# Basics on bilinear maps #

This file provides basics on bilinear maps. The most general form considered are maps that are semilinear in both arguments. They are of type M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P, where M and N are modules over R and S respectively, P is a module over both R₂ and S₂ with commuting actions, and ρ₁₂ : R →+* R₂ and σ₁₂ : S →+* S₂.

## Main declarations #

• LinearMap.mk₂: a constructor for bilinear maps, taking an unbundled function together with proof witnesses of bilinearity
• LinearMap.flip: turns a bilinear map M × N → P into N × M → P
• LinearMap.lcomp and LinearMap.llcomp: composition of linear maps as a bilinear map
• LinearMap.compl₂: composition of a bilinear map M × N → P with a linear map Q → M
• LinearMap.compr₂: composition of a bilinear map M × N → P with a linear map Q → N
• LinearMap.lsmul: scalar multiplication as a bilinear map R × M → M

## Tags #

bilinear

def LinearMap.mk₂'ₛₗ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] (ρ₁₂ : R →+* R₂) (σ₁₂ : S →+* S₂) (f : MNP) (H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n) (H2 : ∀ (c : R) (m : M) (n : N), f (c m) n = ρ₁₂ c f m n) (H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂) (H4 : ∀ (c : S) (m : M) (n : N), f m (c n) = σ₁₂ c f m n) :
M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P

Create a bilinear map from a function that is semilinear in each component. See mk₂' and mk₂ for the linear case.

Equations
• LinearMap.mk₂'ₛₗ ρ₁₂ σ₁₂ f H1 H2 H3 H4 = { toFun := fun (m : M) => { toFun := f m, map_add' := , map_smul' := }, map_add' := , map_smul' := }
Instances For
@[simp]
theorem LinearMap.mk₂'ₛₗ_apply {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : MNP) {H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n} {H2 : ∀ (c : R) (m : M) (n : N), f (c m) n = ρ₁₂ c f m n} {H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂} {H4 : ∀ (c : S) (m : M) (n : N), f m (c n) = σ₁₂ c f m n} (m : M) (n : N) :
((LinearMap.mk₂'ₛₗ ρ₁₂ σ₁₂ f H1 H2 H3 H4) m) n = f m n
def LinearMap.mk₂' (R : Type u_1) [] (S : Type u_2) [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (f : MNPₗ) (H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n) (H2 : ∀ (c : R) (m : M) (n : N), f (c m) n = c f m n) (H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂) (H4 : ∀ (c : S) (m : M) (n : N), f m (c n) = c f m n) :
M →ₗ[R] N →ₗ[S] Pₗ

Create a bilinear map from a function that is linear in each component. See mk₂ for the special case where both arguments come from modules over the same ring.

Equations
Instances For
@[simp]
theorem LinearMap.mk₂'_apply {R : Type u_1} [] {S : Type u_2} [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (f : MNPₗ) {H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n} {H2 : ∀ (c : R) (m : M) (n : N), f (c m) n = c f m n} {H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂} {H4 : ∀ (c : S) (m : M) (n : N), f m (c n) = c f m n} (m : M) (n : N) :
((LinearMap.mk₂' R S f H1 H2 H3 H4) m) n = f m n
theorem LinearMap.ext₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} {f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} {g : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (H : ∀ (m : M) (n : N), (f m) n = (g m) n) :
f = g
theorem LinearMap.congr_fun₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} {f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} {g : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (h : f = g) (x : M) (y : N) :
(f x) y = (g x) y
theorem LinearMap.ext_iff₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} {f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} {g : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} :
f = g ∀ (m : M) (n : N), (f m) n = (g m) n
def LinearMap.flip {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) :
N →ₛₗ[σ₁₂] M →ₛₗ[ρ₁₂] P

Given a linear map from M to linear maps from N to P, i.e., a bilinear map from M × N to P, change the order of variables and get a linear map from N to linear maps from M to P.

Equations
Instances For
@[simp]
theorem LinearMap.flip_apply {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (m : M) (n : N) :
(f.flip n) m = (f m) n
@[simp]
theorem LinearMap.flip_flip {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) :
f.flip.flip = f
theorem LinearMap.flip_inj {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} {f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} {g : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (H : f.flip = g.flip) :
f = g
theorem LinearMap.map_zero₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (y : N) :
(f 0) y = 0
theorem LinearMap.map_neg₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {N : Type u_6} {M' : Type u_13} {P' : Type u_15} [] [] [] [Module S N] [Module R M'] [Module R₂ P'] [Module S₂ P'] [SMulCommClass S₂ R₂ P'] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M' →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P') (x : M') (y : N) :
(f (-x)) y = -(f x) y
theorem LinearMap.map_sub₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {N : Type u_6} {M' : Type u_13} {P' : Type u_15} [] [] [] [Module S N] [Module R M'] [Module R₂ P'] [Module S₂ P'] [SMulCommClass S₂ R₂ P'] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M' →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P') (x : M') (y : M') (z : N) :
(f (x - y)) z = (f x) z - (f y) z
theorem LinearMap.map_add₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (x₁ : M) (x₂ : M) (y : N) :
(f (x₁ + x₂)) y = (f x₁) y + (f x₂) y
theorem LinearMap.map_smul₂ {R : Type u_1} [] {S : Type u_2} [] {S₂ : Type u_4} [Semiring S₂] {M₂ : Type u_8} {N₂ : Type u_9} {P₂ : Type u_10} [] [] [] [Module R M₂] [Module S N₂] [Module R P₂] [Module S₂ P₂] [SMulCommClass S₂ R P₂] {σ₁₂ : S →+* S₂} (f : M₂ →ₗ[R] N₂ →ₛₗ[σ₁₂] P₂) (r : R) (x : M₂) (y : N₂) :
(f (r x)) y = r (f x) y
theorem LinearMap.map_smulₛₗ₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (r : R) (x : M) (y : N) :
(f (r x)) y = ρ₁₂ r (f x) y
theorem LinearMap.map_sum₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} {ι : Type u_16} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (t : ) (x : ιM) (y : N) :
(f (∑ it, x i)) y = it, (f (x i)) y
def LinearMap.domRestrict₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (q : ) :
M →ₛₗ[ρ₁₂] q →ₛₗ[σ₁₂] P

Restricting a bilinear map in the second entry

Equations
• f.domRestrict₂ q = { toFun := fun (m : M) => (f m).domRestrict q, map_add' := , map_smul' := }
Instances For
theorem LinearMap.domRestrict₂_apply {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (q : ) (x : M) (y : q) :
((f.domRestrict₂ q) x) y = (f x) y
def LinearMap.domRestrict₁₂ {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (p : ) (q : ) :
p →ₛₗ[ρ₁₂] q →ₛₗ[σ₁₂] P

Restricting a bilinear map in both components

Equations
• f.domRestrict₁₂ p q = (f.domRestrict p).domRestrict₂ q
Instances For
theorem LinearMap.domRestrict₁₂_apply {R : Type u_1} [] {S : Type u_2} [] {R₂ : Type u_3} [Semiring R₂] {S₂ : Type u_4} [Semiring S₂] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module S N] [Module R₂ P] [Module S₂ P] [SMulCommClass S₂ R₂ P] {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂} (f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P) (p : ) (q : ) (x : p) (y : q) :
((f.domRestrict₁₂ p q) x) y = (f x) y
@[simp]
theorem LinearMap.restrictScalars₁₂_apply_apply {R : Type u_1} [] {S : Type u_2} [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (R' : Type u_16) (S' : Type u_17) [Semiring R'] [Semiring S'] [Module R' M] [Module S' N] [Module R' Pₗ] [Module S' Pₗ] [SMulCommClass S' R' Pₗ] [SMul S' S] [IsScalarTower S' S N] [IsScalarTower S' S Pₗ] [SMul R' R] [IsScalarTower R' R M] [IsScalarTower R' R Pₗ] (B : M →ₗ[R] N →ₗ[S] Pₗ) (m : M) :
∀ (x : N), ((LinearMap.restrictScalars₁₂ R' S' B) m) x = (B m) x
def LinearMap.restrictScalars₁₂ {R : Type u_1} [] {S : Type u_2} [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (R' : Type u_16) (S' : Type u_17) [Semiring R'] [Semiring S'] [Module R' M] [Module S' N] [Module R' Pₗ] [Module S' Pₗ] [SMulCommClass S' R' Pₗ] [SMul S' S] [IsScalarTower S' S N] [IsScalarTower S' S Pₗ] [SMul R' R] [IsScalarTower R' R M] [IsScalarTower R' R Pₗ] (B : M →ₗ[R] N →ₗ[S] Pₗ) :
M →ₗ[R'] N →ₗ[S'] Pₗ

If B : M → N → Pₗ is R-S bilinear and R' and S' are compatible scalar multiplications, then the restriction of scalars is a R'-S' bilinear map.

Equations
Instances For
theorem LinearMap.restrictScalars₁₂_injective {R : Type u_1} [] {S : Type u_2} [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (R' : Type u_16) (S' : Type u_17) [Semiring R'] [Semiring S'] [Module R' M] [Module S' N] [Module R' Pₗ] [Module S' Pₗ] [SMulCommClass S' R' Pₗ] [SMul S' S] [IsScalarTower S' S N] [IsScalarTower S' S Pₗ] [SMul R' R] [IsScalarTower R' R M] [IsScalarTower R' R Pₗ] :
@[simp]
theorem LinearMap.restrictScalars₁₂_inj {R : Type u_1} [] {S : Type u_2} [] {M : Type u_5} {N : Type u_6} {Pₗ : Type u_12} [] [] [] [Module R M] [Module S N] [Module R Pₗ] [Module S Pₗ] [SMulCommClass S R Pₗ] (R' : Type u_16) (S' : Type u_17) [Semiring R'] [Semiring S'] [Module R' M] [Module S' N] [Module R' Pₗ] [Module S' Pₗ] [SMulCommClass S' R' Pₗ] [SMul S' S] [IsScalarTower S' S N] [IsScalarTower S' S Pₗ] [SMul R' R] [IsScalarTower R' R M] [IsScalarTower R' R Pₗ] {B : M →ₗ[R] N →ₗ[S] Pₗ} {B' : M →ₗ[R] N →ₗ[S] Pₗ} :
= B = B'
def LinearMap.mk₂ (R : Type u_1) [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : MNₗPₗ) (H1 : ∀ (m₁ m₂ : M) (n : Nₗ), f (m₁ + m₂) n = f m₁ n + f m₂ n) (H2 : ∀ (c : R) (m : M) (n : Nₗ), f (c m) n = c f m n) (H3 : ∀ (m : M) (n₁ n₂ : Nₗ), f m (n₁ + n₂) = f m n₁ + f m n₂) (H4 : ∀ (c : R) (m : M) (n : Nₗ), f m (c n) = c f m n) :
M →ₗ[R] Nₗ →ₗ[R] Pₗ

Create a bilinear map from a function that is linear in each component.

This is a shorthand for mk₂' for the common case when R = S.

Equations
Instances For
@[simp]
theorem LinearMap.mk₂_apply (R : Type u_1) [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : MNₗPₗ) {H1 : ∀ (m₁ m₂ : M) (n : Nₗ), f (m₁ + m₂) n = f m₁ n + f m₂ n} {H2 : ∀ (c : R) (m : M) (n : Nₗ), f (c m) n = c f m n} {H3 : ∀ (m : M) (n₁ n₂ : Nₗ), f m (n₁ + n₂) = f m n₁ + f m n₂} {H4 : ∀ (c : R) (m : M) (n : Nₗ), f m (c n) = c f m n} (m : M) (n : Nₗ) :
((LinearMap.mk₂ R f H1 H2 H3 H4) m) n = f m n
def LinearMap.lflip {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} :
(M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P) →ₗ[R₃] N →ₛₗ[σ₂₃] M →ₛₗ[σ₁₃] P

Given a linear map from M to linear maps from N to P, i.e., a bilinear map M → N → P, change the order of variables and get a linear map from N to linear maps from M to P.

Equations
• LinearMap.lflip = { toFun := LinearMap.flip, map_add' := , map_smul' := }
Instances For
@[simp]
theorem LinearMap.lflip_apply {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} (f : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P) (m : M) (n : N) :
((LinearMap.lflip f) n) m = (f m) n
def LinearMap.lcomp (R : Type u_1) [] {M : Type u_5} {Nₗ : Type u_10} (Pₗ : Type u_11) [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : M →ₗ[R] Nₗ) :
(Nₗ →ₗ[R] Pₗ) →ₗ[R] M →ₗ[R] Pₗ

Composing a linear map M → N and a linear map N → P to form a linear map M → P.

Equations
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@[simp]
theorem LinearMap.lcomp_apply {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : M →ₗ[R] Nₗ) (g : Nₗ →ₗ[R] Pₗ) (x : M) :
((LinearMap.lcomp R Pₗ f) g) x = g (f x)
theorem LinearMap.lcomp_apply' {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : M →ₗ[R] Nₗ) (g : Nₗ →ₗ[R] Pₗ) :
(LinearMap.lcomp R Pₗ f) g = g ∘ₗ f
def LinearMap.lcompₛₗ {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {M : Type u_5} {N : Type u_6} (P : Type u_7) [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] {σ₁₂ : R →+* R₂} (σ₂₃ : R₂ →+* R₃) {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N) :
(N →ₛₗ[σ₂₃] P) →ₗ[R₃] M →ₛₗ[σ₁₃] P

Composing a semilinear map M → N and a semilinear map N → P to form a semilinear map M → P is itself a linear map.

Equations
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@[simp]
theorem LinearMap.lcompₛₗ_apply {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] N) (g : N →ₛₗ[σ₂₃] P) (x : M) :
((LinearMap.lcompₛₗ P σ₂₃ f) g) x = g (f x)
def LinearMap.llcomp (R : Type u_1) [] (M : Type u_5) (Nₗ : Type u_10) (Pₗ : Type u_11) [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] :
(Nₗ →ₗ[R] Pₗ) →ₗ[R] (M →ₗ[R] Nₗ) →ₗ[R] M →ₗ[R] Pₗ

Composing a linear map M → N and a linear map N → P to form a linear map M → P.

Equations
• LinearMap.llcomp R M Nₗ Pₗ = { toFun := , map_add' := , map_smul' := }.flip
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@[simp]
theorem LinearMap.llcomp_apply {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : Nₗ →ₗ[R] Pₗ) (g : M →ₗ[R] Nₗ) (x : M) :
(((LinearMap.llcomp R M Nₗ Pₗ) f) g) x = f (g x)
theorem LinearMap.llcomp_apply' {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] (f : Nₗ →ₗ[R] Pₗ) (g : M →ₗ[R] Nₗ) :
((LinearMap.llcomp R M Nₗ Pₗ) f) g = f ∘ₗ g
def LinearMap.compl₂ {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {R₄ : Type u_4} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} {Q : Type u_8} [] [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] [Module R₄ Q] {σ₂₃ : R₂ →+* R₃} {σ₄₂ : R₄ →+* R₂} {σ₄₃ : R₄ →+* R₃} [RingHomCompTriple σ₄₂ σ₂₃ σ₄₃] {R₅ : Type u_14} [] [Module R₅ P] [SMulCommClass R₃ R₅ P] {σ₁₅ : R →+* R₅} (h : M →ₛₗ[σ₁₅] N →ₛₗ[σ₂₃] P) (g : Q →ₛₗ[σ₄₂] N) :
M →ₛₗ[σ₁₅] Q →ₛₗ[σ₄₃] P

Composing a linear map Q → N and a bilinear map M → N → P to form a bilinear map M → Q → P.

Equations
• h.compl₂ g = { toFun := fun (a : M) => (LinearMap.lcompₛₗ P σ₂₃ g) (h a), map_add' := , map_smul' := }
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@[simp]
theorem LinearMap.compl₂_apply {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {R₄ : Type u_4} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} {Q : Type u_8} [] [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] [Module R₄ Q] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} {σ₄₂ : R₄ →+* R₂} {σ₄₃ : R₄ →+* R₃} [RingHomCompTriple σ₄₂ σ₂₃ σ₄₃] (f : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P) (g : Q →ₛₗ[σ₄₂] N) (m : M) (q : Q) :
((f.compl₂ g) m) q = (f m) (g q)
@[simp]
theorem LinearMap.compl₂_id {R : Type u_1} [] {R₂ : Type u_2} [] {R₃ : Type u_3} [] {M : Type u_5} {N : Type u_6} {P : Type u_7} [] [] [] [Module R M] [Module R₂ N] [Module R₃ P] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} (f : M →ₛₗ[σ₁₃] N →ₛₗ[σ₂₃] P) :
f.compl₂ LinearMap.id = f
def LinearMap.compl₁₂ {R₂ : Type u_2} [] {N : Type u_6} {Mₗ : Type u_9} {Pₗ : Type u_11} {Qₗ : Type u_12} {Qₗ' : Type u_13} [] [] [] [] [] {R₁ : Type u_14} [] [Module R₂ N] [Module R₂ Pₗ] [Module R₁ Pₗ] [Module R₁ Mₗ] [SMulCommClass R₂ R₁ Pₗ] [Module R₁ Qₗ] [Module R₂ Qₗ'] (f : Mₗ →ₗ[R₁] N →ₗ[R₂] Pₗ) (g : Qₗ →ₗ[R₁] Mₗ) (g' : Qₗ' →ₗ[R₂] N) :
Qₗ →ₗ[R₁] Qₗ' →ₗ[R₂] Pₗ

Composing linear maps Q → M and Q' → N with a bilinear map M → N → P to form a bilinear map Q → Q' → P.

Equations
• f.compl₁₂ g g' = (f ∘ₗ g).compl₂ g'
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@[simp]
theorem LinearMap.compl₁₂_apply {R : Type u_1} [] {Mₗ : Type u_9} {Nₗ : Type u_10} {Pₗ : Type u_11} {Qₗ : Type u_12} {Qₗ' : Type u_13} [] [] [] [] [] [Module R Mₗ] [Module R Nₗ] [Module R Pₗ] [Module R Qₗ] [Module R Qₗ'] (f : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ) (g : Qₗ →ₗ[R] Mₗ) (g' : Qₗ' →ₗ[R] Nₗ) (x : Qₗ) (y : Qₗ') :
((f.compl₁₂ g g') x) y = (f (g x)) (g' y)
@[simp]
theorem LinearMap.compl₁₂_id_id {R : Type u_1} [] {Mₗ : Type u_9} {Nₗ : Type u_10} {Pₗ : Type u_11} [] [] [] [Module R Mₗ] [Module R Nₗ] [Module R Pₗ] (f : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ) :
f.compl₁₂ LinearMap.id LinearMap.id = f
theorem LinearMap.compl₁₂_inj {R : Type u_1} [] {Mₗ : Type u_9} {Nₗ : Type u_10} {Pₗ : Type u_11} {Qₗ : Type u_12} {Qₗ' : Type u_13} [] [] [] [] [] [Module R Mₗ] [Module R Nₗ] [Module R Pₗ] [Module R Qₗ] [Module R Qₗ'] {f₁ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ} {f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ} {g : Qₗ →ₗ[R] Mₗ} {g' : Qₗ' →ₗ[R] Nₗ} (hₗ : ) (hᵣ : ) :
f₁.compl₁₂ g g' = f₂.compl₁₂ g g' f₁ = f₂
def LinearMap.compr₂ {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} {Qₗ : Type u_12} [] [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] [Module R Qₗ] (f : M →ₗ[R] Nₗ →ₗ[R] Pₗ) (g : Pₗ →ₗ[R] Qₗ) :
M →ₗ[R] Nₗ →ₗ[R] Qₗ

Composing a linear map P → Q and a bilinear map M → N → P to form a bilinear map M → N → Q.

Equations
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@[simp]
theorem LinearMap.compr₂_apply {R : Type u_1} [] {M : Type u_5} {Nₗ : Type u_10} {Pₗ : Type u_11} {Qₗ : Type u_12} [] [] [] [] [Module R M] [Module R Nₗ] [Module R Pₗ] [Module R Qₗ] (f : M →ₗ[R] Nₗ →ₗ[R] Pₗ) (g : Pₗ →ₗ[R] Qₗ) (m : M) (n : Nₗ) :
((f.compr₂ g) m) n = g ((f m) n)
def LinearMap.lsmul (R : Type u_1) [] (M : Type u_5) [] [Module R M] :

Scalar multiplication as a bilinear map R → M → M.

Equations
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theorem LinearMap.lsmul_eq_DistribMulAction_toLinearMap {R : Type u_1} [] (M : Type u_5) [] [Module R M] (r : R) :
@[simp]
theorem LinearMap.lsmul_apply {R : Type u_1} [] {M : Type u_5} [] [Module R M] (r : R) (m : M) :
((LinearMap.lsmul R M) r) m = r m
@[reducible, inline]
abbrev LinearMap.BilinMap (R : Type u_1) [] (M : Type u_5) (Nₗ : Type u_10) [] [] [Module R M] [Module R Nₗ] :
Type (max u_10 u_5)

A shorthand for the type of R-bilinear Nₗ-valued maps on M.

Equations
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@[reducible, inline]
abbrev LinearMap.BilinForm (R : Type u_1) [] (M : Type u_5) [] [Module R M] :
Type (max u_1 u_5)

For convenience, a shorthand for the type of bilinear forms from M to R.

Equations
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theorem LinearMap.lsmul_injective {R : Type u_1} {M : Type u_5} [] [] [Module R M] [] {x : R} (hx : x 0) :
theorem LinearMap.ker_lsmul {R : Type u_1} {M : Type u_5} [] [] [Module R M] [] {a : R} (ha : a 0) :