Documentation

Mathlib.LinearAlgebra.Basis.Defs

Bases #

This file defines bases in a module or vector space.

It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light.

Main definitions #

All definitions are given for families of vectors, i.e. v : ι → M where M is the module or vector space and ι : Type* is an arbitrary indexing type.

Basis ι R M is the type of ι-indexed R-bases for a module M, represented by a linear equiv M ≃ₗ[R] ι →₀ R.
the basis vectors of a basis b : Basis ι R M are available as b i, where i : ι
Basis.repr is the isomorphism sending x : M to its coordinates Basis.repr x : ι →₀ R. The converse, turning this isomorphism into a basis, is called Basis.ofRepr.
If ι is finite, there is a variant of repr called Basis.equivFun b : M ≃ₗ[R] ι → R (saving you from having to work with Finsupp). The converse, turning this isomorphism into a basis, is called Basis.ofEquivFun.
Basis.constr b R f constructs a linear map M₁ →ₗ[R] M₂ given the values f : ι → M₂ at the basis elements ⇑b : ι → M₁.
Basis.reindex uses an equiv to map a basis to a different indexing set.
Basis.map uses a linear equiv to map a basis to a different module.

Main statements #

Basis.mk: a linear independent set of vectors spanning the whole module determines a basis
Basis.ext states that two linear maps are equal if they coincide on a basis. Similar results are available for linear equivs (if they coincide on the basis vectors), elements (if their coordinates coincide) and the functions b.repr and ⇑b.

Implementation notes #

We use families instead of sets because it allows us to say that two identical vectors are linearly dependent. For bases, this is useful as well because we can easily derive ordered bases by using an ordered index type ι.

Tags #

basis, bases

structure Basis (ι : Type u_1) (R : Type u_3) (M : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] :

Type (max (max u_1 u_3) u_6)

A Basis ι R M for a module M is the type of ι-indexed R-bases of M.

The basis vectors are available as DFunLike.coe (b : Basis ι R M) : ι → M. To turn a linear independent family of vectors spanning M into a basis, use Basis.mk. They are internally represented as linear equivs M ≃ₗ[R] (ι →₀ R), available as Basis.repr.

ofRepr :: (

repr : M ≃ₗ[R] ι →₀ R
repr is the linear equivalence sending a vector x to its coordinates: the cs such that x = ∑ i, c i.

)

Instances For

instance uniqueBasis {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton R] :

Unique (Basis ι R M)

Equations

uniqueBasis = { default := { repr := default }, uniq := ⋯ }

instance Basis.instInhabitedFinsupp {ι : Type u_1} {R : Type u_3} [Semiring R] :

Inhabited (Basis ι R (ι →₀ R))

Equations

Basis.instInhabitedFinsupp = { default := { repr := LinearEquiv.refl R (ι →₀ R) } }

theorem Basis.repr_injective {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] :

Function.Injective Basis.repr

instance Basis.instFunLike {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] :

FunLike (Basis ι R M) ι M

b i is the ith basis vector.

Equations

Basis.instFunLike = { coe := fun (b : Basis ι R M) (i : ι) => b.repr.symm (Finsupp.single i 1), coe_injective' := ⋯ }

@[simp]

theorem Basis.coe_ofRepr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (e : M ≃ₗ[R] ι →₀ R) :

⇑{ repr := e } = fun (i : ι) => e.symm (Finsupp.single i 1)

theorem Basis.injective {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Nontrivial R] :

Function.Injective ⇑b

theorem Basis.repr_symm_single_one {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :

b.repr.symm (Finsupp.single i 1) = b i

theorem Basis.repr_symm_single {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) (c : R) :

b.repr.symm (Finsupp.single i c) = c • b i

@[simp]

theorem Basis.repr_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :

b.repr (b i) = Finsupp.single i 1

theorem Basis.repr_self_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i j : ι) [Decidable (i = j)] :

(b.repr (b i)) j = if i = j then 1 else 0

@[simp]

theorem Basis.repr_symm_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (v : ι →₀ R) :

b.repr.symm v = (Finsupp.linearCombination R ⇑b) v

@[simp]

theorem Basis.coe_repr_symm {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

↑b.repr.symm = Finsupp.linearCombination R ⇑b

@[simp]

theorem Basis.repr_linearCombination {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (v : ι →₀ R) :

b.repr ((Finsupp.linearCombination R ⇑b) v) = v

@[deprecated Basis.repr_linearCombination]

theorem Basis.repr_total {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (v : ι →₀ R) :

b.repr ((Finsupp.linearCombination R ⇑b) v) = v

Alias of Basis.repr_linearCombination.

@[simp]

theorem Basis.linearCombination_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) :

(Finsupp.linearCombination R ⇑b) (b.repr x) = x

@[deprecated Basis.linearCombination_repr]

theorem Basis.total_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) :

(Finsupp.linearCombination R ⇑b) (b.repr x) = x

Alias of Basis.linearCombination_repr.

theorem Basis.repr_range {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

LinearMap.range ↑b.repr = Finsupp.supported R R Set.univ

theorem Basis.mem_span_repr_support {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (m : M) :

m ∈ Submodule.span R (⇑b '' ↑(b.repr m).support)

theorem Basis.repr_support_subset_of_mem_span {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (s : Set ι) {m : M} (hm : m ∈ Submodule.span R (⇑b '' s)) :

↑(b.repr m).support ⊆ s

theorem Basis.mem_span_image {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {m : M} {s : Set ι} :

m ∈ Submodule.span R (⇑b '' s) ↔ ↑(b.repr m).support ⊆ s

@[simp]

theorem Basis.self_mem_span_image {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Nontrivial R] {i : ι} {s : Set ι} :

b i ∈ Submodule.span R (⇑b '' s) ↔ i ∈ s

def Basis.coord {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :

b.coord i is the linear function giving the i'th coordinate of a vector with respect to the basis b.

b.coord i is an element of the dual space. In particular, for finite-dimensional spaces it is the ιth basis vector of the dual space.

Equations

b.coord i = Finsupp.lapply i ∘ₗ ↑b.repr

Instances For

@[simp]

theorem Basis.coord_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) (a✝ : M) :

(b.coord i) a✝ = (b.repr a✝) i

theorem Basis.forall_coord_eq_zero_iff {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {x : M} :

(∀ (i : ι), (b.coord i) x = 0) ↔ x = 0

noncomputable def Basis.sumCoords {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

The sum of the coordinates of an element m : M with respect to a basis.

Equations

b.sumCoords = ((Finsupp.lsum ℕ) fun (x : ι) => LinearMap.id) ∘ₗ ↑b.repr

Instances For

@[simp]

theorem Basis.coe_sumCoords {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

⇑b.sumCoords = fun (m : M) => (b.repr m).sum fun (x : ι) => id

@[simp]

theorem Basis.coe_sumCoords_of_fintype {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] :

⇑b.sumCoords = ⇑(∑ i : ι, b.coord i)

@[simp]

theorem Basis.sumCoords_self_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :

b.sumCoords (b i) = 1

theorem Basis.dvd_coord_smul {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) (m : M) (r : R) :

r ∣ (b.coord i) (r • m)

theorem Basis.coord_repr_symm {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) (f : ι →₀ R) :

(b.coord i) (b.repr.symm f) = f i

theorem Basis.ext {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R₁ : Type u_10} [Semiring R₁] {σ : R →+* R₁} {M₁ : Type u_11} [AddCommMonoid M₁] [Module R₁ M₁] {f₁ f₂ : M →ₛₗ[σ] M₁} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i)) :

f₁ = f₂

Two linear maps are equal if they are equal on basis vectors.

theorem Basis.ext' {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R₁ : Type u_10} [Semiring R₁] {σ : R →+* R₁} {σ' : R₁ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M₁ : Type u_11} [AddCommMonoid M₁] [Module R₁ M₁] {f₁ f₂ : M ≃ₛₗ[σ] M₁} (h : ∀ (i : ι), f₁ (b i) = f₂ (b i)) :

f₁ = f₂

Two linear equivs are equal if they are equal on basis vectors.

theorem Basis.ext_elem_iff {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {x y : M} :

x = y ↔ ∀ (i : ι), (b.repr x) i = (b.repr y) i

Two elements are equal iff their coordinates are equal.

theorem Basis.ext_elem {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {x y : M} :

(∀ (i : ι), (b.repr x) i = (b.repr y) i) → x = y

Alias of the reverse direction of Basis.ext_elem_iff.

Two elements are equal iff their coordinates are equal.

theorem Basis.repr_eq_iff {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {b : Basis ι R M} {f : M →ₗ[R] ι →₀ R} :

↑b.repr = f ↔ ∀ (i : ι), f (b i) = Finsupp.single i 1

theorem Basis.repr_eq_iff' {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {b : Basis ι R M} {f : M ≃ₗ[R] ι →₀ R} :

b.repr = f ↔ ∀ (i : ι), f (b i) = Finsupp.single i 1

theorem Basis.apply_eq_iff {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {b : Basis ι R M} {x : M} {i : ι} :

b i = x ↔ b.repr x = Finsupp.single i 1

theorem Basis.repr_apply_eq {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (f : M → ι → R) (hadd : ∀ (x y : M), f (x + y) = f x + f y) (hsmul : ∀ (c : R) (x : M), f (c • x) = c • f x) (f_eq : ∀ (i : ι), f (b i) = ⇑(Finsupp.single i 1)) (x : M) (i : ι) :

(b.repr x) i = f x i

An unbundled version of repr_eq_iff

theorem Basis.eq_ofRepr_eq_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {b₁ b₂ : Basis ι R M} (h : ∀ (x : M) (i : ι), (b₁.repr x) i = (b₂.repr x) i) :

b₁ = b₂

Two bases are equal if they assign the same coordinates.

theorem Basis.eq_of_apply_eq {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {b₁ b₂ : Basis ι R M} :

(∀ (i : ι), b₁ i = b₂ i) → b₁ = b₂

Two bases are equal if their basis vectors are the same.

def Basis.map {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (f : M ≃ₗ[R] M') :

Basis ι R M'

Apply the linear equivalence f to the basis vectors.

Equations

b.map f = { repr := f.symm ≪≫ₗ b.repr }

Instances For

@[simp]

theorem Basis.map_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (f : M ≃ₗ[R] M') :

(b.map f).repr = f.symm ≪≫ₗ b.repr

@[simp]

theorem Basis.map_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (f : M ≃ₗ[R] M') (i : ι) :

(b.map f) i = f (b i)

theorem Basis.coe_map {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (f : M ≃ₗ[R] M') :

⇑(b.map f) = ⇑f ∘ ⇑b

instance Basis.instSMul {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G M] [SMulCommClass G R M] :

SMul G (Basis ι R M)

The action on a Basis by acting on each element.

See also Basis.unitsSMul and Basis.groupSMul, for the cases when a different action is applied to each basis element.

Equations

Basis.instSMul = { smul := fun (g : G) (b : Basis ι R M) => b.map (DistribMulAction.toLinearEquiv R M g) }

@[simp]

theorem Basis.smul_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G M] [SMulCommClass G R M] (g : G) (b : Basis ι R M) (i : ι) :

(g • b) i = g • b i

theorem Basis.coe_smul {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G M] [SMulCommClass G R M] (g : G) (b : Basis ι R M) :

⇑(g • b) = g • ⇑b

@[simp]

theorem Basis.smul_eq_map {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (g : M ≃ₗ[R] M) (b : Basis ι R M) :

g • b = b.map g

When the group in question is the automorphisms, • coincides with Basis.map.

@[simp]

theorem Basis.repr_smul {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G M] [SMulCommClass G R M] (g : G) (b : Basis ι R M) :

(g • b).repr = (DistribMulAction.toLinearEquiv R M g).symm ≪≫ₗ b.repr

instance Basis.instMulAction {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} [Group G] [DistribMulAction G M] [SMulCommClass G R M] :

MulAction G (Basis ι R M)

Equations

Basis.instMulAction = Function.Injective.mulAction (fun (f : Basis ι R M) => ⇑f) ⋯ ⋯

instance Basis.instSMulCommClass {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} {G' : Type u_11} [Group G] [Group G'] [DistribMulAction G M] [DistribMulAction G' M] [SMulCommClass G R M] [SMulCommClass G' R M] [SMulCommClass G G' M] :

SMulCommClass G G' (Basis ι R M)

Equations

⋯ = ⋯

instance Basis.instIsScalarTower {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {G : Type u_10} {G' : Type u_11} [Group G] [Group G'] [DistribMulAction G M] [DistribMulAction G' M] [SMulCommClass G R M] [SMulCommClass G' R M] [SMul G G'] [IsScalarTower G G' M] :

IsScalarTower G G' (Basis ι R M)

Equations

⋯ = ⋯

def Basis.mapCoeffs {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R' : Type u_10} [Semiring R'] [Module R' M] (f : R ≃+* R') (h : ∀ (c : R) (x : M), f c • x = c • x) :

Basis ι R' M

If R and R' are isomorphic rings that act identically on a module M, then a basis for M as R-module is also a basis for M as R'-module.

See also Basis.algebraMapCoeffs for the case where f is equal to algebraMap.

Equations

b.mapCoeffs f h = { repr := LinearEquiv.restrictScalars R' b.repr ≪≫ₗ Finsupp.mapRange.linearEquiv (Module.compHom.toLinearEquiv f.symm).symm }

Instances For

@[simp]

theorem Basis.mapCoeffs_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R' : Type u_10} [Semiring R'] [Module R' M] (f : R ≃+* R') (h : ∀ (c : R) (x : M), f c • x = c • x) :

(b.mapCoeffs f h).repr = LinearEquiv.restrictScalars R' b.repr ≪≫ₗ Finsupp.mapRange.linearEquiv (Module.compHom.toLinearEquiv f.symm).symm

theorem Basis.mapCoeffs_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R' : Type u_10} [Semiring R'] [Module R' M] (f : R ≃+* R') (h : ∀ (c : R) (x : M), f c • x = c • x) (i : ι) :

(b.mapCoeffs f h) i = b i

@[simp]

theorem Basis.coe_mapCoeffs {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {R' : Type u_10} [Semiring R'] [Module R' M] (f : R ≃+* R') (h : ∀ (c : R) (x : M), f c • x = c • x) :

⇑(b.mapCoeffs f h) = ⇑b

def Basis.reindex {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (e : ι ≃ ι') :

Basis ι' R M

b.reindex (e : ι ≃ ι') is a basis indexed by ι'

Equations

b.reindex e = { repr := b.repr ≪≫ₗ Finsupp.domLCongr e }

Instances For

theorem Basis.reindex_apply {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (e : ι ≃ ι') (i' : ι') :

(b.reindex e) i' = b (e.symm i')

@[simp]

theorem Basis.coe_reindex {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (e : ι ≃ ι') :

⇑(b.reindex e) = ⇑b ∘ ⇑e.symm

theorem Basis.repr_reindex_apply {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) (e : ι ≃ ι') (i' : ι') :

((b.reindex e).repr x) i' = (b.repr x) (e.symm i')

@[simp]

theorem Basis.repr_reindex {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) (e : ι ≃ ι') :

(b.reindex e).repr x = Finsupp.mapDomain (⇑e) (b.repr x)

@[simp]

theorem Basis.reindex_refl {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

b.reindex (Equiv.refl ι) = b

theorem Basis.range_reindex {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (e : ι ≃ ι') :

Set.range ⇑(b.reindex e) = Set.range ⇑b

simp can prove this as Basis.coe_reindex + EquivLike.range_comp

@[simp]

theorem Basis.sumCoords_reindex {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (e : ι ≃ ι') :

(b.reindex e).sumCoords = b.sumCoords

def Basis.reindexRange {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

Basis (↑(Set.range ⇑b)) R M

b.reindex_range is a basis indexed by range b, the basis vectors themselves.

Equations

b.reindexRange = if h : Nontrivial R then b.reindex (Equiv.ofInjective ⇑b ⋯) else { repr := Module.subsingletonEquiv R M ↑(Set.range ⇑b) }

Instances For

theorem Basis.reindexRange_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) (h : b i ∈ Set.range ⇑b := ⋯) :

b.reindexRange ⟨b i, h⟩ = b i

theorem Basis.reindexRange_repr_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :

b.reindexRange.repr (b i) = Finsupp.single ⟨b i, ⋯⟩ 1

@[simp]

theorem Basis.reindexRange_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : ↑(Set.range ⇑b)) :

b.reindexRange x = ↑x

theorem Basis.reindexRange_repr' {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) {bi : M} {i : ι} (h : b i = bi) :

(b.reindexRange.repr x) ⟨bi, ⋯⟩ = (b.repr x) i

@[simp]

theorem Basis.reindexRange_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) (i : ι) (h : b i ∈ Set.range ⇑b := ⋯) :

(b.reindexRange.repr x) ⟨b i, h⟩ = (b.repr x) i

def Basis.reindexFinsetRange {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] [DecidableEq M] :

Basis { x : M // x ∈ Finset.image (⇑b) Finset.univ } R M

b.reindexFinsetRange is a basis indexed by Finset.univ.image b, the finite set of basis vectors themselves.

Equations

b.reindexFinsetRange = b.reindexRange.reindex ((Equiv.refl M).subtypeEquiv ⋯)

Instances For

theorem Basis.reindexFinsetRange_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] [DecidableEq M] (i : ι) (h : b i ∈ Finset.image (⇑b) Finset.univ := ⋯) :

b.reindexFinsetRange ⟨b i, h⟩ = b i

@[simp]

theorem Basis.reindexFinsetRange_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] [DecidableEq M] (x : { x : M // x ∈ Finset.image (⇑b) Finset.univ }) :

b.reindexFinsetRange x = ↑x

theorem Basis.reindexFinsetRange_repr_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] [DecidableEq M] (i : ι) :

b.reindexFinsetRange.repr (b i) = Finsupp.single ⟨b i, ⋯⟩ 1

@[simp]

theorem Basis.reindexFinsetRange_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Fintype ι] [DecidableEq M] (x : M) (i : ι) (h : b i ∈ Finset.image (⇑b) Finset.univ := ⋯) :

(b.reindexFinsetRange.repr x) ⟨b i, h⟩ = (b.repr x) i

theorem Basis.mem_span {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (x : M) :

x ∈ Submodule.span R (Set.range ⇑b)

@[simp]

theorem Basis.span_eq {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

Submodule.span R (Set.range ⇑b) = ⊤

theorem Basis.index_nonempty {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) [Nontrivial M] :

theorem Basis.mem_submodule_iff {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] {P : Submodule R M} (b : Basis ι R ↥P) {x : M} :

x ∈ P ↔ ∃ (c : ι →₀ R), x = c.sum fun (i : ι) (x : R) => x • ↑(b i)

If the submodule P has a basis, x ∈ P iff it is a linear combination of basis vectors.

def Basis.constr {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] :

(ι → M') ≃ₗ[S] M →ₗ[R] M'

Construct a linear map given the value at the basis, called Basis.constr b S f where b is a basis, f is the value of the linear map over the elements of the basis, and S is an extra semiring (typically S = R or S = ℕ).

This definition is parameterized over an extra Semiring S, such that SMulCommClass R S M' holds. If R is commutative, you can set S := R; if R is not commutative, you can recover an AddEquiv by setting S := ℕ. See library note [bundled maps over different rings].

Equations

One or more equations did not get rendered due to their size.

Instances For

theorem Basis.constr_def {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : ι → M') :

(b.constr S) f = Finsupp.linearCombination R id ∘ₗ Finsupp.lmapDomain R R f ∘ₗ ↑b.repr

theorem Basis.constr_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : ι → M') (x : M) :

((b.constr S) f) x = (b.repr x).sum fun (b : ι) (a : R) => a • f b

@[simp]

theorem Basis.constr_basis {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : ι → M') (i : ι) :

((b.constr S) f) (b i) = f i

theorem Basis.constr_eq {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] {g : ι → M'} {f : M →ₗ[R] M'} (h : ∀ (i : ι), g i = f (b i)) :

(b.constr S) g = f

theorem Basis.constr_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : M →ₗ[R] M') :

((b.constr S) fun (i : ι) => f (b i)) = f

theorem Basis.constr_range {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] {f : ι → M'} :

LinearMap.range ((b.constr S) f) = Submodule.span R (Set.range f)

@[simp]

theorem Basis.constr_comp {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] (f : M' →ₗ[R] M') (v : ι → M') :

(b.constr S) (⇑f ∘ v) = f ∘ₗ (b.constr S) v

def Basis.equiv {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (e : ι ≃ ι') :

If b is a basis for M and b' a basis for M', and the index types are equivalent, b.equiv b' e is a linear equivalence M ≃ₗ[R] M', mapping b i to b' (e i).

Equations

b.equiv b' e = b.repr ≪≫ₗ (b'.reindex e.symm).repr.symm

Instances For

@[simp]

theorem Basis.equiv_apply {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (i : ι) (b' : Basis ι' R M') (e : ι ≃ ι') :

(b.equiv b' e) (b i) = b' (e i)

@[simp]

theorem Basis.equiv_refl {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

b.equiv b (Equiv.refl ι) = LinearEquiv.refl R M

@[simp]

theorem Basis.equiv_symm {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (e : ι ≃ ι') :

(b.equiv b' e).symm = b'.equiv b e.symm

@[simp]

theorem Basis.equiv_trans {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} {M'' : Type u_8} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') [AddCommMonoid M''] [Module R M''] {ι'' : Type u_10} (b'' : Basis ι'' R M'') (e : ι ≃ ι') (e' : ι' ≃ ι'') :

b.equiv b' e ≪≫ₗ b'.equiv b'' e' = b.equiv b'' (e.trans e')

@[simp]

theorem Basis.map_equiv {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (e : ι ≃ ι') :

b.map (b.equiv b' e) = b'.reindex e.symm

def Basis.singleton (ι : Type u_10) (R : Type u_11) [Unique ι] [Semiring R] :

Basis ι R R

Basis.singleton ι R is the basis sending the unique element of ι to 1 : R.

Equations

One or more equations did not get rendered due to their size.

Instances For

@[simp]

theorem Basis.singleton_apply (ι : Type u_10) (R : Type u_11) [Unique ι] [Semiring R] (i : ι) :

(Basis.singleton ι R) i = 1

@[simp]

theorem Basis.singleton_repr (ι : Type u_10) (R : Type u_11) [Unique ι] [Semiring R] (x : R) (i : ι) :

((Basis.singleton ι R).repr x) i = x

def Basis.empty {ι : Type u_1} {R : Type u_3} (M : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton M] [IsEmpty ι] :

Basis ι R M

If M is a subsingleton and ι is empty, this is the unique ι-indexed basis for M.

Equations

Basis.empty M = { repr := 0 }

Instances For

instance Basis.emptyUnique {ι : Type u_1} {R : Type u_3} (M : Type u_6) [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton M] [IsEmpty ι] :

Unique (Basis ι R M)

Equations

Basis.emptyUnique M = { default := Basis.empty M, uniq := ⋯ }

def Basis.equivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (b : Basis ι R M) :

M ≃ₗ[R] ι → R

A module over R with a finite basis is linearly equivalent to functions from its basis to R.

Equations

b.equivFun = b.repr ≪≫ₗ let __src := Finsupp.equivFunOnFinite; { toFun := DFunLike.coe, map_add' := ⋯, map_smul' := ⋯, invFun := __src.invFun, left_inv := ⋯, right_inv := ⋯ }

Instances For

def Module.fintypeOfFintype {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) [Fintype R] :

A module over a finite ring that admits a finite basis is finite.

Equations

Module.fintypeOfFintype b = Fintype.ofEquiv (ι → R) b.equivFun.toEquiv.symm

Instances For

theorem Module.card_fintype {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) [Fintype R] [Fintype M] :

Fintype.card M = Fintype.card R ^ Fintype.card ι

@[simp]

theorem Basis.equivFun_symm_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) (x : ι → R) :

b.equivFun.symm x = ∑ i : ι, x i • b i

Given a basis v indexed by ι, the canonical linear equivalence between ι → R and M maps a function x : ι → R to the linear combination ∑_i x i • v i.

@[simp]

theorem Basis.equivFun_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (b : Basis ι R M) (u : M) :

b.equivFun u = ⇑(b.repr u)

@[simp]

theorem Basis.map_equivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] [Finite ι] (b : Basis ι R M) (f : M ≃ₗ[R] M') :

(b.map f).equivFun = f.symm ≪≫ₗ b.equivFun

theorem Basis.sum_equivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) (u : M) :

∑ i : ι, b.equivFun u i • b i = u

theorem Basis.sum_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) (u : M) :

∑ i : ι, (b.repr u) i • b i = u

@[simp]

theorem Basis.equivFun_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] [DecidableEq ι] (b : Basis ι R M) (i j : ι) :

b.equivFun (b i) j = if i = j then 1 else 0

theorem Basis.repr_sum_self {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] (b : Basis ι R M) (c : ι → R) :

⇑(b.repr (∑ i : ι, c i • b i)) = c

def Basis.ofEquivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (e : M ≃ₗ[R] ι → R) :

Basis ι R M

Define a basis by mapping each vector x : M to its coordinates e x : ι → R, as long as ι is finite.

Equations

Basis.ofEquivFun e = { repr := e ≪≫ₗ (Finsupp.linearEquivFunOnFinite R R ι).symm }

Instances For

@[simp]

theorem Basis.ofEquivFun_repr_apply {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (e : M ≃ₗ[R] ι → R) (x : M) (i : ι) :

((Basis.ofEquivFun e).repr x) i = e x i

@[simp]

theorem Basis.coe_ofEquivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] [DecidableEq ι] (e : M ≃ₗ[R] ι → R) :

⇑(Basis.ofEquivFun e) = fun (i : ι) => e.symm (Pi.single i 1)

@[simp]

theorem Basis.ofEquivFun_equivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (v : Basis ι R M) :

Basis.ofEquivFun v.equivFun = v

@[simp]

theorem Basis.equivFun_ofEquivFun {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (e : M ≃ₗ[R] ι → R) :

(Basis.ofEquivFun e).equivFun = e

@[simp]

theorem Basis.constr_apply_fintype {ι : Type u_1} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] [Fintype ι] (b : Basis ι R M) (f : ι → M') (x : M) :

((b.constr S) f) x = ∑ i : ι, b.equivFun x i • f i

theorem Basis.mem_submodule_iff' {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] {P : Submodule R M} (b : Basis ι R ↥P) {x : M} :

x ∈ P ↔ ∃ (c : ι → R), x = ∑ i : ι, c i • ↑(b i)

If the submodule P has a finite basis, x ∈ P iff it is a linear combination of basis vectors.

theorem Basis.coord_equivFun_symm {ι : Type u_1} {R : Type u_3} {M : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [Finite ι] (b : Basis ι R M) (i : ι) (f : ι → R) :

(b.coord i) (b.equivFun.symm f) = f i

def Basis.equiv' {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (f : M → M') (g : M' → M) (hf : ∀ (i : ι), f (b i) ∈ Set.range ⇑b') (hg : ∀ (i : ι'), g (b' i) ∈ Set.range ⇑b) (hgf : ∀ (i : ι), g (f (b i)) = b i) (hfg : ∀ (i : ι'), f (g (b' i)) = b' i) :

If b is a basis for M and b' a basis for M', and f, g form a bijection between the basis vectors, b.equiv' b' f g hf hg hgf hfg is a linear equivalence M ≃ₗ[R] M', mapping b i to f (b i).

Equations

b.equiv' b' f g hf hg hgf hfg = { toLinearMap := (b.constr R) (f ∘ ⇑b), invFun := ⇑((b'.constr R) (g ∘ ⇑b')), left_inv := ⋯, right_inv := ⋯ }

Instances For

@[simp]

theorem Basis.equiv'_apply {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (f : M → M') (g : M' → M) (hf : ∀ (i : ι), f (b i) ∈ Set.range ⇑b') (hg : ∀ (i : ι'), g (b' i) ∈ Set.range ⇑b) (hgf : ∀ (i : ι), g (f (b i)) = b i) (hfg : ∀ (i : ι'), f (g (b' i)) = b' i) (i : ι) :

(b.equiv' b' f g hf hg hgf hfg) (b i) = f (b i)

@[simp]

theorem Basis.equiv'_symm_apply {ι : Type u_1} {ι' : Type u_2} {R : Type u_3} {M : Type u_6} {M' : Type u_7} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (b : Basis ι R M) (b' : Basis ι' R M') (f : M → M') (g : M' → M) (hf : ∀ (i : ι), f (b i) ∈ Set.range ⇑b') (hg : ∀ (i : ι'), g (b' i) ∈ Set.range ⇑b) (hgf : ∀ (i : ι), g (f (b i)) = b i) (hfg : ∀ (i : ι'), f (g (b' i)) = b' i) (i : ι') :

(b.equiv' b' f g hf hg hgf hfg).symm (b' i) = g (b' i)

theorem Basis.sum_repr_mul_repr {ι : Type u_1} {R : Type u_3} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (b : Basis ι R M) {ι' : Type u_10} [Fintype ι'] (b' : Basis ι' R M) (x : M) (i : ι) :

∑ j : ι', (b.repr (b' j)) i * (b'.repr x) j = (b.repr x) i