mathlib documentation

linear_algebra.finite_dimensional

Finite dimensional vector spaces #

Definition and basic properties of finite dimensional vector spaces, of their dimensions, and of linear maps on such spaces.

Main definitions #

Assume V is a vector space over a field K. There are (at least) three equivalent definitions of finite-dimensionality of V:

We introduce a typeclass finite_dimensional K V capturing this property. For ease of transfer of proof, it is defined using the third point of view, i.e., as is_noetherian. However, we prove that all these points of view are equivalent, with the following lemmas (in the namespace finite_dimensional):

Also defined is finrank, the dimension of a finite dimensional space, returning a nat, as opposed to module.rank, which returns a cardinal. When the space has infinite dimension, its finrank is by convention set to 0.

Preservation of finite-dimensionality and formulas for the dimension are given for

Basic properties of linear maps of a finite-dimensional vector space are given. Notably, the equivalence of injectivity and surjectivity is proved in linear_map.injective_iff_surjective, and the equivalence between left-inverse and right-inverse in mul_eq_one_comm and comp_eq_id_comm.

Implementation notes #

Most results are deduced from the corresponding results for the general dimension (as a cardinal), in dimension.lean. Not all results have been ported yet.

One of the characterizations of finite-dimensionality is in terms of finite generation. This property is currently defined only for submodules, so we express it through the fact that the maximal submodule (which, as a set, coincides with the whole space) is finitely generated. This is not very convenient to use, although there are some helper functions. However, this becomes very convenient when speaking of submodules which are finite-dimensional, as this notion coincides with the fact that the submodule is finitely generated (as a submodule of the whole space). This equivalence is proved in submodule.fg_iff_finite_dimensional.

def finite_dimensional (K : Type u_1) (V : Type u_2) [field K] [add_comm_group V] [module K V] :
Prop

finite_dimensional vector spaces are defined to be noetherian modules. Use finite_dimensional.iff_fg or finite_dimensional.of_fintype_basis to prove finite dimension from a conventional definition.

Equations
def finite_dimensional.fintype_of_fintype (K : Type u) (V : Type v) [field K] [add_comm_group V] [module K V] [fintype K] [is_noetherian K V] :

A finite dimensional vector space over a finite field is finite

Equations
theorem finite_dimensional.of_fintype_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} [fintype ι] (h : basis ι K V) :

If a vector space has a finite basis, then it is finite-dimensional.

theorem finite_dimensional.of_finite_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} {s : set ι} (h : basis s K V) (hs : s.finite) :

If a vector space has a basis indexed by elements of a finite set, then it is finite-dimensional.

theorem finite_dimensional.of_finset_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} {s : finset ι} (h : basis s K V) :

If a vector space has a finite basis, then it is finite-dimensional, finset style.

@[instance]

A subspace of a finite-dimensional space is also finite-dimensional.

@[instance]

A quotient of a finite-dimensional space is also finite-dimensional.

def finite_dimensional.finrank (K : Type u_1) (V : Type u_2) [field K] [add_comm_group V] [module K V] :

The rank of a module as a natural number.

Defined by convention to be 0 if the space has infinite rank.

For a vector space V over a field K, this is the same as the finite dimension of V over K.

Equations

In a finite-dimensional space, its dimension (seen as a cardinal) coincides with its finrank.

theorem finite_dimensional.finrank_eq_of_dim_eq {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {n : } (h : module.rank K V = n) :

We can infer finite_dimensional K V in the presence of [fact (finrank K V = n + 1)]. Declare this as a local instance where needed.

theorem finite_dimensional.dim_eq_card_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} [fintype ι] (h : basis ι K V) :

If a vector space has a finite basis, then its dimension (seen as a cardinal) is equal to the cardinality of the basis.

theorem finite_dimensional.finrank_eq_card_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} [fintype ι] (h : basis ι K V) :

If a vector space has a finite basis, then its dimension is equal to the cardinality of the basis.

theorem finite_dimensional.finrank_eq_card_basis' {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {ι : Type w} (h : basis ι K V) :

If a vector space is finite-dimensional, then the cardinality of any basis is equal to its finrank.

theorem finite_dimensional.finrank_eq_card_finset_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} {b : finset ι} (h : basis b K V) :

If a vector space has a finite basis, then its dimension is equal to the cardinality of the basis. This lemma uses a finset instead of indexed types.

def finite_dimensional.fin_basis (K : Type u) (V : Type v) [field K] [add_comm_group V] [module K V] [finite_dimensional K V] :

A finite dimensional vector space has a basis indexed by fin (finrank K V).

Equations
def finite_dimensional.fin_basis_of_finrank_eq (K : Type u) (V : Type v) [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {n : } (hn : finite_dimensional.finrank K V = n) :
basis (fin n) K V

An n-dimensional vector space has a basis indexed by fin n.

Equations
def finite_dimensional.basis_unique {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (ι : Type u_1) [unique ι] (h : finite_dimensional.finrank K V = 1) :
basis ι K V

A module with dimension 1 has a basis with one element.

Equations
@[simp]
theorem finite_dimensional.basis_unique.repr_eq_zero_iff {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [unique ι] {h : finite_dimensional.finrank K V = 1} {v : V} {i : ι} :
def finite_dimensional.basis_singleton {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (ι : Type u_1) [unique ι] (h : finite_dimensional.finrank K V = 1) (v : V) (hv : v 0) :
basis ι K V

In a vector space with dimension 1, each set {v} is a basis for v ≠ 0.

Equations
@[simp]
theorem finite_dimensional.basis_singleton_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (ι : Type u_1) [unique ι] (h : finite_dimensional.finrank K V = 1) (v : V) (hv : v 0) (i : ι) :
@[simp]
theorem finite_dimensional.range_basis_singleton {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (ι : Type u_1) [unique ι] (h : finite_dimensional.finrank K V = 1) (v : V) (hv : v 0) :
theorem finite_dimensional.cardinal_mk_le_finrank_of_linear_independent {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {ι : Type w} {b : ι → V} (h : linear_independent K b) :
theorem finite_dimensional.fintype_card_le_finrank_of_linear_independent {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {ι : Type u_1} [fintype ι] {b : ι → V} (h : linear_independent K b) :
theorem finite_dimensional.lt_omega_of_linear_independent {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} [finite_dimensional K V] {v : ι → V} (h : linear_independent K v) :
theorem finite_dimensional.not_linear_independent_of_infinite {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type w} [inf : infinite ι] [finite_dimensional K V] (v : ι → V) :
theorem finite_dimensional.finrank_pos_iff_exists_ne_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] :
0 < finite_dimensional.finrank K V ∃ (x : V), x 0

A finite dimensional space has positive finrank iff it has a nonzero element.

A finite dimensional space has positive finrank iff it is nontrivial.

theorem finite_dimensional.finrank_pos {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] [h : nontrivial V] :

A nontrivial finite dimensional space has positive finrank.

A finite dimensional space has zero finrank iff it is a subsingleton. This is the finrank version of dim_zero_iff.

A finite dimensional space that is a subsingleton has zero finrank.

theorem finite_dimensional.exists_nontrivial_relation_of_dim_lt_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {t : finset V} (h : finite_dimensional.finrank K V < t.card) :
∃ (f : V → K), ∑ (e : V) in t, f e e = 0 ∃ (x : V) (H : x t), f x 0

If a finset has cardinality larger than the dimension of the space, then there is a nontrivial linear relation amongst its elements.

theorem finite_dimensional.exists_nontrivial_relation_sum_zero_of_dim_succ_lt_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {t : finset V} (h : finite_dimensional.finrank K V + 1 < t.card) :
∃ (f : V → K), ∑ (e : V) in t, f e e = 0 ∑ (e : V) in t, f e = 0 ∃ (x : V) (H : x t), f x 0

If a finset has cardinality larger than finrank + 1, then there is a nontrivial linear relation amongst its elements, such that the coefficients of the relation sum to zero.

theorem finite_dimensional.exists_relation_sum_zero_pos_coefficient_of_dim_succ_lt_card {L : Type u_1} [linear_ordered_field L] {W : Type v} [add_comm_group W] [module L W] [finite_dimensional L W] {t : finset W} (h : finite_dimensional.finrank L W + 1 < t.card) :
∃ (f : W → L), ∑ (e : W) in t, f e e = 0 ∑ (e : W) in t, f e = 0 ∃ (x : W) (H : x t), 0 < f x

A slight strengthening of exists_nontrivial_relation_sum_zero_of_dim_succ_lt_card available when working over an ordered field: we can ensure a positive coefficient, not just a nonzero coefficient.

theorem finite_dimensional.basis.subset_extend {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} (hs : linear_independent K coe) :
s hs.extend _

If a submodule has maximal dimension in a finite dimensional space, then it is equal to the whole space.

@[simp]

A field is one-dimensional as a vector space over itself.

@[simp]
theorem finite_dimensional.finrank_fintype_fun_eq_card (K : Type u) [field K] {ι : Type v} [fintype ι] :

The vector space of functions on a fintype ι has finrank equal to the cardinality of ι.

@[simp]
theorem finite_dimensional.finrank_fin_fun (K : Type u) [field K] {n : } :

The vector space of functions on fin n has finrank equal to n.

theorem finite_dimensional.span_of_finite (K : Type u) {V : Type v} [field K] [add_comm_group V] [module K V] {A : set V} (hA : A.finite) :

The submodule generated by a finite set is finite-dimensional.

@[instance]

The submodule generated by a single element is finite-dimensional.

theorem finite_dimensional_of_dim_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : module.rank K V = 0) :
theorem finite_dimensional_of_dim_eq_one {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : module.rank K V = 1) :
theorem finrank_eq_zero_of_dim_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (h : module.rank K V = 0) :
theorem finrank_eq_zero_of_basis_imp_not_finite {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : ∀ (s : set V), basis s K V¬s.finite) :
theorem finrank_eq_zero_of_basis_imp_false {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : ∀ (s : finset V), basis s K Vfalse) :
theorem finrank_eq_zero_of_not_exists_basis {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : ¬∃ (s : finset V), nonempty (basis s K V)) :
theorem finrank_eq_zero_of_not_exists_basis_finite {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : ¬∃ (s : set V) (b : basis s K V), s.finite) :
theorem finrank_eq_zero_of_not_exists_basis_finset {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : ¬∃ (s : finset V), nonempty (basis s K V)) :
theorem finite_dimensional_bot (K : Type u) (V : Type v) [field K] [add_comm_group V] [module K V] :
@[simp]
theorem finrank_bot (K : Type u) (V : Type v) [field K] [add_comm_group V] [module K V] :
theorem bot_eq_top_of_dim_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (h : module.rank K V = 0) :
@[simp]
theorem dim_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {S : submodule K V} :
@[simp]
theorem finrank_eq_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {S : submodule K V} [finite_dimensional K S] :
theorem submodule.fg_iff_finite_dimensional {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (s : submodule K V) :

A submodule is finitely generated if and only if it is finite-dimensional

theorem submodule.finite_dimensional_of_le {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {S₁ S₂ : submodule K V} [finite_dimensional K S₂] (h : S₁ S₂) :

A submodule contained in a finite-dimensional submodule is finite-dimensional.

@[instance]
def submodule.finite_dimensional_inf_left {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (S₁ S₂ : submodule K V) [finite_dimensional K S₁] :

The inf of two submodules, the first finite-dimensional, is finite-dimensional.

@[instance]
def submodule.finite_dimensional_inf_right {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (S₁ S₂ : submodule K V) [finite_dimensional K S₂] :

The inf of two submodules, the second finite-dimensional, is finite-dimensional.

@[instance]
def submodule.finite_dimensional_sup {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (S₁ S₂ : submodule K V) [h₁ : finite_dimensional K S₁] [h₂ : finite_dimensional K S₂] :

The sup of two finite-dimensional submodules is finite-dimensional.

In a finite-dimensional vector space, the dimensions of a submodule and of the corresponding quotient add up to the dimension of the space.

The dimension of a submodule is bounded by the dimension of the ambient space.

theorem submodule.finrank_lt {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {s : submodule K V} (h : s < ) :

The dimension of a strict submodule is strictly bounded by the dimension of the ambient space.

The dimension of a quotient is bounded by the dimension of the ambient space.

The sum of the dimensions of s + t and s ∩ t is the sum of the dimensions of s and t

theorem submodule.eq_top_of_disjoint {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (s t : submodule K V) (hdim : finite_dimensional.finrank K s + finite_dimensional.finrank K t = finite_dimensional.finrank K V) (hdisjoint : disjoint s t) :
s t =
theorem linear_equiv.finite_dimensional {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] (f : V ≃ₗ[K] V₂) [finite_dimensional K V] :

Finite dimensionality is preserved under linear equivalence.

theorem linear_equiv.finrank_eq {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] (f : V ≃ₗ[K] V₂) [finite_dimensional K V] :

The dimension of a finite dimensional space is preserved under linear equivalence.

@[instance]
def finite_dimensional_finsupp {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] [finite_dimensional K V] :
theorem finite_dimensional.nonempty_linear_equiv_of_finrank_eq {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] (cond : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂) :
nonempty (V ≃ₗ[K] V₂)

Two finite-dimensional vector spaces are isomorphic if they have the same (finite) dimension.

Two finite-dimensional vector spaces are isomorphic if and only if they have the same (finite) dimension.

def finite_dimensional.linear_equiv.of_finrank_eq {K : Type u} (V : Type v) [field K] [add_comm_group V] [module K V] (V₂ : Type v') [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] (cond : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂) :
V ≃ₗ[K] V₂

Two finite-dimensional vector spaces are isomorphic if they have the same (finite) dimension.

Equations
theorem finite_dimensional.eq_of_le_of_finrank_le {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {S₁ S₂ : submodule K V} [finite_dimensional K S₂] (hle : S₁ S₂) (hd : finite_dimensional.finrank K S₂ finite_dimensional.finrank K S₁) :
S₁ = S₂
theorem finite_dimensional.eq_of_le_of_finrank_eq {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {S₁ S₂ : submodule K V} [finite_dimensional K S₂] (hle : S₁ S₂) (hd : finite_dimensional.finrank K S₁ = finite_dimensional.finrank K S₂) :
S₁ = S₂

If a submodule is less than or equal to a finite-dimensional submodule with the same dimension, they are equal.

def finite_dimensional.linear_equiv.quot_equiv_of_equiv {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] {p : subspace K V} {q : subspace K V₂} (f₁ : p ≃ₗ[K] q) (f₂ : V ≃ₗ[K] V₂) :

Given isomorphic subspaces p q of vector spaces V and V₁ respectively, p.quotient is isomorphic to q.quotient.

Equations

Given the subspaces p q, if p.quotient ≃ₗ[K] q, then q.quotient ≃ₗ[K] p

Equations
theorem linear_map.surjective_of_injective {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {f : V →ₗ[K] V} (hinj : function.injective f) :

On a finite-dimensional space, an injective linear map is surjective.

On a finite-dimensional space, a linear map is injective if and only if it is surjective.

theorem linear_map.ker_eq_bot_iff_range_eq_top {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {f : V →ₗ[K] V} :
theorem linear_map.mul_eq_one_of_mul_eq_one {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {f g : V →ₗ[K] V} (hfg : f * g = 1) :
g * f = 1

In a finite-dimensional space, if linear maps are inverse to each other on one side then they are also inverse to each other on the other side.

theorem linear_map.mul_eq_one_comm {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {f g : V →ₗ[K] V} :
f * g = 1 g * f = 1

In a finite-dimensional space, linear maps are inverse to each other on one side if and only if they are inverse to each other on the other side.

theorem linear_map.comp_eq_id_comm {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {f g : V →ₗ[K] V} :

In a finite-dimensional space, linear maps are inverse to each other on one side if and only if they are inverse to each other on the other side.

theorem linear_map.finite_dimensional_of_surjective {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [h : finite_dimensional K V] (f : V →ₗ[K] V₂) (hf : f.range = ) :

The image under an onto linear map of a finite-dimensional space is also finite-dimensional.

@[instance]
def linear_map.finite_dimensional_range {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [h : finite_dimensional K V] (f : V →ₗ[K] V₂) :

The range of a linear map defined on a finite-dimensional space is also finite-dimensional.

rank-nullity theorem : the dimensions of the kernel and the range of a linear map add up to the dimension of the source space.

def linear_equiv.of_injective_endo {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : V →ₗ[K] V) (h_inj : f.ker = ) :

The linear equivalence corresponging to an injective endomorphism.

Equations
@[simp]
theorem linear_equiv.coe_of_injective_endo {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : V →ₗ[K] V) (h_inj : f.ker = ) :
@[simp]
theorem linear_equiv.of_injective_endo_right_inv {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : V →ₗ[K] V) (h_inj : f.ker = ) :
@[simp]
theorem linear_equiv.of_injective_endo_left_inv {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : V →ₗ[K] V) (h_inj : f.ker = ) :
theorem linear_map.is_unit_iff {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : V →ₗ[K] V) :
@[simp]
theorem finrank_top {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] :
theorem finrank_zero_iff_forall_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] :
finite_dimensional.finrank K V = 0 ∀ (x : V), x = 0
def basis_of_finrank_zero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] {ι : Type u_1} [is_empty ι] (hV : finite_dimensional.finrank K V = 0) :
basis ι K V

If ι is an empty type and V is zero-dimensional, there is a unique ι-indexed basis.

Equations
theorem linear_map.ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] (H : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂) {f : V →ₗ[K] V₂} :
theorem linear_map.finrank_le_finrank_of_injective {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] {f : V →ₗ[K] V₂} (hf : function.injective f) :
def linear_map.linear_equiv_of_ker_eq_bot {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] (f : V →ₗ[K] V₂) (hf : f.ker = ) (hdim : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂) :
V ≃ₗ[K] V₂

Given a linear map f between two vector spaces with the same dimension, if ker f = ⊥ then linear_equiv_of_ker_eq_bot is the induced isomorphism between the two vector spaces.

Equations
@[simp]
theorem linear_map.linear_equiv_of_ker_eq_bot_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {V₂ : Type v'} [add_comm_group V₂] [module K V₂] [finite_dimensional K V] [finite_dimensional K V₂] {f : V →ₗ[K] V₂} (hf : f.ker = ) (hdim : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂) (x : V) :
theorem alg_hom.bijective {F : Type u_1} [field F] {E : Type u_2} [field E] [algebra F E] [finite_dimensional F E] (ϕ : E →ₐ[F] E) :
def alg_equiv_equiv_alg_hom (F : Type u) [field F] (E : Type v) [field E] [algebra F E] [finite_dimensional F E] :
(E ≃ₐ[F] E) (E →ₐ[F] E)

Bijection between algebra equivalences and algebra homomorphisms

Equations
def field_of_finite_dimensional (F : Type u_1) (K : Type u_2) [field F] [integral_domain K] [algebra F K] [finite_dimensional F K] :

An integral domain that is module-finite as an algebra over a field is a field.

Equations
theorem submodule.finrank_mono {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] :
theorem submodule.lt_of_le_of_finrank_lt_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s t : submodule K V} (le : s t) (lt : finite_dimensional.finrank K s < finite_dimensional.finrank K t) :
s < t
theorem finrank_span_le_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (s : set V) [fin : fintype s] :
theorem finrank_span_finset_le_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (s : finset V) :
theorem finrank_span_eq_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] {b : ι → V} (hb : linear_independent K b) :
theorem finrank_span_set_eq_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (s : set V) [fin : fintype s] (hs : linear_independent K coe) :
theorem finrank_span_finset_eq_card {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (s : finset V) (hs : linear_independent K coe) :
theorem span_lt_of_subset_of_card_lt_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} [fintype s] {t : submodule K V} (subset : s t) (card_lt : s.to_finset.card < finite_dimensional.finrank K t) :
theorem span_lt_top_of_card_lt_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} [fintype s] (card_lt : s.to_finset.card < finite_dimensional.finrank K V) :
theorem finrank_span_singleton {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {v : V} (hv : v 0) :
theorem linear_independent_of_span_eq_top_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] {b : ι → V} (span_eq : submodule.span K (set.range b) = ) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
theorem linear_independent_iff_card_eq_finrank_span {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] {b : ι → V} :

A finite family of vectors is linearly independent if and only if its cardinality equals the dimension of its span.

def basis_of_span_eq_top_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] (b : ι → V) (span_eq : submodule.span K (set.range b) = ) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
basis ι K V

A family of finrank K V vectors forms a basis if they span the whole space.

Equations
@[simp]
theorem coe_basis_of_span_eq_top_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [fintype ι] (b : ι → V) (span_eq : submodule.span K (set.range b) = ) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
def finset_basis_of_span_eq_top_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : finset V} (span_eq : submodule.span K s = ) (card_eq : s.card = finite_dimensional.finrank K V) :

A finset of finrank K V vectors forms a basis if they span the whole space.

Equations
@[simp]
theorem finset_basis_of_span_eq_top_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : finset V} (span_eq : submodule.span K s = ) (card_eq : s.card = finite_dimensional.finrank K V) (ᾰ : s →₀ K) :
def set_basis_of_span_eq_top_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} [fintype s] (span_eq : submodule.span K s = ) (card_eq : s.to_finset.card = finite_dimensional.finrank K V) :
basis s K V

A set of finrank K V vectors forms a basis if they span the whole space.

Equations
@[simp]
theorem set_basis_of_span_eq_top_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} [fintype s] (span_eq : submodule.span K s = ) (card_eq : s.to_finset.card = finite_dimensional.finrank K V) (ᾰ : s →₀ K) :
theorem span_eq_top_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [hι : nonempty ι] [fintype ι] {b : ι → V} (lin_ind : linear_independent K b) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
def basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι → V} (lin_ind : linear_independent K b) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
basis ι K V

A linear independent family of finrank K V vectors forms a basis.

Equations
@[simp]
theorem basis_of_linear_independent_of_card_eq_finrank_repr_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι → V} (lin_ind : linear_independent K b) (card_eq : fintype.card ι = finite_dimensional.finrank K V) (ᾰ : V) :
@[simp]
theorem basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι → V} (lin_ind : linear_independent K b) (card_eq : fintype.card ι = finite_dimensional.finrank K V) (ᾰ : ι →₀ K) :
@[simp]
theorem coe_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι → V} (lin_ind : linear_independent K b) (card_eq : fintype.card ι = finite_dimensional.finrank K V) :
@[simp]
theorem finset_basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : finset V} (hs : s.nonempty) (lin_ind : linear_independent K coe) (card_eq : s.card = finite_dimensional.finrank K V) (ᾰ : {x // x s} →₀ K) :
def finset_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : finset V} (hs : s.nonempty) (lin_ind : linear_independent K coe) (card_eq : s.card = finite_dimensional.finrank K V) :
basis s K V

A linear independent finset of finrank K V vectors forms a basis.

Equations
@[simp]
def set_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {s : set V} [nonempty s] [fintype s] (lin_ind : linear_independent K coe) (card_eq : s.to_finset.card = finite_dimensional.finrank K V) :
basis s K V

A linear independent set of finrank K V vectors forms a basis.

Equations

We now give characterisations of finrank K V = 1 and finrank K V ≤ 1.

theorem finrank_eq_one {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (v : V) (n : v 0) (h : ∀ (w : V), ∃ (c : K), c v = w) :

If there is a nonzero vector and every other vector is a multiple of it, then the module has dimension one.

theorem finrank_le_one {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (v : V) (h : ∀ (w : V), ∃ (c : K), c v = w) :

If every vector is a multiple of some v : V, then V has dimension at most one.

theorem finrank_eq_one_iff_of_nonzero {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (v : V) (nz : v 0) :

A vector space with a nonzero vector v has dimension 1 iff v spans.

theorem finrank_eq_one_iff_of_nonzero' {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (v : V) (nz : v 0) :
finite_dimensional.finrank K V = 1 ∀ (w : V), ∃ (c : K), c v = w

A module with a nonzero vector v has dimension 1 iff every vector is a multiple of v.

theorem finrank_eq_one_iff {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] (ι : Type u_1) [unique ι] :

A module has dimension 1 iff there is some v : V so {v} is a basis.

theorem finrank_eq_one_iff' {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] :
finite_dimensional.finrank K V = 1 ∃ (v : V) (n : v 0), ∀ (w : V), ∃ (c : K), c v = w

A module has dimension 1 iff there is some nonzero v : V so every vector is a multiple of v.

theorem finrank_le_one_iff {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] :
finite_dimensional.finrank K V 1 ∃ (v : V), ∀ (w : V), ∃ (c : K), c v = w

A finite dimensional module has dimension at most 1 iff there is some v : V so every vector is a multiple of v.

theorem subalgebra.dim_eq_one_of_eq_bot {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} (h : S = ) :
@[simp]
theorem subalgebra.dim_bot {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] :
theorem subalgebra_top_dim_eq_submodule_top_dim {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] :
theorem subalgebra.dim_top {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] :
theorem subalgebra.finite_dimensional_bot {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] :
@[simp]
theorem subalgebra.finrank_bot {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] :
theorem subalgebra.finrank_eq_one_of_eq_bot {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} (h : S = ) :
theorem subalgebra.eq_bot_of_finrank_one {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} (h : finite_dimensional.finrank F S = 1) :
S =
theorem subalgebra.eq_bot_of_dim_one {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} (h : module.rank F S = 1) :
S =
@[simp]
theorem subalgebra.bot_eq_top_of_dim_eq_one {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] (h : module.rank F E = 1) :
@[simp]
theorem subalgebra.bot_eq_top_of_finrank_eq_one {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] (h : finite_dimensional.finrank F E = 1) :
@[simp]
theorem subalgebra.dim_eq_one_iff {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} :
@[simp]
theorem subalgebra.finrank_eq_one_iff {F : Type u_1} {E : Type u_2} [field F] [field E] [algebra F E] {S : subalgebra F E} :
theorem module.End.exists_ker_pow_eq_ker_pow_succ {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] [finite_dimensional K V] (f : module.End K V) :
theorem module.End.ker_pow_constant {K : Type u} {V : Type v} [field K] [add_comm_group V] [module K V] {f : module.End K V} {k : } (h : linear_map.ker (f ^ k) = linear_map.ker (f ^ k.succ)) (m : ) :