σ : R →+* S} {f : M →ₑ+[σ] M₂} {g : M →ₑ+[σ] M₂} (h : f = g) :
                                                    f = g
                                                    def IsLinearMap.mk' {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : MM₂) (lin : IsLinearMap R f) :
                                                    M →ₗ[R] M₂

                                                    Convert an IsLinearMap predicate to a LinearMap

                                                    Equations
                                                    Instances For
                                                      @[simp]
                                                      theorem IsLinearMap.mk'_apply {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {f : MM₂} (lin : IsLinearMap R f) (x : M) :
                                                      (IsLinearMap.mk' f lin) x = f x
                                                      theorem IsLinearMap.isLinearMap_smul {R : Type u_17} {M : Type u_18} [CommSemiring R] [AddCommMonoid M] [Module R M] (c : R) :
                                                      IsLinearMap R fun (z : M) => c z
                                                      theorem IsLinearMap.isLinearMap_smul' {R : Type u_17} {M : Type u_18} [Semiring R] [AddCommMonoid M] [Module R M] (a : M) :
                                                      IsLinearMap R fun (c : R) => c a
                                                      theorem IsLinearMap.map_zero {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {f : MM₂} (lin : IsLinearMap R f) :
                                                      f 0 = 0
                                                      theorem IsLinearMap.isLinearMap_neg {R : Type u_1} {M : Type u_9} [Semiring R] [AddCommGroup M] [Module R M] :
                                                      IsLinearMap R fun (z : M) => -z
                                                      theorem IsLinearMap.map_neg {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R M₂] {f : MM₂} (lin : IsLinearMap R f) (x : M) :
                                                      f (-x) = -f x
                                                      theorem IsLinearMap.map_sub {R : Type u_1} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R M₂] {f : MM₂} (lin : IsLinearMap R f) (x : M) (y : M) :
                                                      f (x - y) = f x - f y
                                                      def AddMonoidHom.toNatLinearMap {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] (f : M →+ M₂) :

                                                      Reinterpret an additive homomorphism as an -linear map.

                                                      Equations
                                                      • f.toNatLinearMap = { toFun := f, map_add' := , map_smul' := }
                                                      Instances For
                                                        theorem AddMonoidHom.toNatLinearMap_injective {M : Type u_9} {M₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid M₂] :
                                                        Function.Injective AddMonoidHom.toNatLinearMap
                                                        def AddMonoidHom.toIntLinearMap {M : Type u_9} {M₂ : Type u_11} [AddCommGroup M] [AddCommGroup M₂] (f : M →+ M₂) :

                                                        Reinterpret an additive homomorphism as a -linear map.

                                                        Equations
                                                        • f.toIntLinearMap = { toFun := f, map_add' := , map_smul' := }
                                                        Instances For
                                                          theorem AddMonoidHom.toIntLinearMap_injective {M : Type u_9} {M₂ : Type u_11} [AddCommGroup M] [AddCommGroup M₂] :
                                                          Function.Injective AddMonoidHom.toIntLinearMap
                                                          @[simp]
                                                          theorem AddMonoidHom.coe_toIntLinearMap {M : Type u_9} {M₂ : Type u_11} [AddCommGroup M] [AddCommGroup M₂] (f : M →+ M₂) :
                                                          f.toIntLinearMap = f
                                                          instance LinearMap.instSMul {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] :
                                                          SMul S (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • LinearMap.instSMul = { smul := fun (a : S) (f : M →ₛₗ[σ₁₂] M₂) => { toFun := a f, map_add' := , map_smul' := } }
                                                          @[simp]
                                                          theorem LinearMap.smul_apply {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] (a : S) (f : M →ₛₗ[σ₁₂] M₂) (x : M) :
                                                          (a f) x = a f x
                                                          theorem LinearMap.coe_smul {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] (a : S) (f : M →ₛₗ[σ₁₂] M₂) :
                                                          (a f) = a f
                                                          instance LinearMap.instSMulCommClass {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {T : Type u_8} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] [Monoid T] [DistribMulAction T M₂] [SMulCommClass R₂ T M₂] [SMulCommClass S T M₂] :
                                                          SMulCommClass S T (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • =
                                                          instance LinearMap.instIsScalarTower {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {T : Type u_8} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] [Monoid T] [DistribMulAction T M₂] [SMulCommClass R₂ T M₂] [SMul S T] [IsScalarTower S T M₂] :
                                                          IsScalarTower S T (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • =
                                                          instance LinearMap.instIsCentralScalar {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] [DistribMulAction Sᵐᵒᵖ M₂] [SMulCommClass R₂ Sᵐᵒᵖ M₂] [IsCentralScalar S M₂] :
                                                          IsCentralScalar S (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • =

                                                          Arithmetic on the codomain #

                                                          instance LinearMap.instZero {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                          Zero (M →ₛₗ[σ₁₂] M₂)

                                                          The constant 0 map is linear.

                                                          Equations
                                                          • LinearMap.instZero = { zero := { toFun := 0, map_add' := , map_smul' := } }
                                                          @[simp]
                                                          theorem LinearMap.zero_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} (x : M) :
                                                          0 x = 0
                                                          @[simp]
                                                          theorem LinearMap.comp_zero {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →ₛₗ[σ₂₃] M₃) :
                                                          g.comp 0 = 0
                                                          @[simp]
                                                          theorem LinearMap.zero_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) :
                                                          instance LinearMap.instInhabited {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                          Inhabited (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • LinearMap.instInhabited = { default := 0 }
                                                          @[simp]
                                                          theorem LinearMap.default_def {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                          default = 0
                                                          instance LinearMap.uniqueOfLeft {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} [Subsingleton M] :
                                                          Unique (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          instance LinearMap.uniqueOfRight {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} [Subsingleton M₂] :
                                                          Unique (M →ₛₗ[σ₁₂] M₂)
                                                          Equations
                                                          • LinearMap.uniqueOfRight = .unique
                                                          instance LinearMap.instAdd {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                          Add (M →ₛₗ[σ₁₂] M₂)

                                                          The sum of two linear maps is linear.

                                                          Equations
                                                          • LinearMap.instAdd = { add := fun (f g : M →ₛₗ[σ₁₂] M₂) => { toFun := f + g, map_add' := , map_smul' := } }
                                                          @[simp]
                                                          theorem LinearMap.add_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (g : M →ₛₗ[σ₁₂] M₂) (x : M) :
                                                          (f + g) x = f x + g x
                                                          theorem LinearMap.add_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₂ →ₛₗ[σ₂₃] M₃) :
                                                          (h + g).comp f = h.comp f + g.comp f
                                                          theorem LinearMap.comp_add {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M →ₛₗ[σ₁₂] M₂) (h : M₂ →ₛₗ[σ₂₃] M₃) :
                                                          h.comp (f + g) = h.comp f + h.comp g
                                                          instance LinearMap.addCommMonoid {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                          AddCommMonoid (M →ₛₗ[σ₁₂] M₂)

                                                          The type of linear maps is an additive monoid.

                                                          Equations
                                                          instance LinearMap.instNeg {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {N₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommGroup N₂] [Module R₁ M] [Module R₂ N₂] {σ₁₂ : R₁ →+* R₂} :
                                                          Neg (M →ₛₗ[σ₁₂] N₂)

                                                          The negation of a linear map is linear.

                                                          Equations
                                                          • LinearMap.instNeg = { neg := fun (f : M →ₛₗ[σ₁₂] N₂) => { toFun := -f, map_add' := , map_smul' := } }
                                                          @[simp]
                                                          theorem LinearMap.neg_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {N₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommGroup N₂] [Module R₁ M] [Module R₂ N₂] {σ₁₂ : R₁ →+* R₂} (f : M →ₛₗ[σ₁₂] N₂) (x : M) :
                                                          (-f) x = -f x
                                                          @[simp]
                                                          theorem LinearMap.neg_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {N₃ : Type u_15} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommGroup N₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ N₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] N₃) :
                                                          (-g).comp f = -g.comp f
                                                          @[simp]
                                                          theorem LinearMap.comp_neg {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {N₂ : Type u_14} {N₃ : Type u_15} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommGroup N₂] [AddCommGroup N₃] [Module R₁ M] [Module R₂ N₂] [Module R₃ N₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N₂) (g : N₂ →ₛₗ[σ₂₃] N₃) :
                                                          g.comp (-f) = -g.comp f
                                                          instance LinearMap.instSub {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {N₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommGroup N₂] [Module R₁ M] [Module R₂ N₂] {σ₁₂ : R₁ →+* R₂} :
                                                          Sub (M →ₛₗ[σ₁₂] N₂)

                                                          The subtraction of two linear maps is linear.

                                                          Equations
                                                          • LinearMap.instSub = { sub := fun (f g : M →ₛₗ[σ₁₂] N₂) => { toFun := f - g, map_add' := , map_smul' := } }
                                                          @[simp]
                                                          theorem LinearMap.sub_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {N₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommGroup N₂] [Module R₁ M] [Module R₂ N₂] {σ₁₂ : R₁ →+* R₂} (f : M →ₛₗ[σ₁₂] N₂) (g : M →ₛₗ[σ₁₂] N₂) (x : M) :
                                                          (f - g) x = f x - g x
                                                          theorem LinearMap.sub_comp {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_11} {N₃ : Type u_15} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommGroup N₃] [Module R₁ M] [Module R₂ M₂] [Module R₃ N₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] N₃) (h : M₂ →ₛₗ[σ₂₃] N₃) :
                                                          (g - h).comp f = g.comp f - h.comp f
                                                          theorem LinearMap.comp_sub {R₁ : Type u_2} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {N₂ : Type u_14} {N₃ : Type u_15} [Semiring R₁] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommGroup N₂] [AddCommGroup N₃] [Module R₁ M] [Module R₂ N₂] [Module R₃ N₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N₂) (g : M →ₛₗ[σ₁₂] N₂) (h : N₂ →ₛₗ[σ₂₃] N₃) :
                                                          h.comp (g - f) = h.comp g - h.comp f
                                                          instance LinearMap.addCommGroup {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {N₂ : Type u_14} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommGroup N₂] [Module R₁ M] [Module R₂ N₂] {σ₁₂ : R₁ →+* R₂} :
                                                          AddCommGroup (M →ₛₗ[σ₁₂] N₂)

                                                          The type of linear maps is an additive group.

                                                          Equations
                                                          @[simp]
                                                          theorem LinearMap.evalAddMonoidHom_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} (a : M) (f : M →ₛₗ[σ₁₂] M₂) :
                                                          def LinearMap.evalAddMonoidHom {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} (a : M) :
                                                          (M →ₛₗ[σ₁₂] M₂) →+ M₂

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

                                                          Equations
                                                          Instances For
                                                            @[simp]
                                                            theorem LinearMap.toAddMonoidHom'_apply {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) :
                                                            LinearMap.toAddMonoidHom' f = f.toAddMonoidHom
                                                            def LinearMap.toAddMonoidHom' {R₁ : Type u_2} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_11} [Semiring R₁] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R₁ M] [Module R₂ M₂] {σ₁₂ : R₁ →+* R₂} :
                                                            (M →ₛₗ[σ₁₂] M₂) →+ M →+ M₂

                                                            LinearMap.toAddMonoidHom promoted to an AddMonoidHom.

                                                            Equations
                                                            • LinearMap.toAddMonoidHom' = { toFun := LinearMap.toAddMonoidHom, map_zero' := , map_add' := }
                                                            Instances For
                                                              @[simp]
                                                              theorem LinearMap.identityMapOfZeroModuleIsZero {R₁ : Type u_2} {M : Type u_9} [Semiring R₁] [AddCommMonoid M] [Module R₁ M] [Subsingleton M] :
                                                              LinearMap.id = 0

                                                              If M is the zero module, then the identity map of M is the zero map.

                                                              instance LinearMap.instDistribMulAction {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] :
                                                              DistribMulAction S (M →ₛₗ[σ₁₂] M₂)
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                                                              theorem LinearMap.smul_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {S₃ : Type u_7} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [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] [Monoid S₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃] (a : S₃) (g : M₂ →ₛₗ[σ₂₃] M₃) (f : M →ₛₗ[σ₁₂] M₂) :
                                                              (a g).comp f = a g.comp f
                                                              theorem LinearMap.comp_smul {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} {M₃ : Type u_12} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Monoid S] [DistribMulAction S M₂] [Module R M₂] [Module R M₃] [SMulCommClass R S M₂] [DistribMulAction S M₃] [SMulCommClass R S M₃] [LinearMap.CompatibleSMul M₃ M₂ S R] (g : M₃ →ₗ[R] M₂) (a : S) (f : M →ₗ[R] M₃) :
                                                              g ∘ₗ (a f) = a g ∘ₗ f
                                                              instance LinearMap.module {R : Type u_1} {R₂ : Type u_3} {S : Type u_6} {M : Type u_9} {M₂ : Type u_11} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Semiring S] [Module S M₂] [SMulCommClass R₂ S M₂] :
                                                              Module S (M →ₛₗ[σ₁₂] M₂)
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                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_zero (R : Type u_17) (S : Type u_18) (M : Type u_19) (N : Type u_20) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] :
                                                              R 0 = 0
                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_add {R : Type u_17} {S : Type u_18} {M : Type u_19} {N : Type u_20} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (f : M →ₗ[S] N) (g : M →ₗ[S] N) :
                                                              R (f + g) = R f + R g
                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_neg {R : Type u_17} {S : Type u_18} [Semiring R] [Semiring S] {M : Type u_22} {N : Type u_23} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (f : M →ₗ[S] N) :
                                                              R (-f) = -R f
                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_smul {R : Type u_17} {S : Type u_18} {M : Type u_19} {N : Type u_20} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] {R₁ : Type u_22} [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] (c : R₁) (f : M →ₗ[S] N) :
                                                              R (c f) = c R f
                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_comp {R : Type u_17} {S : Type u_18} {M : Type u_19} {N : Type u_20} {P : Type u_21} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] [AddCommMonoid P] [Module S P] [Module R P] [LinearMap.CompatibleSMul N P R S] [LinearMap.CompatibleSMul M P R S] (f : N →ₗ[S] P) (g : M →ₗ[S] N) :
                                                              R (f ∘ₗ g) = R f ∘ₗ R g
                                                              @[simp]
                                                              theorem LinearMap.restrictScalars_trans {R : Type u_17} {S : Type u_18} {M : Type u_19} {N : Type u_20} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] {T : Type u_23} [CommSemiring T] [Module T M] [Module T N] [LinearMap.CompatibleSMul M N S T] [LinearMap.CompatibleSMul M N R T] (f : M →ₗ[T] N) :
                                                              R (S f) = R f
                                                              @[simp]
                                                              theorem LinearMap.restrictScalarsₗ_apply (R : Type u_17) (S : Type u_18) (M : Type u_19) (N : Type u_20) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (R₁ : Type u_22) [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] (fₗ : M →ₗ[S] N) :
                                                              (LinearMap.restrictScalarsₗ R S M N R₁) fₗ = R fₗ
                                                              def LinearMap.restrictScalarsₗ (R : Type u_17) (S : Type u_18) (M : Type u_19) (N : Type u_20) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (R₁ : Type u_22) [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] :
                                                              (M →ₗ[S] N) →ₗ[R₁] M →ₗ[R] N

                                                              LinearMap.restrictScalars as a LinearMap.

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