mathlib documentation

linear_algebra.basic

Linear algebra #

This file defines the basics of linear algebra. It sets up the "categorical/lattice structure" of modules over a ring, submodules, and linear maps.

Many of the relevant definitions, including module, submodule, and linear_map, are found in src/algebra/module.

Main definitions #

See linear_algebra.span for the span of a set (as a submodule), and linear_algebra.quotient for quotients by submodules.

Main theorems #

See linear_algebra.isomorphisms for Noether's three isomorphism theorems for modules.

Notations #

Implementation notes #

We note that, when constructing linear maps, it is convenient to use operations defined on bundled maps (linear_map.prod, linear_map.coprod, arithmetic operations like +) instead of defining a function and proving it is linear.

TODO #

Tags #

linear algebra, vector space, module

theorem finsupp.smul_sum {α : Type u_1} {β : Type u_2} {R : Type u_3} {M : Type u_4} [has_zero β] [monoid R] [add_comm_monoid M] [distrib_mul_action R M] {v : α →₀ β} {c : R} {h : α → β → M} :
c v.sum h = v.sum (λ (a : α) (b : β), c h a b)
@[simp]
theorem finsupp.sum_smul_index_linear_map' {α : Type u_1} {R : Type u_2} {M : Type u_3} {M₂ : Type u_4} [semiring R] [add_comm_monoid M] [module R M] [add_comm_monoid M₂] [module R M₂] {v : α →₀ M} {c : R} {h : α → (M →ₗ[R] M₂)} :
(c v).sum (λ (a : α), (h a)) = c v.sum (λ (a : α), (h a))
@[simp]
theorem finsupp.linear_equiv_fun_on_fintype_apply (R : Type u_1) (M : Type u_9) (α : Type u_20) [fintype α] [add_comm_monoid M] [semiring R] [module R M] (x : α →₀ M) (ᾰ : α) :
noncomputable def finsupp.linear_equiv_fun_on_fintype (R : Type u_1) (M : Type u_9) (α : Type u_20) [fintype α] [add_comm_monoid M] [semiring R] [module R M] :
→₀ M) ≃ₗ[R] α → M

Given fintype α, linear_equiv_fun_on_fintype R is the natural R-linear equivalence between α →₀ β and α → β.

Equations
@[simp]
theorem finsupp.linear_equiv_fun_on_fintype_single (R : Type u_1) (M : Type u_9) (α : Type u_20) [fintype α] [add_comm_monoid M] [semiring R] [module R M] [decidable_eq α] (x : α) (m : M) :
@[simp]
theorem finsupp.linear_equiv_fun_on_fintype_symm_single (R : Type u_1) (M : Type u_9) (α : Type u_20) [fintype α] [add_comm_monoid M] [semiring R] [module R M] [decidable_eq α] (x : α) (m : M) :
@[simp]
theorem finsupp.linear_equiv_fun_on_fintype_symm_coe (R : Type u_1) (M : Type u_9) (α : Type u_20) [fintype α] [add_comm_monoid M] [semiring R] [module R M] (f : α →₀ M) :
theorem pi_eq_sum_univ {ι : Type u_1} [fintype ι] [decidable_eq ι] {R : Type u_2} [semiring R] (x : ι → R) :
x = finset.univ.sum (λ (i : ι), x i λ (j : ι), ite (i = j) 1 0)

decomposing x : ι → R as a sum along the canonical basis

Properties of linear maps #

theorem linear_map.comp_assoc {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {R₄ : Type u_5} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} {M₄ : Type u_14} [semiring R] [semiring R₂] [semiring R₃] [semiring R₄] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [add_comm_monoid M₄] [module R M] [module R₂ M₂] [module R₃ M₃] [module R₄ M₄] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₃₄ : R₃ →+* R₄} {σ₁₃ : R →+* R₃} {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R →+* R₄} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [ring_hom_comp_triple σ₂₃ σ₃₄ σ₂₄] [ring_hom_comp_triple σ₁₃ σ₃₄ σ₁₄] [ring_hom_comp_triple σ₁₂ σ₂₄ σ₁₄] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (h : M₃ →ₛₗ[σ₃₄] M₄) :
(h.comp g).comp f = h.comp (g.comp f)
def linear_map.dom_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
p →ₛₗ[σ₁₂] M₂

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

Equations
@[simp]
theorem linear_map.dom_restrict_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) (x : p) :
def linear_map.cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (p : submodule R₂ M₂) (f : M →ₛₗ[σ₁₂] M₂) (h : ∀ (c : M), f c p) :
M →ₛₗ[σ₁₂] p

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

Equations
@[simp]
theorem linear_map.cod_restrict_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (p : submodule R₂ M₂) (f : M →ₛₗ[σ₁₂] M₂) {h : ∀ (c : M), f c p} (x : M) :
@[simp]
theorem linear_map.comp_cod_restrict {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : submodule R₃ M₃) (h : ∀ (b : M₂), g b p) :
@[simp]
theorem linear_map.subtype_comp_cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ (b : M), f b p) :
def linear_map.restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) {p : submodule R M} (hf : ∀ (x : M), x pf x p) :

Restrict domain and codomain of an endomorphism.

Equations
theorem linear_map.restrict_apply {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f : M →ₗ[R] M} {p : submodule R M} (hf : ∀ (x : M), x pf x p) (x : p) :
(f.restrict hf) x = f x, _⟩
theorem linear_map.subtype_comp_restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f : M →ₗ[R] M} {p : submodule R M} (hf : ∀ (x : M), x pf x p) :
theorem linear_map.restrict_eq_cod_restrict_dom_restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f : M →ₗ[R] M} {p : submodule R M} (hf : ∀ (x : M), x pf x p) :
theorem linear_map.restrict_eq_dom_restrict_cod_restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f : M →ₗ[R] M} {p : submodule R M} (hf : ∀ (x : M), f x p) :
@[protected, instance]
def linear_map.unique_of_left {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [subsingleton M] :
unique (M →ₛₗ[σ₁₂] M₂)
Equations
@[protected, instance]
def linear_map.unique_of_right {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [subsingleton M₂] :
unique (M →ₛₗ[σ₁₂] M₂)
Equations
def linear_map.eval_add_monoid_hom {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (a : M) :
(M →ₛₗ[σ₁₂] M₂) →+ M₂

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

Equations
def linear_map.to_add_monoid_hom' {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} :
(M →ₛₗ[σ₁₂] M₂) →+ M →+ M₂

linear_map.to_add_monoid_hom promoted to an add_monoid_hom

Equations
theorem linear_map.sum_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (t : finset ι) (f : ι → (M →ₛₗ[σ₁₂] M₂)) (b : M) :
(t.sum (λ (d : ι), f d)) b = t.sum (λ (d : ι), (f d) b)
def linear_map.smul_right {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₁ : Type u_11} [semiring R] [add_comm_monoid M] [add_comm_monoid M₁] [module R M] [module R M₁] [semiring S] [module R S] [module S M] [is_scalar_tower R S M] (f : M₁ →ₗ[R] S) (x : M) :
M₁ →ₗ[R] M

When f is an R-linear map taking values in S, then λb, f b • x is an R-linear map.

Equations
@[simp]
theorem linear_map.coe_smul_right {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₁ : Type u_11} [semiring R] [add_comm_monoid M] [add_comm_monoid M₁] [module R M] [module R M₁] [semiring S] [module R S] [module S M] [is_scalar_tower R S M] (f : M₁ →ₗ[R] S) (x : M) :
(f.smul_right x) = λ (c : M₁), f c x
theorem linear_map.smul_right_apply {R : Type u_1} {S : Type u_6} {M : Type u_9} {M₁ : Type u_11} [semiring R] [add_comm_monoid M] [add_comm_monoid M₁] [module R M] [module R M₁] [semiring S] [module R S] [module S M] [is_scalar_tower R S M] (f : M₁ →ₗ[R] S) (x : M) (c : M₁) :
(f.smul_right x) c = f c x
@[protected, instance]
def linear_map.module.End.nontrivial {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] [nontrivial M] :
@[simp, norm_cast]
theorem linear_map.coe_fn_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {ι : Type u_2} (t : finset ι) (f : ι → (M →ₛₗ[σ₁₂] M₂)) :
(t.sum (λ (i : ι), f i)) = t.sum (λ (i : ι), (f i))
@[simp]
theorem linear_map.pow_apply {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) (n : ) (m : M) :
(f ^ n) m = f^[n] m
theorem linear_map.pow_map_zero_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f : module.End R M} {m : M} {k l : } (hk : k l) (hm : (f ^ k) m = 0) :
(f ^ l) m = 0
theorem linear_map.commute_pow_left_of_commute {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} {g : module.End R M} {g₂ : module.End R₂ M₂} (h : linear_map.comp g₂ f = f.comp g) (k : ) :
linear_map.comp (g₂ ^ k) f = f.comp (g ^ k)
theorem linear_map.submodule_pow_eq_zero_of_pow_eq_zero {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {N : submodule R M} {g : module.End R N} {G : module.End R M} (h : linear_map.comp G N.subtype = N.subtype.comp g) {k : } (hG : G ^ k = 0) :
g ^ k = 0
theorem linear_map.coe_pow {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) (n : ) :
(f ^ n) = (f^[n])
@[simp]
theorem linear_map.id_pow {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (n : ) :
theorem linear_map.iterate_succ {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} (n : ) :
f' ^ (n + 1) = (f' ^ n).comp f'
theorem linear_map.iterate_surjective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} (h : function.surjective f') (n : ) :
theorem linear_map.iterate_injective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} (h : function.injective f') (n : ) :
theorem linear_map.iterate_bijective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} (h : function.bijective f') (n : ) :
theorem linear_map.injective_of_iterate_injective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} {n : } (hn : n 0) (h : function.injective (f' ^ n)) :
theorem linear_map.surjective_of_iterate_surjective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} {n : } (hn : n 0) (h : function.surjective (f' ^ n)) :
theorem linear_map.pow_apply_mem_of_forall_mem {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} {p : submodule R M} (n : ) (h : ∀ (x : M), x pf' x p) (x : M) (hx : x p) :
(f' ^ n) x p
theorem linear_map.pow_restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {f' : M →ₗ[R] M} {p : submodule R M} (n : ) (h : ∀ (x : M), x pf' x p) (h' : (∀ (x : M), x p(f' ^ n) x p) := _) :
f'.restrict h ^ n = (f' ^ n).restrict h'
theorem linear_map.pi_apply_eq_sum_univ {R : Type u_1} {M : Type u_9} {ι : Type u_17} [semiring R] [add_comm_monoid M] [module R M] [fintype ι] [decidable_eq ι] (f : (ι → R) →ₗ[R] M) (x : ι → R) :
f x = finset.univ.sum (λ (i : ι), x i f (λ (j : ι), ite (i = j) 1 0))

A linear map f applied to x : ι → R can be computed using the image under f of elements of the canonical basis.

@[simp]
theorem linear_map.applyₗ'_apply_apply {R : Type u_1} (S : Type u_6) {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring S] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] [module S M₂] [smul_comm_class R S M₂] (v : M) (f : M →ₗ[R] M₂) :
def linear_map.applyₗ' {R : Type u_1} (S : Type u_6) {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring S] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] [module S M₂] [smul_comm_class R S M₂] :
M →+ (M →ₗ[R] M₂) →ₗ[S] M₂

Applying a linear map at v : M, seen as S-linear map from M →ₗ[R] M₂ to M₂.

See linear_map.applyₗ for a version where S = R.

Equations
@[simp]
theorem linear_map.ring_lmap_equiv_self_apply (R : Type u_1) (S : Type u_6) (M : Type u_9) [semiring R] [semiring S] [add_comm_monoid M] [module R M] [module S M] [smul_comm_class R S M] (f : R →ₗ[R] M) :
@[simp]
theorem linear_map.ring_lmap_equiv_self_symm_apply (R : Type u_1) (S : Type u_6) (M : Type u_9) [semiring R] [semiring S] [add_comm_monoid M] [module R M] [module S M] [smul_comm_class R S M] (x : M) :
def linear_map.ring_lmap_equiv_self (R : Type u_1) (S : Type u_6) (M : Type u_9) [semiring R] [semiring S] [add_comm_monoid M] [module R M] [module S M] [smul_comm_class R S M] :
(R →ₗ[R] M) ≃ₗ[S] M

The equivalence between R-linear maps from R to M, and points of M itself. This says that the forgetful functor from R-modules to types is representable, by R.

This as an S-linear equivalence, under the assumption that S acts on M commuting with R. When R is commutative, we can take this to be the usual action with S = R. Otherwise, S = ℕ shows that the equivalence is additive. See note [bundled maps over different rings].

Equations
def linear_map.comp_right {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R M₂] [module R M₃] (f : M₂ →ₗ[R] M₃) :
(M →ₗ[R] M₂) →ₗ[R] M →ₗ[R] M₃

Composition by f : M₂ → M₃ is a linear map from the space of linear maps M → M₂ to the space of linear maps M₂ → M₃.

Equations
@[simp]
theorem linear_map.comp_right_apply {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R M₂] [module R M₃] (f : M₂ →ₗ[R] M₃) (g : M →ₗ[R] M₂) :
(f.comp_right) g = f.comp g
@[simp]
theorem linear_map.applyₗ_apply_apply {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] (v : M) (f : M →ₗ[R] M₂) :
def linear_map.applyₗ {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] :
M →ₗ[R] (M →ₗ[R] M₂) →ₗ[R] M₂

Applying a linear map at v : M, seen as a linear map from M →ₗ[R] M₂ to M₂. See also linear_map.applyₗ' for a version that works with two different semirings.

This is the linear_map version of add_monoid_hom.eval.

Equations
def linear_map.dom_restrict' {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] (p : submodule R M) :
(M →ₗ[R] M₂) →ₗ[R] p →ₗ[R] M₂

Alternative version of dom_restrict as a linear map.

Equations
@[simp]
theorem linear_map.dom_restrict'_apply {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] (f : M →ₗ[R] M₂) (p : submodule R M) (x : p) :
def linear_map.smul_rightₗ {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] :
(M₂ →ₗ[R] R) →ₗ[R] M →ₗ[R] M₂ →ₗ[R] M

The family of linear maps M₂ → M parameterised by f ∈ M₂ → R, x ∈ M, is linear in f, x.

Equations
@[simp]
theorem linear_map.smul_rightₗ_apply {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [comm_semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] (f : M₂ →ₗ[R] R) (x : M) (c : M₂) :
@[simp]
theorem add_monoid_hom_lequiv_nat_apply {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_monoid A] [add_comm_monoid B] [module R B] (f : A →+ B) :
@[simp]
theorem add_monoid_hom_lequiv_nat_symm_apply {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_monoid A] [add_comm_monoid B] [module R B] (f : A →ₗ[] B) :
def add_monoid_hom_lequiv_nat {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_monoid A] [add_comm_monoid B] [module R B] :

The R-linear equivalence between additive morphisms A →+ B and -linear morphisms A →ₗ[ℕ] B.

Equations
@[simp]
theorem add_monoid_hom_lequiv_int_symm_apply {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_group A] [add_comm_group B] [module R B] (f : A →ₗ[] B) :
def add_monoid_hom_lequiv_int {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_group A] [add_comm_group B] [module R B] :

The R-linear equivalence between additive morphisms A →+ B and -linear morphisms A →ₗ[ℤ] B.

Equations
@[simp]
theorem add_monoid_hom_lequiv_int_apply {A : Type u_1} {B : Type u_2} (R : Type u_3) [semiring R] [add_comm_group A] [add_comm_group B] [module R B] (f : A →+ B) :

Properties of submodules #

def submodule.of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : p p') :

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

Equations
@[simp]
theorem submodule.coe_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : p p') (x : p) :
theorem submodule.of_le_apply {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : p p') (x : p) :
theorem submodule.of_le_injective {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : p p') :
theorem submodule.subtype_comp_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p q : submodule R M) (h : p q) :
@[simp]
theorem submodule.subsingleton_iff (R : Type u_1) {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] :
@[simp]
theorem submodule.nontrivial_iff (R : Type u_1) {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] :
@[protected, instance]
def submodule.unique {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] [subsingleton M] :
Equations
@[protected, instance]
def submodule.unique' {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] [subsingleton R] :
Equations
@[protected, instance]
def submodule.nontrivial {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] [nontrivial M] :
theorem submodule.mem_right_iff_eq_zero_of_disjoint {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : disjoint p p') {x : p} :
x p' x = 0
theorem submodule.mem_left_iff_eq_zero_of_disjoint {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : disjoint p p') {x : p'} :
x p x = 0
def submodule.map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
submodule R₂ M₂

The pushforward of a submodule p ⊆ M by f : M → M₂

Equations
Instances for submodule.map
@[simp]
theorem submodule.map_coe {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
theorem submodule.map_to_add_submonoid {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
theorem submodule.map_to_add_submonoid' {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
@[simp]
theorem submodule.mem_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p : submodule R M} {x : M₂} :
x submodule.map f p ∃ (y : M), y p f y = x
theorem submodule.mem_map_of_mem {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p : submodule R M} {r : M} (h : r p) :
theorem submodule.apply_coe_mem_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) {p : submodule R M} (r : p) :
@[simp]
theorem submodule.map_id {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
theorem submodule.map_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [ring_hom_surjective σ₁₂] [ring_hom_surjective σ₂₃] [ring_hom_surjective σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : submodule R M) :
theorem submodule.map_mono {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p p' : submodule R M} :
@[simp]
theorem submodule.map_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (p : submodule R M) [ring_hom_surjective σ₁₂] :
theorem submodule.map_add_le {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (p : submodule R M) [ring_hom_surjective σ₁₂] (f g : M →ₛₗ[σ₁₂] M₂) :
theorem submodule.range_map_nonempty {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (N : submodule R M) :
(set.range (λ (ϕ : M →ₛₗ[σ₁₂] M₂), submodule.map ϕ N)).nonempty
noncomputable def submodule.equiv_map_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (i : function.injective f) (p : submodule R M) :

The pushforward of a submodule by an injective linear map is linearly equivalent to the original submodule. See also linear_equiv.submodule_map for a computable version when f has an explicit inverse.

Equations
@[simp]
theorem submodule.coe_equiv_map_of_injective_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (i : function.injective f) (p : submodule R M) (x : p) :
def submodule.comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R₂ M₂) :

The pullback of a submodule p ⊆ M₂ along f : M → M₂

Equations
@[simp]
theorem submodule.comap_coe {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R₂ M₂) :
@[simp]
theorem submodule.mem_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {x : M} {f : M →ₛₗ[σ₁₂] M₂} {p : submodule R₂ M₂} :
@[simp]
theorem submodule.comap_id {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
theorem submodule.comap_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] M₃) (p : submodule R₃ M₃) :
theorem submodule.comap_mono {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} {q q' : submodule R₂ M₂} :
theorem submodule.le_comap_pow_of_le_comap {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) {f : M →ₗ[R] M} (h : p submodule.comap f p) (k : ) :
theorem submodule.map_le_iff_le_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p : submodule R M} {q : submodule R₂ M₂} :
theorem submodule.gc_map_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) :
@[simp]
theorem submodule.map_bot {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) :
@[simp]
theorem submodule.map_sup {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (p p' : submodule R M) [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) :
@[simp]
theorem submodule.map_supr {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {ι : Sort u_2} (f : M →ₛₗ[σ₁₂] M₂) (p : ι → submodule R M) :
submodule.map f (⨆ (i : ι), p i) = ⨆ (i : ι), submodule.map f (p i)
@[simp]
theorem submodule.comap_top {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) :
@[simp]
theorem submodule.comap_inf {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (q q' : submodule R₂ M₂) (f : M →ₛₗ[σ₁₂] M₂) :
@[simp]
theorem submodule.comap_infi {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {ι : Sort u_2} (f : M →ₛₗ[σ₁₂] M₂) (p : ι → submodule R₂ M₂) :
submodule.comap f (⨅ (i : ι), p i) = ⨅ (i : ι), submodule.comap f (p i)
@[simp]
theorem submodule.comap_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} (q : submodule R₂ M₂) :
theorem submodule.map_comap_le {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (q : submodule R₂ M₂) :
theorem submodule.le_comap_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : submodule R M) :
def submodule.gi_map_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] :

map f and comap f form a galois_insertion when f is surjective.

Equations
theorem submodule.map_comap_eq_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] (p : submodule R₂ M₂) :
theorem submodule.map_surjective_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] :
theorem submodule.comap_injective_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] :
theorem submodule.map_sup_comap_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] (p q : submodule R₂ M₂) :
theorem submodule.map_supr_comap_of_sujective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] {ι : Sort u_2} (S : ι → submodule R₂ M₂) :
submodule.map f (⨆ (i : ι), submodule.comap f (S i)) = supr S
theorem submodule.map_inf_comap_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] (p q : submodule R₂ M₂) :
theorem submodule.map_infi_comap_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] {ι : Sort u_2} (S : ι → submodule R₂ M₂) :
submodule.map f (⨅ (i : ι), submodule.comap f (S i)) = infi S
theorem submodule.comap_le_comap_iff_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] (p q : submodule R₂ M₂) :
theorem submodule.comap_strict_mono_of_surjective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} (hf : function.surjective f) [ring_hom_surjective σ₁₂] :
def submodule.gci_map_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) :

map f and comap f form a galois_coinsertion when f is injective.

Equations
theorem submodule.comap_map_eq_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) (p : submodule R M) :
theorem submodule.comap_surjective_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) :
theorem submodule.map_injective_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) :
theorem submodule.comap_inf_map_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) (p q : submodule R M) :
theorem submodule.comap_infi_map_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) {ι : Sort u_2} (S : ι → submodule R M) :
submodule.comap f (⨅ (i : ι), submodule.map f (S i)) = infi S
theorem submodule.comap_sup_map_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) (p q : submodule R M) :
theorem submodule.comap_supr_map_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) {ι : Sort u_2} (S : ι → submodule R M) :
submodule.comap f (⨆ (i : ι), submodule.map f (S i)) = supr S
theorem submodule.map_le_map_iff_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) (p q : submodule R M) :
theorem submodule.map_strict_mono_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : function.injective f) :
theorem submodule.map_inf_eq_map_inf_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₁₂ : R →+* R₂} [ring_hom_surjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p : submodule R M} {p' : submodule R₂ M₂} :
theorem submodule.map_comap_subtype {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p p' : submodule R M) :
theorem submodule.eq_zero_of_bot_submodule {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (b : ) :
b = 0
theorem linear_map.infi_invariant {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {σ : R →+* R} [ring_hom_surjective σ] {ι : Sort u_2} (f : M →ₛₗ[σ] M) {p : ι → submodule R M} (hf : ∀ (i : ι) (v : M), v p if v p i) (v : M) (H : v infi p) :
f v infi p

The infimum of a family of invariant submodule of an endomorphism is also an invariant submodule.

@[simp]
theorem submodule.neg_coe {R : Type u_1} {M : Type u_9} [ring R] [add_comm_group M] [module R M] (p : submodule R M) :
@[protected, simp]
theorem submodule.map_neg {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [ring R] [add_comm_group M] [module R M] (p : submodule R M) [add_comm_group M₂] [module R M₂] (f : M →ₗ[R] M₂) :
theorem submodule.comap_smul {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (p : submodule K V₂) (a : K) (h : a 0) :
theorem submodule.map_smul {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (p : submodule K V) (a : K) (h : a 0) :
theorem submodule.comap_smul' {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (p : submodule K V₂) (a : K) :
submodule.comap (a f) p = ⨅ (h : a 0), submodule.comap f p
theorem submodule.map_smul' {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (p : submodule K V) (a : K) :
submodule.map (a f) p = ⨆ (h : a 0), submodule.map f p

Properties of linear maps #

@[simp]
theorem linear_map.map_finsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : Type u_20} [has_zero γ] (f : M →ₛₗ[σ₁₂] M₂) {t : ι →₀ γ} {g : ι → γ → M} :
f (t.sum g) = t.sum (λ (i : ι) (d : γ), f (g i d))
theorem linear_map.coe_finsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : Type u_20} [has_zero γ] (t : ι →₀ γ) (g : ι → γ → (M →ₛₗ[σ₁₂] M₂)) :
(t.sum g) = t.sum (λ (i : ι) (d : γ), (g i d))
@[simp]
theorem linear_map.finsupp_sum_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : Type u_20} [has_zero γ] (t : ι →₀ γ) (g : ι → γ → (M →ₛₗ[σ₁₂] M₂)) (b : M) :
(t.sum g) b = t.sum (λ (i : ι) (d : γ), (g i d) b)
@[simp]
theorem linear_map.map_dfinsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), has_zero (γ i)] [Π (i : ι) (x : γ i), decidable (x 0)] (f : M →ₛₗ[σ₁₂] M₂) {t : Π₀ (i : ι), γ i} {g : Π (i : ι), γ i → M} :
f (t.sum g) = t.sum (λ (i : ι) (d : γ i), f (g i d))
theorem linear_map.coe_dfinsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), has_zero (γ i)] [Π (i : ι) (x : γ i), decidable (x 0)] (t : Π₀ (i : ι), γ i) (g : Π (i : ι), γ i(M →ₛₗ[σ₁₂] M₂)) :
(t.sum g) = t.sum (λ (i : ι) (d : γ i), (g i d))
@[simp]
theorem linear_map.dfinsupp_sum_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), has_zero (γ i)] [Π (i : ι) (x : γ i), decidable (x 0)] (t : Π₀ (i : ι), γ i) (g : Π (i : ι), γ i(M →ₛₗ[σ₁₂] M₂)) (b : M) :
(t.sum g) b = t.sum (λ (i : ι) (d : γ i), (g i d) b)
@[simp]
theorem linear_map.map_dfinsupp_sum_add_hom {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {σ₁₂ : R →+* R₂} [module R M] [module R₂ M₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), add_zero_class (γ i)] (f : M →ₛₗ[σ₁₂] M₂) {t : Π₀ (i : ι), γ i} {g : Π (i : ι), γ i →+ M} :
theorem linear_map.map_cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₂₁ : R₂ →+* R} [ring_hom_surjective σ₂₁] (p : submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (h : ∀ (c : M₂), f c p) (p' : submodule R₂ M₂) :
theorem linear_map.comap_cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {σ₂₁ : R₂ →+* R} (p : submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (hf : ∀ (c : M₂), f c p) (p' : submodule R p) :
def linear_map.range {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
submodule R₂ M₂

The range of a linear map f : M → M₂ is a submodule of M₂. See Note [range copy pattern].

Equations
Instances for linear_map.range
theorem linear_map.range_coe {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.range_to_add_submonoid {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
@[simp]
theorem linear_map.mem_range {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {x : M₂} :
x f.range ∃ (y : M), f y = x
theorem linear_map.range_eq_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.mem_range_self {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) (x : M) :
@[simp]
theorem linear_map.range_id {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] :
theorem linear_map.range_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] [ring_hom_surjective τ₁₂] [ring_hom_surjective τ₂₃] [ring_hom_surjective τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
theorem linear_map.range_comp_le_range {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] [ring_hom_surjective τ₂₃] [ring_hom_surjective τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
theorem linear_map.range_eq_top {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} :
theorem linear_map.range_le_iff_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R₂ M₂} :
theorem linear_map.map_le_range {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R M} :
@[simp]
theorem linear_map.range_neg {R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [semiring R] [ring R₂] [add_comm_monoid M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
(-f).range = f.range
@[simp]
theorem linear_map.iterate_range_coe {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) (n : ) :
def linear_map.iterate_range {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) :

The decreasing sequence of submodules consisting of the ranges of the iterates of a linear map.

Equations
def linear_map.range_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
M →ₛₗ[τ₁₂] (f.range)

Restrict the codomain of a linear map f to f.range.

This is the bundled version of set.range_factorization.

Equations
@[protected, instance]
def linear_map.fintype_range {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [fintype M] [decidable_eq M₂] [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :

The range of a linear map is finite if the domain is finite. Note: this instance can form a diamond with subtype.fintype in the presence of fintype M₂.

Equations
def linear_map.ker {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) :

The kernel of a linear map f : M → M₂ is defined to be comap f ⊥. This is equivalent to the set of x : M such that f x = 0. The kernel is a submodule of M.

Equations
@[simp]
theorem linear_map.mem_ker {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {y : M} :
y f.ker f y = 0
@[simp]
theorem linear_map.ker_id {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] :
@[simp]
theorem linear_map.map_coe_ker {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) (x : (f.ker)) :
f x = 0
theorem linear_map.ker_to_add_submonoid {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.comp_ker_subtype {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.ker_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
theorem linear_map.ker_le_ker_comp {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) :
f.ker (g.comp f).ker
theorem linear_map.disjoint_ker {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R M} :
disjoint p f.ker ∀ (x : M), x pf x = 0x = 0
theorem linear_map.ker_eq_bot' {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} :
f.ker = ∀ (m : M), f m = 0m = 0
theorem linear_map.ker_eq_bot_of_inverse {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [ring_hom_inv_pair τ₁₂ τ₂₁] {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₁] M} (h : g.comp f = linear_map.id) :
theorem linear_map.le_ker_iff_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R M} :
theorem linear_map.ker_cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₂₁ : R₂ →+* R} (p : submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf : ∀ (c : M₂), f c p) :
theorem linear_map.range_cod_restrict {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₂₁ : R₂ →+* R} [ring_hom_surjective τ₂₁] (p : submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf : ∀ (c : M₂), f c p) :
theorem linear_map.ker_restrict {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p : submodule R M} {f : M →ₗ[R] M} (hf : ∀ (x : M), x pf x p) :
theorem submodule.map_comap_eq {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) (q : submodule R₂ M₂) :
theorem submodule.map_comap_eq_self {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {q : submodule R₂ M₂} (h : q f.range) :
@[simp]
theorem linear_map.ker_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} :
@[simp]
theorem linear_map.range_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] :
theorem linear_map.ker_eq_top {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} :
f.ker = f = 0
theorem linear_map.range_le_bot_iff {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
f.range f = 0
theorem linear_map.range_eq_bot {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} :
f.range = f = 0
theorem linear_map.range_le_ker_iff {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₃] M₃} :
f.range g.ker g.comp f = 0
theorem linear_map.comap_le_comap_iff {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} (hf : f.range = ) {p p' : submodule R₂ M₂} :
theorem linear_map.comap_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} (hf : f.range = ) :
theorem linear_map.ker_eq_bot_of_injective {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} (hf : function.injective f) :
@[simp]
theorem linear_map.iterate_ker_coe {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) (n : ) :
(f.iterate_ker) n = (f ^ n).ker
def linear_map.iterate_ker {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (f : M →ₗ[R] M) :

The increasing sequence of submodules consisting of the kernels of the iterates of a linear map.

Equations
theorem linear_map.range_to_add_subgroup {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.ker_to_add_subgroup {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) :
theorem linear_map.sub_mem_ker_iff {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {x y : M} :
x - y f.ker f x = f y
theorem linear_map.disjoint_ker' {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R M} :
disjoint p f.ker ∀ (x : M), x p∀ (y : M), y pf x = f yx = y
theorem linear_map.inj_of_disjoint_ker {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} {p : submodule R M} {s : set M} (h : s p) (hd : disjoint p f.ker) (x : M) (H : x s) (y : M) (H_1 : y s) :
f x = f yx = y
theorem linear_map.ker_eq_bot {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} :
theorem linear_map.ker_le_iff {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [ring R] [ring R₂] [add_comm_group M] [add_comm_group M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} [ring_hom_surjective τ₁₂] {p : submodule R M} :
f.ker p ∃ (y : M₂) (H : y f.range), f ⁻¹' {y} p
theorem linear_map.ker_smul {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a 0) :
(a f).ker = f.ker
theorem linear_map.ker_smul' {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (a : K) :
(a f).ker = ⨅ (h : a 0), f.ker
theorem linear_map.range_smul {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (a : K) (h : a 0) :
(a f).range = f.range
theorem linear_map.range_smul' {K : Type u_7} {V : Type u_18} {V₂ : Type u_19} [field K] [add_comm_group V] [module K V] [add_comm_group V₂] [module K V₂] (f : V →ₗ[K] V₂) (a : K) :
(a f).range = ⨆ (h : a 0), f.range
theorem is_linear_map.is_linear_map_add {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] :
is_linear_map R (λ (x : M × M), x.fst + x.snd)
theorem is_linear_map.is_linear_map_sub {R : Type u_1} {M : Type u_2} [semiring R] [add_comm_group M] [module R M] :
is_linear_map R (λ (x : M × M), x.fst - x.snd)
@[simp]
theorem submodule.map_top {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} [ring_hom_surjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) :
@[simp]
theorem submodule.comap_bot {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) :
@[simp]
theorem submodule.ker_subtype {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
@[simp]
theorem submodule.range_subtype {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
theorem submodule.map_subtype_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) (p' : submodule R p) :
@[simp]
theorem submodule.map_subtype_top {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :

Under the canonical linear map from a submodule p to the ambient space M, the image of the maximal submodule of p is just p.

@[simp]
theorem submodule.comap_subtype_eq_top {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} :
@[simp]
theorem submodule.comap_subtype_self {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
@[simp]
theorem submodule.ker_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p p' : submodule R M) (h : p p') :
theorem submodule.range_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p q : submodule R M) (h : p q) :
@[simp]
theorem submodule.map_subtype_range_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p p' : submodule R M} (h : p p') :
theorem submodule.disjoint_iff_comap_eq_bot {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {p q : submodule R M} :
def submodule.map_subtype.rel_iso {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :
submodule R p ≃o {p' // p' p}

If N ⊆ M then submodules of N are the same as submodules of M contained in N

Equations
def submodule.map_subtype.order_embedding {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) :

If p ⊆ M is a submodule, the ordering of submodules of p is embedded in the ordering of submodules of M.

Equations
@[simp]
theorem submodule.map_subtype_embedding_eq {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] (p : submodule R M) (p' : submodule R p) :
theorem linear_map.ker_eq_bot_of_cancel {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} (h : ∀ (u v : (f.ker) →ₗ[R] M), f.comp u = f.comp vu = v) :

A monomorphism is injective.

theorem linear_map.range_comp_of_range_eq_top {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] [ring_hom_surjective τ₁₂] [ring_hom_surjective τ₂₃] [ring_hom_surjective τ₁₃] {f : M →ₛₗ[τ₁₂] M₂} (g : M₂ →ₛₗ[τ₂₃] M₃) (hf : f.range = ) :
(g.comp f).range = g.range
theorem linear_map.ker_comp_of_ker_eq_bot {R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {M : Type u_9} {M₂ : Type u_12} {M₃ : Type u_13} [semiring R] [semiring R₂] [semiring R₃] [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [module R M] [module R₂ M₂] [module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [ring_hom_comp_triple τ₁₂ τ₂₃ τ₁₃] (f : M →ₛₗ[τ₁₂] M₂) {g : M₂ →ₛₗ[τ₂₃] M₃} (hg : g.ker = ) :
(g.comp f).ker = f.ker
def linear_map.submodule_image {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {M' : Type u_2} [add_comm_monoid M'] [module R M'] {O : submodule R M} (ϕ : O →ₗ[R] M') (N : submodule R M) :

If O is a submodule of M, and Φ : O →ₗ M' is a linear map, then (ϕ : O →ₗ M').submodule_image N is ϕ(N) as a submodule of M'

Equations
@[simp]
theorem linear_map.mem_submodule_image {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {M' : Type u_2} [add_comm_monoid M'] [module R M'] {O : submodule R M} {ϕ : O →ₗ[R] M'} {N : submodule R M} {x : M'} :
x ϕ.submodule_image N ∃ (y : M) (yO : y O) (yN : y N), ϕ y, yO⟩ = x
theorem linear_map.mem_submodule_image_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {M' : Type u_2} [add_comm_monoid M'] [module R M'] {O : submodule R M} {ϕ : O →ₗ[R] M'} {N : submodule R M} (hNO : N O) {x : M'} :
x ϕ.submodule_image N ∃ (y : M) (yN : y N), ϕ y, _⟩ = x
theorem linear_map.submodule_image_apply_of_le {R : Type u_1} {M : Type u_9} [semiring R] [add_comm_monoid M] [module R M] {M' : Type u_2} [add_comm_group M'] [module R M'] {O : submodule R M} (ϕ : O →ₗ[R] M') (N : submodule R M) (hNO : N O) :
@[simp]
theorem linear_map.range_range_restrict {R : Type u_1} {M : Type u_9} {M₂ : Type u_12} [semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] (f : M →ₗ[R] M₂) :

Linear equivalences #

@[protected, instance]
def linear_equiv.has_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [subsingleton M] [subsingleton M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] :
has_zero (M ≃ₛₗ[σ₁₂] M₂)

Between two zero modules, the zero map is an equivalence.

Equations
@[simp]
theorem linear_equiv.zero_symm {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [subsingleton M] [subsingleton M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] :
0.symm = 0
@[simp]
theorem linear_equiv.coe_zero {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [subsingleton M] [subsingleton M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] :
0 = 0
theorem linear_equiv.zero_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [subsingleton M] [subsingleton M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] (x : M) :
0 x = 0
@[protected, instance]
def linear_equiv.unique {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [subsingleton M] [subsingleton M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] :
unique (M ≃ₛₗ[σ₁₂] M₂)

Between two zero modules, the zero map is the only equivalence.

Equations
theorem linear_equiv.map_eq_comap {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {module_M : module R M} {module_M₂ : module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : ring_hom_inv_pair σ₁₂ σ₂₁} {re₂₁ : ring_hom_inv_pair σ₂₁ σ₁₂} (e : M ≃ₛₗ[σ₁₂] M₂) {p : submodule R M} :
def linear_equiv.submodule_map {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {module_M : module R M} {module_M₂ : module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : ring_hom_inv_pair σ₁₂ σ₂₁} {re₂₁ : ring_hom_inv_pair σ₂₁ σ₁₂} (e : M ≃ₛₗ[σ₁₂] M₂) (p : submodule R M) :

A linear equivalence of two modules restricts to a linear equivalence from any submodule p of the domain onto the image of that submodule.

This is the linear version of add_equiv.submonoid_map and add_equiv.subgroup_map.

This is linear_equiv.of_submodule' but with map on the right instead of comap on the left.

Equations
@[simp]
theorem linear_equiv.submodule_map_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {module_M : module R M} {module_M₂ : module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : ring_hom_inv_pair σ₁₂ σ₂₁} {re₂₁ : ring_hom_inv_pair σ₂₁ σ₁₂} (e : M ≃ₛₗ[σ₁₂] M₂) (p : submodule R M) (x : p) :
@[simp]
theorem linear_equiv.submodule_map_symm_apply {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] {module_M : module R M} {module_M₂ : module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : ring_hom_inv_pair σ₁₂ σ₂₁} {re₂₁ : ring_hom_inv_pair σ₂₁ σ₁₂} (e : M ≃ₛₗ[σ₁₂] M₂) (p : submodule R M) (x : (submodule.map e p)) :
@[simp]
theorem linear_equiv.map_finsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} {γ : Type u_20} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] [has_zero γ] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [ring_hom_inv_pair τ₁₂ τ₂₁] [ring_hom_inv_pair τ₂₁ τ₁₂] (f : M ≃ₛₗ[τ₁₂] M₂) {t : ι →₀ γ} {g : ι → γ → M} :
f (t.sum g) = t.sum (λ (i : ι) (d : γ), f (g i d))
@[simp]
theorem linear_equiv.map_dfinsupp_sum {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [ring_hom_inv_pair τ₁₂ τ₂₁] [ring_hom_inv_pair τ₂₁ τ₁₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), has_zero (γ i)] [Π (i : ι) (x : γ i), decidable (x 0)] (f : M ≃ₛₗ[τ₁₂] M₂) (t : Π₀ (i : ι), γ i) (g : Π (i : ι), γ i → M) :
f (t.sum g) = t.sum (λ (i : ι) (d : γ i), f (g i d))
@[simp]
theorem linear_equiv.map_dfinsupp_sum_add_hom {R : Type u_1} {R₂ : Type u_3} {M : Type u_9} {M₂ : Type u_12} {ι : Type u_17} [semiring R] [semiring R₂] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R₂ M₂] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [ring_hom_inv_pair τ₁₂ τ₂₁] [ring_hom_inv_pair τ₂₁ τ₁₂] {γ : ι → Type u_20} [decidable_eq ι] [Π (i : ι), add_zero_class (γ i)] (f : M ≃ₛₗ[τ₁₂] M₂) (t : Π₀ (i : ι), γ i) (g : Π (i : ι), γ i →+ M) :
@[protected]
def linear_equiv.curry (R : Type u_1) (V : Type u_18) (V₂ : Type u_19) [semiring R] :
(V × V₂ → R) ≃ₗ[R] V → V₂ → R

Linear equivalence between a curried and uncurried function. Differs from tensor_product.curry.

Equations
@[simp]
theorem linear_equiv.coe_curry (R : Type u_1) (V : Type u_18) (V₂ : Type u_19) [semiring R] :