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_indexstates that a finite-dimensional vector space has a finite basisfinite_dimensional.fin_basisandfinite_dimensional.fin_basis_of_finrank_eqare bases for finite dimensional vector spaces, where the index type isfinof_fintype_basisstates that the existence of a basis indexed by a finite type implies finite-dimensionalityof_finite_basisstates that the existence of a basis indexed by a finite set implies finite-dimensionalityis_noetherian.iff_fgstates 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)
[division_ring K]
[add_comm_group V]
[module K 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
- finite_dimensional K V = module.finite K V
theorem
finite_dimensional.of_injective
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
(f : V →ₗ[K] V₂)
(w : function.injective ⇑f)
[finite_dimensional K 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}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
(f : V →ₗ[K] V₂)
(w : function.surjective ⇑f)
[finite_dimensional K V] :
finite_dimensional K V₂
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)
[division_ring K]
{ι : Type u_1}
[finite ι] :
finite_dimensional K (ι → K)
@[protected, instance]
def
finite_dimensional.finite_dimensional_pi'
(K : Type u)
[division_ring K]
{ι : Type u_1}
[finite ι]
(M : ι → Type u_2)
[Π (i : ι), add_comm_group (M i)]
[Π (i : ι), module K (M i)]
[I : ∀ (i : ι), finite_dimensional K (M i)] :
finite_dimensional K (Π (i : ι), M i)
noncomputable
def
finite_dimensional.fintype_of_fintype
(K : Type u)
(V : Type v)
[division_ring K]
[add_comm_group V]
[module K V]
[fintype K]
[finite_dimensional K V] :
fintype 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)
[division_ring K]
[add_comm_group V]
[module K V]
[finite K]
[finite_dimensional K V] :
finite V
theorem
finite_dimensional.of_fintype_basis
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Type w}
[finite ι]
(h : basis ι 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}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Type u_1}
[finite_dimensional K V]
(b : basis ι K V) :
fintype ι
If a vector space is finite_dimensional, all bases are indexed by a finite type
Equations
- finite_dimensional.fintype_basis_index b = let _inst : is_noetherian K V := finite_dimensional.fintype_basis_index._proof_1 in is_noetherian.fintype_basis_index b
@[protected, instance]
noncomputable
def
finite_dimensional.basis.of_vector_space_index.fintype
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
If a vector space is finite_dimensional, basis.of_vector_space is indexed by
a finite type.
Equations
- finite_dimensional.basis.of_vector_space_index.fintype = let _inst : is_noetherian K V := finite_dimensional.basis.of_vector_space_index.fintype._proof_1 in is_noetherian.basis.of_vector_space_index.fintype
theorem
finite_dimensional.of_finite_basis
{K : Type u}
{V : Type v}
[division_ring 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.
@[protected, instance]
def
finite_dimensional.finite_dimensional_submodule
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(S : submodule K 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(S : submodule K V) :
finite_dimensional K (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)
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
↑(finite_dimensional.finrank K V) = module.rank K 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}
[division_ring K]
[add_comm_group V]
[module K V]
(h : ¬finite_dimensional K V) :
finite_dimensional.finrank K V = 0
theorem
finite_dimensional.finite_dimensional_of_finrank
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(h : 0 < finite_dimensional.finrank K V) :
theorem
finite_dimensional.finite_dimensional_of_finrank_eq_succ
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{n : ℕ}
(hn : finite_dimensional.finrank K V = n.succ) :
theorem
finite_dimensional.fact_finite_dimensional_of_finrank_eq_succ
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(n : ℕ)
[fact (finite_dimensional.finrank K V = 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}
[division_ring K]
[add_comm_group V]
[module K V]
{W : Type v}
[add_comm_group W]
[module K W]
{n : ℕ}
(hn : n ≠ 0)
(hVW : module.rank K V = n • module.rank K W) :
finite_dimensional K V ↔ finite_dimensional K W
theorem
finite_dimensional.finrank_eq_card_basis'
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{ι : Type w}
(h : basis ι K V) :
↑(finite_dimensional.finrank K V) = cardinal.mk ι
If a vector space is finite-dimensional, then the cardinality of any basis is equal to its
finrank.
noncomputable
def
finite_dimensional.fin_basis
(K : Type u)
(V : Type v)
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
basis (fin (finite_dimensional.finrank K V)) K V
A finite dimensional vector space has a basis indexed by fin (finrank K V).
Equations
- finite_dimensional.fin_basis K V = have h : fintype.card ↥(is_noetherian.finset_basis_index K V) = finite_dimensional.finrank K V, from _, (is_noetherian.finset_basis K V).reindex (fintype.equiv_fin_of_card_eq h)
noncomputable
def
finite_dimensional.fin_basis_of_finrank_eq
(K : Type u)
(V : Type v)
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{n : ℕ}
(hn : finite_dimensional.finrank K V = n) :
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}
[division_ring 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}
[division_ring 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} :
theorem
finite_dimensional.cardinal_mk_le_finrank_of_linear_independent
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{ι : Type w}
{b : ι → V}
(h : linear_independent K b) :
cardinal.mk ι ≤ ↑(finite_dimensional.finrank K V)
theorem
finite_dimensional.fintype_card_le_finrank_of_linear_independent
{K : Type u}
{V : Type v}
[division_ring 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.finset_card_le_finrank_of_linear_independent
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{b : finset V}
(h : linear_independent K (λ (x : ↥b), ↑x)) :
b.card ≤ finite_dimensional.finrank K V
theorem
finite_dimensional.lt_aleph_0_of_linear_independent
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Type w}
[finite_dimensional K V]
{v : ι → V}
(h : linear_independent K v) :
theorem
linear_independent.finite
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{b : set V}
(h : linear_independent K (λ (x : ↥b), ↑x)) :
b.finite
theorem
finite_dimensional.not_linear_independent_of_infinite
{K : Type u}
{V : Type v}
[division_ring 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
0 < finite_dimensional.finrank K V ↔ nontrivial V
A finite dimensional space has positive finrank iff it is nontrivial.
theorem
finite_dimensional.finrank_pos
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
[h : nontrivial V] :
0 < finite_dimensional.finrank K V
A nontrivial finite dimensional space has positive finrank.
theorem
finite_dimensional.finrank_zero_iff
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
finite_dimensional.finrank K V = 0 ↔ subsingleton 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{S : submodule K V}
(h : finite_dimensional.finrank K ↥S = finite_dimensional.finrank K V) :
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}
[division_ring K]
[add_comm_group V]
[module K V]
{A : set V}
(hA : A.finite) :
finite_dimensional K ↥(submodule.span K A)
The submodule generated by a finite set is finite-dimensional.
@[protected, instance]
def
finite_dimensional.span_singleton
(K : Type u)
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(x : V) :
finite_dimensional K ↥(submodule.span K {x})
The submodule generated by a single element is finite-dimensional.
@[protected, instance]
def
finite_dimensional.span_finset
(K : Type u)
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(s : finset V) :
finite_dimensional K ↥(submodule.span K ↑s)
The submodule generated by a finset is finite-dimensional.
@[protected, instance]
def
finite_dimensional.submodule.map.finite_dimensional
(K : Type u)
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
(f : V →ₗ[K] V₂)
(p : submodule K V)
[h : finite_dimensional K ↥p] :
finite_dimensional K ↥(submodule.map f 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{ι : Type w}
{p : ι → submodule K V}
(hp : complete_lattice.independent p) :
cardinal.mk {i // p i ≠ ⊥} ≤ ↑(finite_dimensional.finrank K V)
noncomputable
def
complete_lattice.independent.fintype_ne_bot_of_finite_dimensional
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{ι : Type w}
{p : ι → submodule K V}
(hp : complete_lattice.independent p) :
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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{ι : Type w}
{p : ι → submodule K V}
(hp : complete_lattice.independent p)
[fintype {i // p i ≠ ⊥}] :
fintype.card {i // p i ≠ ⊥} ≤ finite_dimensional.finrank K V
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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{t : finset V}
(h : finite_dimensional.finrank K V < t.card) :
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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{t : finset V}
(h : finite_dimensional.finrank K V + 1 < t.card) :
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}
[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) :
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}
[division_ring 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)
(w : V) :
⇑((finite_dimensional.basis_singleton ι h v hv).repr) w = finsupp.single inhabited.default (⇑(⇑((finite_dimensional.basis_unique ι h).repr) w) inhabited.default / ⇑(⇑((finite_dimensional.basis_unique ι h).repr) v) inhabited.default)
@[simp]
theorem
finite_dimensional.basis_singleton_repr_symm_apply
{K : Type u}
{V : Type v}
[division_ring 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)
(f : ι →₀ K) :
⇑((finite_dimensional.basis_singleton ι h v hv).repr.symm) f = ⇑f inhabited.default • v
noncomputable
def
finite_dimensional.basis_singleton
{K : Type u}
{V : Type v}
[division_ring 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
- finite_dimensional.basis_singleton ι h v hv = let b : basis ι K V := finite_dimensional.basis_unique ι h, h_1 : ⇑(⇑(b.repr) v) inhabited.default ≠ 0 := _ in {repr := {to_fun := λ (w : V), finsupp.single inhabited.default (⇑(⇑(b.repr) w) inhabited.default / ⇑(⇑(b.repr) v) inhabited.default), map_add' := _, map_smul' := _, inv_fun := λ (f : ι →₀ K), ⇑f inhabited.default • v, left_inv := _, right_inv := _}}
@[simp]
theorem
finite_dimensional.basis_singleton_apply
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑(finite_dimensional.basis_singleton ι h v hv) i = v
@[simp]
theorem
finite_dimensional.range_basis_singleton
{K : Type u}
{V : Type v}
[division_ring 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) :
set.range ⇑(finite_dimensional.basis_singleton ι h v hv) = {v}
theorem
finite_dimensional_of_rank_eq_nat
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{n : ℕ}
(h : module.rank K V = ↑n) :
theorem
finite_dimensional_of_rank_eq_zero
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(h : module.rank K V = 0) :
theorem
finite_dimensional_of_rank_eq_one
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(h : module.rank K V = 1) :
theorem
finrank_eq_zero_of_rank_eq_zero
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(h : module.rank K V = 0) :
finite_dimensional.finrank K V = 0
@[protected, instance]
def
finite_dimensional_bot
(K : Type u)
(V : Type v)
[division_ring K]
[add_comm_group V]
[module K V] :
theorem
bot_eq_top_of_rank_eq_zero
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(h : module.rank K V = 0) :
@[simp]
theorem
rank_eq_zero
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{S : submodule K V} :
@[simp]
theorem
finrank_eq_zero
{K : Type u}
{V : Type v}
[division_ring 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}
[division_ring K]
[add_comm_group V]
[module K V]
(s : submodule K V) :
s.fg ↔ finite_dimensional K ↥s
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}
[division_ring K]
[add_comm_group V]
[module K V]
{S₁ S₂ : submodule K V}
[finite_dimensional K ↥S₂]
(h : S₁ ≤ S₂) :
finite_dimensional K ↥S₁
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}
[division_ring K]
[add_comm_group V]
[module K V]
(S₁ S₂ : submodule K V)
[finite_dimensional K ↥S₁] :
finite_dimensional K ↥(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}
[division_ring K]
[add_comm_group V]
[module K V]
(S₁ S₂ : submodule K V)
[finite_dimensional K ↥S₂] :
finite_dimensional K ↥(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}
[division_ring K]
[add_comm_group V]
[module K V]
(S₁ S₂ : submodule K V)
[h₁ : finite_dimensional K ↥S₁]
[h₂ : finite_dimensional K ↥S₂] :
finite_dimensional K ↥(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}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Type u_1}
(s : finset ι)
(S : ι → submodule K V)
[∀ (i : ι), finite_dimensional K ↥(S i)] :
finite_dimensional K ↥(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}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Sort u_1}
[finite ι]
(S : ι → submodule K V)
[∀ (i : ι), finite_dimensional K ↥(S i)] :
finite_dimensional K (↥⨆ (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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(s : submodule K V) :
finite_dimensional.finrank K (V ⧸ s) + finite_dimensional.finrank K ↥s = finite_dimensional.finrank K 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}
[division_ring 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.
theorem
submodule.finrank_sup_add_finrank_inf_eq
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
(s t : submodule K V)
[finite_dimensional K ↥s]
[finite_dimensional K ↥t] :
finite_dimensional.finrank K ↥(s ⊔ t) + finite_dimensional.finrank K ↥(s ⊓ t) = finite_dimensional.finrank K ↥s + finite_dimensional.finrank K ↥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}
[division_ring K]
[add_comm_group V]
[module K V]
(s t : submodule K V)
[finite_dimensional K ↥s]
[finite_dimensional K ↥t] :
finite_dimensional.finrank K ↥(s ⊔ t) ≤ finite_dimensional.finrank K ↥s + finite_dimensional.finrank K ↥t
theorem
submodule.eq_top_of_disjoint
{K : Type u}
{V : Type v}
[division_ring 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) :
@[protected]
theorem
linear_equiv.finite_dimensional
{K : Type u}
{V : Type v}
[division_ring 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_dimensional K V₂
Finite dimensionality is preserved under linear equivalence.
@[protected, instance]
def
finite_dimensional_finsupp
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{ι : Type u_1}
[finite ι]
[h : finite_dimensional K V] :
finite_dimensional K (ι →₀ V)
theorem
finite_dimensional.eq_of_le_of_finrank_le
{K : Type u}
{V : Type v}
[division_ring 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}
[division_ring 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.
noncomputable
def
finite_dimensional.linear_equiv.quot_equiv_of_equiv
{K : Type u}
{V : Type v}
[division_ring 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
- finite_dimensional.linear_equiv.quot_equiv_of_equiv f₁ f₂ = linear_equiv.of_finrank_eq (V ⧸ p) (V₂ ⧸ q) _
noncomputable
def
finite_dimensional.linear_equiv.quot_equiv_of_quot_equiv
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{p q : subspace K V}
(f : (V ⧸ p) ≃ₗ[K] ↥q) :
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}
[division_ring 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.
theorem
linear_map.finite_dimensional_of_surjective
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
[finite_dimensional K V]
(f : V →ₗ[K] V₂)
(hf : linear_map.range f = ⊤) :
finite_dimensional K V₂
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}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
[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.
theorem
linear_map.injective_iff_surjective
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{f : V →ₗ[K] V} :
linear_map.ker f = ⊥ ↔ linear_map.range f = ⊤
theorem
linear_map.mul_eq_one_of_mul_eq_one
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{f g : V →ₗ[K] V}
(hfg : f * g = 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}
[division_ring 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.comp_eq_id_comm
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{f g : V →ₗ[K] V} :
f.comp g = linear_map.id ↔ g.comp f = linear_map.id
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}
[division_ring K]
[add_comm_group V]
[module K V]
{V₂ : Type v'}
[add_comm_group V₂]
[module K V₂]
[finite_dimensional K 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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V)
(h_inj : function.injective ⇑f) :
The linear equivalence corresponging to an injective endomorphism.
Equations
- linear_equiv.of_injective_endo f h_inj = linear_equiv.of_bijective f _
@[simp]
theorem
linear_equiv.coe_of_injective_endo
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V)
(h_inj : function.injective ⇑f) :
⇑(linear_equiv.of_injective_endo f h_inj) = ⇑f
@[simp]
theorem
linear_equiv.of_injective_endo_right_inv
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V)
(h_inj : function.injective ⇑f) :
f * ↑((linear_equiv.of_injective_endo f h_inj).symm) = 1
@[simp]
theorem
linear_equiv.of_injective_endo_left_inv
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V)
(h_inj : function.injective ⇑f) :
↑((linear_equiv.of_injective_endo f h_inj).symm) * f = 1
theorem
linear_map.is_unit_iff_ker_eq_bot
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V) :
is_unit f ↔ linear_map.ker f = ⊥
theorem
linear_map.is_unit_iff_range_eq_top
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : V →ₗ[K] V) :
is_unit f ↔ linear_map.range f = ⊤
theorem
finrank_zero_iff_forall_zero
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
finite_dimensional.finrank K V = 0 ↔ ∀ (x : V), x = 0
noncomputable
def
basis_of_finrank_zero
{K : Type u}
{V : Type v}
[division_ring 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.injective_iff_surjective_of_finrank_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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.ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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} :
linear_map.ker f = ⊥ ↔ linear_map.range f = ⊤
noncomputable
def
linear_map.linear_equiv_of_injective
{K : Type u}
{V : Type v}
[division_ring 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)
(hdim : finite_dimensional.finrank K V = finite_dimensional.finrank 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
- f.linear_equiv_of_injective hf hdim = linear_equiv.of_bijective f _
@[simp]
theorem
linear_map.linear_equiv_of_injective_apply
{K : Type u}
{V : Type v}
[division_ring 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)
(hdim : finite_dimensional.finrank K V = finite_dimensional.finrank K V₂)
(x : V) :
⇑(f.linear_equiv_of_injective hf 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]
[algebra F K]
[finite_dimensional F K] :
A domain that is module-finite as an algebra over a field is a division ring.
Equations
- division_ring_of_finite_dimensional F K = {add := ring.add _inst_2, add_assoc := _, zero := ring.zero _inst_2, zero_add := _, add_zero := _, nsmul := ring.nsmul _inst_2, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg _inst_2, sub := ring.sub _inst_2, sub_eq_add_neg := _, zsmul := ring.zsmul _inst_2, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := ring.int_cast _inst_2, nat_cast := ring.nat_cast _inst_2, one := ring.one _inst_2, nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := ring.mul _inst_2, mul_assoc := _, one_mul := _, mul_one := _, npow := ring.npow _inst_2, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, inv := λ (x : K), dite (x = 0) (λ (H : x = 0), 0) (λ (H : ¬x = 0), classical.some _), div := div_inv_monoid.div._default ring.mul ring.mul_assoc ring.one ring.one_mul ring.mul_one ring.npow ring.npow_zero' ring.npow_succ' (λ (x : K), dite (x = 0) (λ (H : x = 0), 0) (λ (H : ¬x = 0), classical.some _)), div_eq_mul_inv := _, zpow := div_inv_monoid.zpow._default ring.mul ring.mul_assoc ring.one ring.one_mul ring.mul_one ring.npow ring.npow_zero' ring.npow_succ' (λ (x : K), dite (x = 0) (λ (H : x = 0), 0) (λ (H : ¬x = 0), classical.some _)), zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, exists_pair_ne := _, rat_cast := rat.cast_rec (div_inv_monoid.to_has_inv K), mul_inv_cancel := _, inv_zero := _, rat_cast_mk := _, qsmul := qsmul_rec rat.cast_rec (distrib.to_has_mul K), qsmul_eq_mul' := _}
noncomputable
def
field_of_finite_dimensional
(F : Type u_1)
(K : Type u_2)
[field F]
[comm_ring K]
[is_domain K]
[algebra F K]
[finite_dimensional F K] :
field K
An integral domain that is module-finite as an algebra over a field is a field.
Equations
- field_of_finite_dimensional F K = {add := division_ring.add (division_ring_of_finite_dimensional F K), add_assoc := _, zero := division_ring.zero (division_ring_of_finite_dimensional F K), zero_add := _, add_zero := _, nsmul := division_ring.nsmul (division_ring_of_finite_dimensional F K), nsmul_zero' := _, nsmul_succ' := _, neg := division_ring.neg (division_ring_of_finite_dimensional F K), sub := division_ring.sub (division_ring_of_finite_dimensional F K), sub_eq_add_neg := _, zsmul := division_ring.zsmul (division_ring_of_finite_dimensional F K), zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := division_ring.int_cast (division_ring_of_finite_dimensional F K), nat_cast := division_ring.nat_cast (division_ring_of_finite_dimensional F K), one := division_ring.one (division_ring_of_finite_dimensional F K), nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := division_ring.mul (division_ring_of_finite_dimensional F K), mul_assoc := _, one_mul := _, mul_one := _, npow := division_ring.npow (division_ring_of_finite_dimensional F K), npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _, inv := division_ring.inv (division_ring_of_finite_dimensional F K), div := division_ring.div (division_ring_of_finite_dimensional F K), div_eq_mul_inv := _, zpow := division_ring.zpow (division_ring_of_finite_dimensional F K), zpow_zero' := _, zpow_succ' := _, zpow_neg' := _, exists_pair_ne := _, rat_cast := division_ring.rat_cast (division_ring_of_finite_dimensional F K), mul_inv_cancel := _, inv_zero := _, rat_cast_mk := _, qsmul := division_ring.qsmul (division_ring_of_finite_dimensional F K), qsmul_eq_mul' := _}
theorem
submodule.eq_top_of_finrank_eq
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{S : submodule K V}
(h : finite_dimensional.finrank K ↥S = finite_dimensional.finrank K V) :
theorem
submodule.finrank_mono
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
monotone (λ (s : submodule K V), finite_dimensional.finrank K ↥s)
theorem
submodule.finrank_lt_finrank_of_lt
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{s t : submodule K V}
[finite_dimensional K ↥t]
(hst : s < t) :
theorem
submodule.finrank_strict_mono
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
strict_mono (λ (s : submodule K V), finite_dimensional.finrank K ↥s)
theorem
submodule.finrank_add_eq_of_is_compl
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{U W : submodule K V}
(h : is_compl U W) :
theorem
finrank_span_singleton
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{v : V}
(hv : v ≠ 0) :
finite_dimensional.finrank K ↥(submodule.span K {v}) = 1
theorem
set.finrank_mono
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{s t : set V}
(h : s ⊆ t) :
set.finrank K s ≤ set.finrank K t
theorem
span_eq_top_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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) :
submodule.span K (set.range b) = ⊤
noncomputable
def
basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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
- basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq = basis.mk lin_ind _
@[simp]
theorem
basis_of_linear_independent_of_card_eq_finrank_repr_apply
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑((basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq).repr) ᾰ = ⇑(lin_ind.repr) (⇑(linear_map.cod_restrict (submodule.span K (set.range b)) linear_map.id _) ᾰ)
@[simp]
theorem
basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑((basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq).repr.symm) ᾰ = ⇑(finsupp.total ι V K b) ᾰ
@[simp]
theorem
coe_basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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_of_linear_independent_of_card_eq_finrank lin_ind card_eq) = b
@[simp]
theorem
finset_basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑((finset_basis_of_linear_independent_of_card_eq_finrank hs lin_ind card_eq).repr.symm) ᾰ = ⇑(finsupp.total {x // x ∈ s} V K coe) ᾰ
@[simp]
theorem
finset_basis_of_linear_independent_of_card_eq_finrank_repr_apply
{K : Type u}
{V : Type v}
[division_ring 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 : V) :
⇑((finset_basis_of_linear_independent_of_card_eq_finrank hs lin_ind card_eq).repr) ᾰ = ⇑(lin_ind.repr) (⇑(linear_map.cod_restrict (submodule.span K (set.range coe)) linear_map.id _) ᾰ)
noncomputable
def
finset_basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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) :
A linear independent finset of finrank K V vectors forms a basis.
Equations
- finset_basis_of_linear_independent_of_card_eq_finrank hs lin_ind card_eq = basis_of_linear_independent_of_card_eq_finrank lin_ind _
@[simp]
theorem
coe_finset_basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑(finset_basis_of_linear_independent_of_card_eq_finrank hs lin_ind card_eq) = coe
noncomputable
def
set_basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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) :
A linear independent set of finrank K V vectors forms a basis.
Equations
- set_basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq = basis_of_linear_independent_of_card_eq_finrank lin_ind _
@[simp]
theorem
set_basis_of_linear_independent_of_card_eq_finrank_repr_symm_apply
{K : Type u}
{V : Type v}
[division_ring 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)
(ᾰ : ↥s →₀ K) :
⇑((set_basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq).repr.symm) ᾰ = ⇑(finsupp.total ↥s V K coe) ᾰ
@[simp]
theorem
set_basis_of_linear_independent_of_card_eq_finrank_repr_apply
{K : Type u}
{V : Type v}
[division_ring 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)
(ᾰ : V) :
⇑((set_basis_of_linear_independent_of_card_eq_finrank lin_ind card_eq).repr) ᾰ = ⇑(lin_ind.repr) (⇑(linear_map.cod_restrict (submodule.span K (set.range coe)) linear_map.id _) ᾰ)
@[simp]
theorem
coe_set_basis_of_linear_independent_of_card_eq_finrank
{K : Type u}
{V : Type v}
[division_ring 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) :
⇑(set_basis_of_linear_independent_of_card_eq_finrank lin_ind 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}
[division_ring K]
[add_comm_group V]
[module K V]
(v : V)
(nz : v ≠ 0) :
finite_dimensional.finrank K V = 1 ↔ submodule.span K {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}
[division_ring K]
[add_comm_group V]
[module K V]
(v : V)
(nz : v ≠ 0) :
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}
[division_ring K]
[add_comm_group V]
[module K V]
(ι : Type u_1)
[unique ι] :
finite_dimensional.finrank K V = 1 ↔ 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}
[division_ring K]
[add_comm_group V]
[module K V] :
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}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
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}
[division_ring K]
[add_comm_group V]
[module K V]
(W : submodule K V)
[finite_dimensional K ↥W] :
finite_dimensional.finrank K ↥W ≤ 1 ↔ W.is_principal
theorem
module.finrank_le_one_iff_top_is_principal
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
theorem
surjective_of_nonzero_of_finrank_eq_one
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{W : Type u_1}
{A : Type u_2}
[semiring A]
[module A V]
[add_comm_group W]
[module K W]
[module A W]
[linear_map.compatible_smul V W K A]
(h : finite_dimensional.finrank K W = 1)
{f : V →ₗ[A] W}
(w : f ≠ 0) :
theorem
is_simple_module_of_finrank_eq_one
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{A : Type u_1}
[semiring A]
[module A V]
[has_smul K A]
[is_scalar_tower K A V]
(h : finite_dimensional.finrank K V = 1) :
is_simple_order (submodule A V)
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]
[algebra F E]
{S : subalgebra F 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]
[algebra F E]
{S : subalgebra F 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]
[algebra F E]
{S : subalgebra F 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]
[algebra F E]
[finite_dimensional F E]
(S : subalgebra F E) :
theorem
subalgebra.eq_bot_of_rank_le_one
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F E]
{S : subalgebra F E}
(h : module.rank F ↥S ≤ 1) :
theorem
subalgebra.eq_bot_of_finrank_one
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F E]
{S : subalgebra F E}
(h : finite_dimensional.finrank F ↥S = 1) :
@[simp]
theorem
subalgebra.rank_eq_one_iff
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F E]
[nontrivial E]
{S : subalgebra F E} :
@[simp]
theorem
subalgebra.finrank_eq_one_iff
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F E]
[nontrivial E]
{S : subalgebra F E} :
theorem
subalgebra.bot_eq_top_iff_rank_eq_one
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F E]
[nontrivial E] :
theorem
subalgebra.bot_eq_top_iff_finrank_eq_one
{F : Type u_1}
{E : Type u_2}
[field F]
[ring E]
[algebra F 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]
[algebra F E]
[nontrivial E] :
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]
[algebra F 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]
[algebra F E]
(hr : finite_dimensional.finrank F E = 2) :
is_simple_order (subalgebra F E)
theorem
module.End.exists_ker_pow_eq_ker_pow_succ
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : module.End K V) :
∃ (k : ℕ), k ≤ finite_dimensional.finrank K V ∧ linear_map.ker (f ^ k) = linear_map.ker (f ^ k.succ)
theorem
module.End.ker_pow_constant
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
{f : module.End K V}
{k : ℕ}
(h : linear_map.ker (f ^ k) = linear_map.ker (f ^ k.succ))
(m : ℕ) :
linear_map.ker (f ^ k) = linear_map.ker (f ^ (k + m))
theorem
module.End.ker_pow_eq_ker_pow_finrank_of_le
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
{f : module.End K V}
{m : ℕ}
(hm : finite_dimensional.finrank K V ≤ m) :
linear_map.ker (f ^ m) = linear_map.ker (f ^ finite_dimensional.finrank K V)
theorem
module.End.ker_pow_le_ker_pow_finrank
{K : Type u}
{V : Type v}
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V]
(f : module.End K V)
(m : ℕ) :
linear_map.ker (f ^ m) ≤ linear_map.ker (f ^ finite_dimensional.finrank K V)
theorem
cardinal_mk_eq_cardinal_mk_field_pow_rank
(K V : Type u)
[division_ring K]
[add_comm_group V]
[module K V]
[finite_dimensional K V] :
cardinal.mk V = cardinal.mk K ^ module.rank K V
theorem
cardinal_lt_aleph_0_of_finite_dimensional
(K V : Type u)
[division_ring K]
[add_comm_group V]
[module K V]
[finite K]
[finite_dimensional K V] :