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linear_algebra.finite_dimensional

Finite dimensional vector spaces #

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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 division ring `K`. There are (at least) three equivalent definitions of finite-dimensionality of `V`:

• it admits a finite basis.
• it is finitely generated.
• it is noetherian, i.e., every subspace is finitely generated.

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

• `fintype_basis_index` states that a finite-dimensional vector space has a finite basis
• `finite_dimensional.fin_basis` and `finite_dimensional.fin_basis_of_finrank_eq` are bases for finite dimensional vector spaces, where the index type is `fin`
• `of_fintype_basis` states that the existence of a basis indexed by a finite type implies finite-dimensionality
• `of_finite_basis` states that the existence of a basis indexed by a finite set implies finite-dimensionality
• `is_noetherian.iff_fg` states that the space is finite-dimensional if and only if it is noetherian

We make use of `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`. `finrank` is not defined using `finite_dimensional`. For basic results that do not need the `finite_dimensional` class, import `linear_algebra.finrank`.

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

• submodules
• quotients (for the dimension of a quotient, see `finrank_quotient_add_finrank`)
• linear equivs, in `linear_equiv.finite_dimensional`
• image under a linear map (the rank-nullity formula is in `finrank_range_add_finrank_ker`)

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 `linear_map.mul_eq_one_comm` and `linear_map.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.

You should not assume that there has been any effort to state lemmas as generally as possible.

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`.

@[reducible]
def finite_dimensional (K : Type u_1) (V : Type u_2) [ V] :
Prop

`finite_dimensional` vector spaces are defined to be finite modules. Use `finite_dimensional.of_fintype_basis` to prove finite dimension from another definition.

Equations
theorem finite_dimensional.of_injective {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] (f : V →ₗ[K] V₂) (w : function.injective f) [ V₂] :

If the codomain of an injective linear map is finite dimensional, the domain must be as well.

theorem finite_dimensional.of_surjective {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] (f : V →ₗ[K] V₂) (w : function.surjective f) [ V] :

If the domain of a surjective linear map is finite dimensional, the codomain must be as well.

@[protected, instance]
def finite_dimensional.finite_dimensional_pi (K : Type u) {ι : Type u_1} [finite ι] :
K)
@[protected, instance]
def finite_dimensional.finite_dimensional_pi' (K : Type u) {ι : Type u_1} [finite ι] (M : ι Type u_2) [Π (i : ι), add_comm_group (M i)] [Π (i : ι), (M i)] [I : (i : ι), (M i)] :
(Π (i : ι), M i)
noncomputable def finite_dimensional.fintype_of_fintype (K : Type u) (V : Type v) [ V] [fintype K] [ V] :

A finite dimensional vector space over a finite field is finite

Equations
theorem finite_dimensional.finite_of_finite (K : Type u) (V : Type v) [ V] [finite K] [ V] :
theorem finite_dimensional.of_fintype_basis {K : Type u} {V : Type v} [ V] {ι : Type w} [finite ι] (h : K V) :

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

noncomputable def finite_dimensional.fintype_basis_index {K : Type u} {V : Type v} [ V] {ι : Type u_1} [ V] (b : K V) :

If a vector space is `finite_dimensional`, all bases are indexed by a finite type

Equations
• = let _inst : := finite_dimensional.fintype_basis_index._proof_1 in
@[protected, instance]
noncomputable def finite_dimensional.basis.of_vector_space_index.fintype {K : Type u} {V : Type v} [ V] [ V] :

If a vector space is `finite_dimensional`, `basis.of_vector_space` is indexed by a finite type.

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

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

@[protected, instance]
def finite_dimensional.finite_dimensional_submodule {K : Type u} {V : Type v} [ V] [ V] (S : V) :

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

@[protected, instance]
def finite_dimensional.finite_dimensional_quotient {K : Type u} {V : Type v} [ V] [ V] (S : V) :
(V S)

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

theorem finite_dimensional.finrank_eq_rank' (K : Type u) (V : Type v) [ V] [ V] :
= V

In a finite-dimensional space, its dimension (seen as a cardinal) coincides with its `finrank`. This is a copy of `finrank_eq_rank _ _` which creates easier typeclass searches.

theorem finite_dimensional.finrank_of_infinite_dimensional {K : Type u} {V : Type v} [ V] (h : ¬) :
theorem finite_dimensional.finite_dimensional_of_finrank {K : Type u} {V : Type v} [ V] (h : 0 < ) :
theorem finite_dimensional.finite_dimensional_of_finrank_eq_succ {K : Type u} {V : Type v} [ V] {n : } (hn : = n.succ) :
theorem finite_dimensional.fact_finite_dimensional_of_finrank_eq_succ {K : Type u} {V : Type v} [ V] (n : ) [fact = n + 1)] :

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.finite_dimensional_iff_of_rank_eq_nsmul {K : Type u} {V : Type v} [ V] {W : Type v} [ W] {n : } (hn : n 0) (hVW : V = n W) :
theorem finite_dimensional.finrank_eq_card_basis' {K : Type u} {V : Type v} [ V] [ V] {ι : Type w} (h : K V) :

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

noncomputable def basis.unique {K : Type u} {ι : Type u_1} (b : K K) :

Given a basis of a division ring over itself indexed by a type `ι`, then `ι` is `unique`.

Equations
noncomputable def finite_dimensional.fin_basis (K : Type u) (V : Type v) [ V] [ V] :
basis (fin K V

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

Equations
• = have h : , from _,
noncomputable def finite_dimensional.fin_basis_of_finrank_eq (K : Type u) (V : Type v) [ V] [ V] {n : } (hn : = n) :
basis (fin n) K V

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

Equations
noncomputable def finite_dimensional.basis_unique {K : Type u} {V : Type v} [ V] (ι : Type u_1) [unique ι] (h : = 1) :
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} [ V] {ι : Type u_1} [unique ι] {h : = 1} {v : V} {i : ι} :
(.repr) v) i = 0 v = 0
theorem finite_dimensional.cardinal_mk_le_finrank_of_linear_independent {K : Type u} {V : Type v} [ V] [ V] {ι : Type w} {b : ι V} (h : b) :
theorem finite_dimensional.fintype_card_le_finrank_of_linear_independent {K : Type u} {V : Type v} [ V] [ V] {ι : Type u_1} [fintype ι] {b : ι V} (h : b) :
theorem finite_dimensional.finset_card_le_finrank_of_linear_independent {K : Type u} {V : Type v} [ V] [ V] {b : finset V} (h : (λ (x : b), x)) :
theorem finite_dimensional.lt_aleph_0_of_linear_independent {K : Type u} {V : Type v} [ V] {ι : Type w} [ V] {v : ι V} (h : v) :
theorem linear_independent.finite {K : Type u} {V : Type v} [ V] [ V] {b : set V} (h : (λ (x : b), x)) :
theorem finite_dimensional.not_linear_independent_of_infinite {K : Type u} {V : Type v} [ V] {ι : Type w} [inf : infinite ι] [ V] (v : ι V) :
theorem finite_dimensional.finrank_pos_iff_exists_ne_zero {K : Type u} {V : Type v} [ V] [ V] :
(x : V), x 0

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

theorem finite_dimensional.finrank_pos_iff {K : Type u} {V : Type v} [ V] [ V] :

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

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

A nontrivial finite dimensional space has positive `finrank`.

theorem finite_dimensional.finrank_zero_iff {K : Type u} {V : Type v} [ V] [ V] :

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

theorem finite_dimensional.eq_top_of_finrank_eq {K : Type u} {V : Type v} [ V] [ V] {S : V}  :
S =

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

@[protected, instance]
theorem finite_dimensional.span_of_finite (K : Type u) {V : Type v} [ V] {A : set V} (hA : A.finite) :

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

@[protected, instance]
def finite_dimensional.span_singleton (K : Type u) {V : Type v} [ V] (x : V) :
{x})

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

@[protected, instance]
def finite_dimensional.span_finset (K : Type u) {V : Type v} [ V] (s : finset V) :

The submodule generated by a finset is finite-dimensional.

@[protected, instance]
def finite_dimensional.submodule.map.finite_dimensional (K : Type u) {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] (f : V →ₗ[K] V₂) (p : V) [h : p] :
p)

Pushforwards of finite-dimensional submodules are finite-dimensional.

theorem complete_lattice.independent.subtype_ne_bot_le_finrank_aux {K : Type u} {V : Type v} [ V] [ V] {ι : Type w} {p : ι V} (hp : complete_lattice.independent p) :
cardinal.mk {i // p i }
noncomputable def complete_lattice.independent.fintype_ne_bot_of_finite_dimensional {K : Type u} {V : Type v} [ V] [ V] {ι : Type w} {p : ι V} (hp : complete_lattice.independent p) :
fintype {i // p i }

If `p` is an independent family of subspaces of a finite-dimensional space `V`, then the number of nontrivial subspaces in the family `p` is finite.

Equations
theorem complete_lattice.independent.subtype_ne_bot_le_finrank {K : Type u} {V : Type v} [ V] [ V] {ι : Type w} {p : ι V} (hp : complete_lattice.independent p) [fintype {i // p i }] :
fintype.card {i // p i }

If `p` is an independent family of subspaces of a finite-dimensional space `V`, then the number of nontrivial subspaces in the family `p` is bounded above by the dimension of `V`.

Note that the `fintype` hypothesis required here can be provided by `complete_lattice.independent.fintype_ne_bot_of_finite_dimensional`.

theorem finite_dimensional.exists_nontrivial_relation_of_rank_lt_card {K : Type u} {V : Type v} [ V] [ V] {t : finset V} (h : < t.card) :
(f : V K), t.sum (λ (e : V), 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_rank_succ_lt_card {K : Type u} {V : Type v} [ V] [ V] {t : finset V} (h : < t.card) :
(f : V K), t.sum (λ (e : V), f e e) = 0 t.sum (λ (e : V), 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_rank_succ_lt_card {L : Type u_1} {W : Type v} [ W] [ W] {t : finset W} (h : < t.card) :
(f : W L), t.sum (λ (e : W), f e e) = 0 t.sum (λ (e : W), f e) = 0 (x : W) (H : x t), 0 < f x

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

@[simp]
theorem finite_dimensional.basis_singleton_repr_apply {K : Type u} {V : Type v} [ V] (ι : Type u_1) [unique ι] (h : = 1) (v : V) (hv : v 0) (w : V) :
hv).repr) w =
@[simp]
theorem finite_dimensional.basis_singleton_repr_symm_apply {K : Type u} {V : Type v} [ V] (ι : Type u_1) [unique ι] (h : = 1) (v : V) (hv : v 0) (f : ι →₀ K) :
hv).repr.symm) f =
noncomputable def finite_dimensional.basis_singleton {K : Type u} {V : Type v} [ V] (ι : Type u_1) [unique ι] (h : = 1) (v : V) (hv : v 0) :
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} [ V] (ι : Type u_1) [unique ι] (h : = 1) (v : V) (hv : v 0) (i : ι) :
hv) i = v
@[simp]
theorem finite_dimensional.range_basis_singleton {K : Type u} {V : Type v} [ V] (ι : Type u_1) [unique ι] (h : = 1) (v : V) (hv : v 0) :
set.range hv) = {v}
theorem finite_dimensional_of_rank_eq_nat {K : Type u} {V : Type v} [ V] {n : } (h : V = n) :
theorem finite_dimensional_of_rank_eq_zero {K : Type u} {V : Type v} [ V] (h : V = 0) :
theorem finite_dimensional_of_rank_eq_one {K : Type u} {V : Type v} [ V] (h : V = 1) :
theorem finrank_eq_zero_of_rank_eq_zero {K : Type u} {V : Type v} [ V] [ V] (h : V = 0) :
@[protected, instance]
def finite_dimensional_bot (K : Type u) (V : Type v) [ V] :
theorem bot_eq_top_of_rank_eq_zero {K : Type u} {V : Type v} [ V] (h : V = 0) :
@[simp]
theorem rank_eq_zero {K : Type u} {V : Type v} [ V] {S : V} :
S = 0 S =
@[simp]
theorem finrank_eq_zero {K : Type u} {V : Type v} [ V] {S : V} [ S] :
S =
theorem submodule.fg_iff_finite_dimensional {K : Type u} {V : Type v} [ V] (s : V) :
s.fg

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} [ V] {S₁ S₂ : V} [ S₂] (h : S₁ S₂) :

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

@[protected, instance]
def submodule.finite_dimensional_inf_left {K : Type u} {V : Type v} [ V] (S₁ S₂ : V) [ S₁] :
(S₁ S₂)

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

@[protected, instance]
def submodule.finite_dimensional_inf_right {K : Type u} {V : Type v} [ V] (S₁ S₂ : V) [ S₂] :
(S₁ S₂)

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

@[protected, instance]
def submodule.finite_dimensional_sup {K : Type u} {V : Type v} [ V] (S₁ S₂ : V) [h₁ : S₁] [h₂ : S₂] :
(S₁ S₂)

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

@[protected, instance]
def submodule.finite_dimensional_finset_sup {K : Type u} {V : Type v} [ V] {ι : Type u_1} (s : finset ι) (S : ι V) [ (i : ι), (S i)] :
(s.sup S)

The submodule generated by a finite supremum of finite dimensional submodules is finite-dimensional.

Note that strictly this only needs `∀ i ∈ s, finite_dimensional K (S i)`, but that doesn't work well with typeclass search.

@[protected, instance]
def submodule.finite_dimensional_supr {K : Type u} {V : Type v} [ V] {ι : Sort u_1} [finite ι] (S : ι V) [ (i : ι), (S i)] :
( (i : ι), S i)

The submodule generated by a supremum of finite dimensional submodules, indexed by a finite sort is finite-dimensional.

theorem submodule.finrank_quotient_add_finrank {K : Type u} {V : Type v} [ V] [ V] (s : V) :

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

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

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

theorem submodule.finrank_sup_add_finrank_inf_eq {K : Type u} {V : Type v} [ V] (s t : V) [ s] [ t] :
(s t) + (s t) =

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

theorem submodule.finrank_add_le_finrank_add_finrank {K : Type u} {V : Type v} [ V] (s t : V) [ s] [ t] :
theorem submodule.eq_top_of_disjoint {K : Type u} {V : Type v} [ V] [ V] (s t : V) (hdim : = ) (hdisjoint : t) :
s t =
@[protected]
theorem linear_equiv.finite_dimensional {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] (f : V ≃ₗ[K] V₂) [ V] :

Finite dimensionality is preserved under linear equivalence.

@[protected, instance]
def finite_dimensional_finsupp {K : Type u} {V : Type v} [ V] {ι : Type u_1} [finite ι] [h : V] :
→₀ V)
theorem finite_dimensional.eq_of_le_of_finrank_le {K : Type u} {V : Type v} [ V] {S₁ S₂ : V} [ S₂] (hle : S₁ S₂) (hd : ) :
S₁ = S₂
theorem finite_dimensional.eq_of_le_of_finrank_eq {K : Type u} {V : Type v} [ V] {S₁ S₂ : V} [ S₂] (hle : S₁ S₂) (hd : = ) :
S₁ = S₂

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

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

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

Equations
noncomputable def finite_dimensional.linear_equiv.quot_equiv_of_quot_equiv {K : Type u} {V : Type v} [ V] [ V] {p q : V} (f : (V p) ≃ₗ[K] q) :
(V q) ≃ₗ[K] p

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} [ V] [ V] {f : V →ₗ[K] V} (hinj : function.injective f) :

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

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

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

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

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

theorem linear_map.injective_iff_surjective {K : Type u} {V : Type v} [ V] [ V] {f : V →ₗ[K] V} :

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} [ V] [ V] {f : V →ₗ[K] V} :
theorem linear_map.mul_eq_one_of_mul_eq_one {K : Type u} {V : Type v} [ V] [ 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} [ V] [ 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} [ V] [ 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.finrank_range_add_finrank_ker {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] [ V] (f : V →ₗ[K] V₂) :

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

noncomputable def linear_equiv.of_injective_endo {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) (h_inj : function.injective f) :

The linear equivalence corresponging to an injective endomorphism.

Equations
@[simp]
theorem linear_equiv.coe_of_injective_endo {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) (h_inj : function.injective f) :
h_inj) = f
@[simp]
theorem linear_equiv.of_injective_endo_right_inv {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) (h_inj : function.injective f) :
f * h_inj).symm) = 1
@[simp]
theorem linear_equiv.of_injective_endo_left_inv {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) (h_inj : function.injective f) :
h_inj).symm) * f = 1
theorem linear_map.is_unit_iff_ker_eq_bot {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) :
theorem linear_map.is_unit_iff_range_eq_top {K : Type u} {V : Type v} [ V] [ V] (f : V →ₗ[K] V) :
theorem finrank_zero_iff_forall_zero {K : Type u} {V : Type v} [ V] [ V] :
(x : V), x = 0
noncomputable def basis_of_finrank_zero {K : Type u} {V : Type v} [ V] [ V] {ι : Type u_1} [is_empty ι] (hV : = 0) :
K V

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

Equations
theorem linear_map.injective_iff_surjective_of_finrank_eq_finrank {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] [ V] [ V₂] (H : = ) {f : V →ₗ[K] V₂} :
theorem linear_map.ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] [ V] [ V₂] (H : = ) {f : V →ₗ[K] V₂} :
noncomputable def linear_map.linear_equiv_of_injective {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] [ V] [ V₂] (f : V →ₗ[K] V₂) (hf : function.injective f) (hdim : = ) :
V ≃ₗ[K] V₂

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

Equations
• hdim =
@[simp]
theorem linear_map.linear_equiv_of_injective_apply {K : Type u} {V : Type v} [ V] {V₂ : Type v'} [add_comm_group V₂] [ V₂] [ V] [ V₂] {f : V →ₗ[K] V₂} (hf : function.injective f) (hdim : = ) (x : V) :
hdim) x = f x
noncomputable def division_ring_of_finite_dimensional (F : Type u_1) (K : Type u_2) [field F] [ring K] [is_domain K] [ K] [ K] :

A domain that is module-finite as an algebra over a field is a division ring.

Equations
noncomputable def field_of_finite_dimensional (F : Type u_1) (K : Type u_2) [field F] [comm_ring K] [is_domain K] [ K] [ K] :

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

Equations
theorem submodule.eq_top_of_finrank_eq {K : Type u} {V : Type v} [ V] [ V] {S : V}  :
S =
theorem submodule.finrank_mono {K : Type u} {V : Type v} [ V] [ V] :
monotone (λ (s : V),
theorem submodule.finrank_lt_finrank_of_lt {K : Type u} {V : Type v} [ V] {s t : V} [ t] (hst : s < t) :
theorem submodule.finrank_strict_mono {K : Type u} {V : Type v} [ V] [ V] :
strict_mono (λ (s : V),
theorem submodule.finrank_add_eq_of_is_compl {K : Type u} {V : Type v} [ V] [ V] {U W : V} (h : W) :
theorem finrank_span_singleton {K : Type u} {V : Type v} [ V] {v : V} (hv : v 0) :
= 1
theorem set.finrank_mono {K : Type u} {V : Type v} [ V] [ V] {s t : set V} (h : s t) :
s t
theorem span_eq_top_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {ι : Type u_1} [hι : nonempty ι] [fintype ι] {b : ι V} (lin_ind : b) (card_eq : = ) :
noncomputable def basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι V} (lin_ind : b) (card_eq : = ) :
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} [ V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι V} (lin_ind : b) (card_eq : = ) (ᾰ : V) :
card_eq).repr) = (lin_ind.repr) ( ᾰ)
@[simp]
theorem basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [ V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι V} (lin_ind : b) (card_eq : = ) (ᾰ : ι →₀ K) :
card_eq).repr.symm) = V K b)
@[simp]
theorem coe_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {ι : Type u_1} [nonempty ι] [fintype ι] {b : ι V} (lin_ind : b) (card_eq : = ) :
card_eq) = b
@[simp]
theorem finset_basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [ V] {s : finset V} (hs : s.nonempty) (lin_ind : coe) (card_eq : s.card = ) (ᾰ : {x // x s} →₀ K) :
card_eq).repr.symm) = (finsupp.total {x // x s} V K coe)
@[simp]
theorem finset_basis_of_linear_independent_of_card_eq_finrank_repr_apply {K : Type u} {V : Type v} [ V] {s : finset V} (hs : s.nonempty) (lin_ind : coe) (card_eq : s.card = ) (ᾰ : V) :
card_eq).repr) = (lin_ind.repr) ᾰ)
noncomputable def finset_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {s : finset V} (hs : s.nonempty) (lin_ind : coe) (card_eq : s.card = ) :
K V

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

Equations
• card_eq =
@[simp]
theorem coe_finset_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {s : finset V} (hs : s.nonempty) (lin_ind : coe) (card_eq : s.card = ) :
card_eq) = coe
noncomputable def set_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {s : set V} [nonempty s] [fintype s] (lin_ind : coe) (card_eq : s.to_finset.card = ) :
K V

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

Equations
• card_eq =
@[simp]
theorem set_basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply {K : Type u} {V : Type v} [ V] {s : set V} [nonempty s] [fintype s] (lin_ind : coe) (card_eq : s.to_finset.card = ) (ᾰ : s →₀ K) :
card_eq).repr.symm) = V K coe)
@[simp]
theorem set_basis_of_linear_independent_of_card_eq_finrank_repr_apply {K : Type u} {V : Type v} [ V] {s : set V} [nonempty s] [fintype s] (lin_ind : coe) (card_eq : s.to_finset.card = ) (ᾰ : V) :
card_eq).repr) = (lin_ind.repr) ᾰ)
@[simp]
theorem coe_set_basis_of_linear_independent_of_card_eq_finrank {K : Type u} {V : Type v} [ V] {s : set V} [nonempty s] [fintype s] (lin_ind : coe) (card_eq : s.to_finset.card = ) :
card_eq) = coe

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

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

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} [ V] (v : V) (nz : v 0) :
(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} [ V] (ι : Type u_1) [unique ι] :
nonempty (basis ι K V)

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} [ V] :
(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} [ V] [ V] :
(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 submodule.finrank_le_one_iff_is_principal {K : Type u} {V : Type v} [ V] (W : V) [ W] :
theorem module.finrank_le_one_iff_top_is_principal {K : Type u} {V : Type v} [ V] [ V] :
theorem surjective_of_nonzero_of_finrank_eq_one {K : Type u} {V : Type v} [ V] {W : Type u_1} {A : Type u_2} [semiring A] [ V] [ W] [ W] [ A] (h : = 1) {f : V →ₗ[A] W} (w : f 0) :
theorem is_simple_module_of_finrank_eq_one {K : Type u} {V : Type v} [ V] {A : Type u_1} [semiring A] [ V] [ A] [ V] (h : = 1) :

Any `K`-algebra module that is 1-dimensional over `K` is simple.

theorem subalgebra.finite_dimensional_to_submodule {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] {S : E} :

A `subalgebra` is `finite_dimensional` iff it is finite_dimensional as a submodule.

theorem finite_dimensional.subalgebra_to_submodule {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] {S : E} :

Alias of the reverse direction of `subalgebra.finite_dimensional_to_submodule`.

theorem finite_dimensional.of_subalgebra_to_submodule {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] {S : E} :

Alias of the forward direction of `subalgebra.finite_dimensional_to_submodule`.

@[protected, instance]
def finite_dimensional.finite_dimensional_subalgebra {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [ E] (S : E) :
@[protected, instance]
def subalgebra.finite_dimensional_bot {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] :
theorem subalgebra.eq_bot_of_rank_le_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] {S : E} (h : S 1) :
S =
theorem subalgebra.eq_bot_of_finrank_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] {S : E} (h : = 1) :
S =
@[simp]
theorem subalgebra.rank_eq_one_iff {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] {S : E} :
S = 1 S =
@[simp]
theorem subalgebra.finrank_eq_one_iff {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] {S : E} :
S =
theorem subalgebra.bot_eq_top_iff_rank_eq_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] :
E = 1
theorem subalgebra.bot_eq_top_iff_finrank_eq_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] :
@[simp]
theorem subalgebra.bot_eq_top_of_rank_eq_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] :
E = 1

Alias of the reverse direction of `subalgebra.bot_eq_top_iff_rank_eq_one`.

@[simp]
theorem subalgebra.bot_eq_top_of_finrank_eq_one {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] [nontrivial E] :

Alias of the reverse direction of `subalgebra.bot_eq_top_iff_finrank_eq_one`.

theorem subalgebra.is_simple_order_of_finrank {F : Type u_1} {E : Type u_2} [field F] [ring E] [ E] (hr : = 2) :
theorem module.End.exists_ker_pow_eq_ker_pow_succ {K : Type u} {V : Type v} [ V] [ V] (f : V) :
theorem module.End.ker_pow_constant {K : Type u} {V : Type v} [ V] {f : V} {k : } (h : linear_map.ker (f ^ k) = linear_map.ker (f ^ k.succ)) (m : ) :
theorem module.End.ker_pow_eq_ker_pow_finrank_of_le {K : Type u} {V : Type v} [ V] [ V] {f : V} {m : } (hm : m) :
theorem module.End.ker_pow_le_ker_pow_finrank {K : Type u} {V : Type v} [ V] [ V] (f : V) (m : ) :
theorem cardinal_mk_eq_cardinal_mk_field_pow_rank (K V : Type u) [ V] [ V] :
= ^ V
theorem cardinal_lt_aleph_0_of_finite_dimensional (K V : Type u) [ V] [finite K] [ V] :