fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) :
TensorProduct.map (dualTensorHom R M P (f ⊗ₜ[R] p)) (dualTensorHom R N Q (g ⊗ₜ[R] q)) =
dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q) (dualDistrib R M N (f ⊗ₜ g) ⊗ₜ[R] (p ⊗ₜ[R] q)) := by
ext m n
simp only [compr₂_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, ←
smul_tmul_smul]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | map_dualTensorHom | null |
comp_dualTensorHom (f : Module.Dual R M) (n : N) (g : Module.Dual R N) (p : P) :
dualTensorHom R N P (g ⊗ₜ[R] p) ∘ₗ dualTensorHom R M N (f ⊗ₜ[R] n) =
g n • dualTensorHom R M P (f ⊗ₜ p) := by
ext m
simp only [coe_comp, Function.comp_apply, dualTensorHom_apply, LinearMap.map_smul,
LinearMap.smul_apply]
rw [smul_comm] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | comp_dualTensorHom | null |
toMatrix_dualTensorHom {m : Type*} {n : Type*} [Fintype m] [Finite n] [DecidableEq m]
[DecidableEq n] (bM : Basis m R M) (bN : Basis n R N) (j : m) (i : n) :
toMatrix bM bN (dualTensorHom R M N (bM.coord j ⊗ₜ bN i)) = single i j 1 := by
ext i' j'
by_cases hij : i = i' ∧ j = j' <;>
simp [LinearMap.toMatrix_apply, Finsupp.single_eq_pi_single, hij]
rw [and_iff_not_or_not, Classical.not_not] at hij
rcases hij with hij | hij <;> simp [hij] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | toMatrix_dualTensorHom | As a matrix, `dualTensorHom` evaluated on a basis element of `M* ⊗ N` is a matrix with a
single one and zeros elsewhere |
@[simp]
noncomputable dualTensorHomEquiv : Module.Dual R M ⊗[R] N ≃ₗ[R] M →ₗ[R] N :=
dualTensorHomEquivOfBasis (Module.Free.chooseBasis R M) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | dualTensorHomEquiv | If `M` is free, the natural linear map $M^* ⊗ N → Hom(M, N)$ is an equivalence. This function
provides this equivalence in return for a basis of `M`. -/
-- We manually create simp-lemmas because `@[simps]` generates a malformed lemma
noncomputable def dualTensorHomEquivOfBasis : Module.Dual R M ⊗[R] N ≃ₗ[R] M →ₗ[R] N :=
LinearEquiv.ofLinear (dualTensorHom R M N)
(∑ i, TensorProduct.mk R _ N (b.dualBasis i) ∘ₗ (LinearMap.applyₗ (R := R) (b i)))
(by
ext f m
simp only [applyₗ_apply_apply, coeFn_sum, dualTensorHom_apply, mk_apply, id_coe, _root_.id,
Fintype.sum_apply, Function.comp_apply, Basis.coe_dualBasis, coe_comp, Basis.coord_apply, ←
f.map_smul, _root_.map_sum (dualTensorHom R M N), ← _root_.map_sum f, b.sum_repr])
(by
ext f m
simp only [applyₗ_apply_apply, coeFn_sum, dualTensorHom_apply, mk_apply, id_coe, _root_.id,
Fintype.sum_apply, Function.comp_apply, Basis.coe_dualBasis, coe_comp, compr₂_apply,
tmul_smul, smul_tmul', ← sum_tmul, Basis.sum_dual_apply_smul_coord])
@[simp]
theorem dualTensorHomEquivOfBasis_apply (x : Module.Dual R M ⊗[R] N) :
dualTensorHomEquivOfBasis b x = dualTensorHom R M N x := by
ext; rfl
@[simp]
theorem dualTensorHomEquivOfBasis_toLinearMap :
(dualTensorHomEquivOfBasis b).toLinearMap = dualTensorHom R M N :=
rfl
@[simp]
theorem dualTensorHomEquivOfBasis_symm_cancel_left (x : Module.Dual R M ⊗[R] N) :
(dualTensorHomEquivOfBasis b).symm (dualTensorHom R M N x) = x := by
rw [← dualTensorHomEquivOfBasis_apply b,
LinearEquiv.symm_apply_apply <| dualTensorHomEquivOfBasis (N := N) b]
@[simp]
theorem dualTensorHomEquivOfBasis_symm_cancel_right (x : M →ₗ[R] N) :
dualTensorHom R M N ((dualTensorHomEquivOfBasis b).symm x) = x := by
rw [← dualTensorHomEquivOfBasis_apply b, LinearEquiv.apply_symm_apply]
variable (R M N P Q)
variable [Module.Free R M] [Module.Finite R M]
/-- If `M` is finite free, the natural map $M^* ⊗ N → Hom(M, N)$ is an
equivalence. |
noncomputable lTensorHomEquivHomLTensor : P ⊗[R] (M →ₗ[R] Q) ≃ₗ[R] M →ₗ[R] P ⊗[R] Q :=
congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q).symm ≪≫ₗ
TensorProduct.leftComm R P _ Q ≪≫ₗ
dualTensorHomEquiv R M _ | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | lTensorHomEquivHomLTensor | When `M` is a finite free module, the map `lTensorHomToHomLTensor` is an equivalence. Note
that `lTensorHomEquivHomLTensor` is not defined directly in terms of
`lTensorHomToHomLTensor`, but the equivalence between the two is given by
`lTensorHomEquivHomLTensor_toLinearMap` and `lTensorHomEquivHomLTensor_apply`. |
noncomputable rTensorHomEquivHomRTensor : (M →ₗ[R] P) ⊗[R] Q ≃ₗ[R] M →ₗ[R] P ⊗[R] Q :=
congr (dualTensorHomEquiv R M P).symm (LinearEquiv.refl R Q) ≪≫ₗ TensorProduct.assoc R _ P Q ≪≫ₗ
dualTensorHomEquiv R M _
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | rTensorHomEquivHomRTensor | When `M` is a finite free module, the map `rTensorHomToHomRTensor` is an equivalence. Note
that `rTensorHomEquivHomRTensor` is not defined directly in terms of
`rTensorHomToHomRTensor`, but the equivalence between the two is given by
`rTensorHomEquivHomRTensor_toLinearMap` and `rTensorHomEquivHomRTensor_apply`. |
lTensorHomEquivHomLTensor_toLinearMap :
(lTensorHomEquivHomLTensor R M P Q).toLinearMap = lTensorHomToHomLTensor R M P Q := by
let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)
have h : Function.Surjective e.toLinearMap := e.surjective
refine (cancel_right h).1 ?_
ext f q m
simp only [e, lTensorHomEquivHomLTensor, dualTensorHomEquiv, LinearEquiv.comp_coe, compr₂_apply,
mk_apply, LinearEquiv.coe_coe, LinearEquiv.trans_apply, congr_tmul, LinearEquiv.refl_apply,
dualTensorHomEquivOfBasis_apply, dualTensorHomEquivOfBasis_symm_cancel_left, leftComm_tmul,
dualTensorHom_apply, coe_comp, Function.comp_apply, lTensorHomToHomLTensor_apply, tmul_smul]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | lTensorHomEquivHomLTensor_toLinearMap | null |
rTensorHomEquivHomRTensor_toLinearMap :
(rTensorHomEquivHomRTensor R M P Q).toLinearMap = rTensorHomToHomRTensor R M P Q := by
let e := congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)
have h : Function.Surjective e.toLinearMap := e.surjective
refine (cancel_right h).1 ?_
ext f p q m
simp only [e, rTensorHomEquivHomRTensor, dualTensorHomEquiv, compr₂_apply, mk_apply, coe_comp,
LinearEquiv.coe_toLinearMap, Function.comp_apply,
dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul,
dualTensorHomEquivOfBasis_symm_cancel_left, LinearEquiv.refl_apply, assoc_tmul,
dualTensorHom_apply, rTensorHomToHomRTensor_apply, smul_tmul']
variable {R M N P Q}
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | rTensorHomEquivHomRTensor_toLinearMap | null |
lTensorHomEquivHomLTensor_apply (x : P ⊗[R] (M →ₗ[R] Q)) :
lTensorHomEquivHomLTensor R M P Q x = lTensorHomToHomLTensor R M P Q x := by
rw [← LinearEquiv.coe_toLinearMap, lTensorHomEquivHomLTensor_toLinearMap]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | lTensorHomEquivHomLTensor_apply | null |
rTensorHomEquivHomRTensor_apply (x : (M →ₗ[R] P) ⊗[R] Q) :
rTensorHomEquivHomRTensor R M P Q x = rTensorHomToHomRTensor R M P Q x := by
rw [← LinearEquiv.coe_toLinearMap, rTensorHomEquivHomRTensor_toLinearMap]
variable (R M N P Q) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | rTensorHomEquivHomRTensor_apply | null |
noncomputable homTensorHomEquiv : (M →ₗ[R] P) ⊗[R] (N →ₗ[R] Q) ≃ₗ[R] M ⊗[R] N →ₗ[R] P ⊗[R] Q :=
rTensorHomEquivHomRTensor R M P _ ≪≫ₗ
(LinearEquiv.refl R M).arrowCongr (lTensorHomEquivHomLTensor R N _ Q) ≪≫ₗ
lift.equiv R M N _
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | homTensorHomEquiv | When `M` and `N` are free `R` modules, the map `homTensorHomMap` is an equivalence. Note that
`homTensorHomEquiv` is not defined directly in terms of `homTensorHomMap`, but the equivalence
between the two is given by `homTensorHomEquiv_toLinearMap` and `homTensorHomEquiv_apply`. |
homTensorHomEquiv_toLinearMap :
(homTensorHomEquiv R M N P Q).toLinearMap = homTensorHomMap R M N P Q := by
ext m n
simp only [homTensorHomEquiv, compr₂_apply, mk_apply, LinearEquiv.coe_toLinearMap,
LinearEquiv.trans_apply, lift.equiv_apply, LinearEquiv.arrowCongr_apply, LinearEquiv.refl_symm,
LinearEquiv.refl_apply, rTensorHomEquivHomRTensor_apply, lTensorHomEquivHomLTensor_apply,
lTensorHomToHomLTensor_apply, rTensorHomToHomRTensor_apply, homTensorHomMap_apply,
map_tmul]
variable {R M N P Q}
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | homTensorHomEquiv_toLinearMap | null |
homTensorHomEquiv_apply (x : (M →ₗ[R] P) ⊗[R] (N →ₗ[R] Q)) :
homTensorHomEquiv R M N P Q x = homTensorHomMap R M N P Q x := by
rw [← LinearEquiv.coe_toLinearMap, homTensorHomEquiv_toLinearMap] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Dual.Lemmas",
"Mathlib.LinearAlgebra.Matrix.ToLin"
] | Mathlib/LinearAlgebra/Contraction.lean | homTensorHomEquiv_apply | null |
crossProduct : (Fin 3 → R) →ₗ[R] (Fin 3 → R) →ₗ[R] Fin 3 → R := by
apply LinearMap.mk₂ R fun a b : Fin 3 → R =>
![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0]
· intros
simp_rw [vec3_add, Pi.add_apply]
apply vec3_eq <;> ring
· intros
simp_rw [smul_vec3, Pi.smul_apply, smul_sub, smul_mul_assoc]
· intros
simp_rw [vec3_add, Pi.add_apply]
apply vec3_eq <;> ring
· intros
simp_rw [smul_vec3, Pi.smul_apply, smul_sub, mul_smul_comm]
@[inherit_doc] scoped[Matrix] infixl:74 " ⨯₃ " => crossProduct | def | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | crossProduct | The cross product of two vectors in $R^3$ for $R$ a commutative ring. |
@[simp]
dot_self_cross (v w : Fin 3 → R) : v ⬝ᵥ v ⨯₃ w = 0 := by
rw [cross_apply, vec3_dotProduct]
dsimp only [Matrix.cons_val]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | dot_self_cross | A deprecated notation for `⨯₃`. -/
@[deprecated «term_⨯₃_» (since := "2025-07-11")]
scoped syntax:74 (name := _root_.«term_×₃_») term:74 " ×₃ " term:75 : term
end Matrix
open Lean Elab Meta.Tactic Term in
@[term_elab Matrix._root_.«term_×₃_», inherit_doc «term_×₃_»]
def elabDeprecatedCross : TermElab
| `($x ×₃%$tk $y) => fun ty? => do
logWarningAt tk <| .tagged ``Linter.deprecatedAttr <| m!"The ×₃ notation has been deprecated"
TryThis.addSuggestion tk { suggestion := "⨯₃" }
elabTerm (← `($x ⨯₃ $y)) ty?
| _ => fun _ => throwUnsupportedSyntax
theorem cross_apply (a b : Fin 3 → R) :
a ⨯₃ b = ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0] := rfl
section ProductsProperties
@[simp]
theorem cross_anticomm (v w : Fin 3 → R) : -(v ⨯₃ w) = w ⨯₃ v := by
simp [cross_apply, mul_comm]
alias neg_cross := cross_anticomm
@[simp]
theorem cross_anticomm' (v w : Fin 3 → R) : v ⨯₃ w + w ⨯₃ v = 0 := by
rw [add_eq_zero_iff_eq_neg, cross_anticomm]
@[simp]
theorem cross_self (v : Fin 3 → R) : v ⨯₃ v = 0 := by
simp [cross_apply, mul_comm]
/-- The cross product of two vectors is perpendicular to the first vector. |
@[simp]
dot_cross_self (v w : Fin 3 → R) : w ⬝ᵥ v ⨯₃ w = 0 := by
rw [← cross_anticomm, dotProduct_neg, dot_self_cross, neg_zero] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | dot_cross_self | The cross product of two vectors is perpendicular to the second vector. |
triple_product_permutation (u v w : Fin 3 → R) : u ⬝ᵥ v ⨯₃ w = v ⬝ᵥ w ⨯₃ u := by
simp_rw [cross_apply, vec3_dotProduct]
dsimp only [Matrix.cons_val]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | triple_product_permutation | Cyclic permutations preserve the triple product. See also `triple_product_eq_det`. |
triple_product_eq_det (u v w : Fin 3 → R) : u ⬝ᵥ v ⨯₃ w = Matrix.det ![u, v, w] := by
rw [vec3_dotProduct, cross_apply, det_fin_three]
dsimp only [Matrix.cons_val]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | triple_product_eq_det | The triple product of `u`, `v`, and `w` is equal to the determinant of the matrix
with those vectors as its rows. |
cross_dot_cross (u v w x : Fin 3 → R) :
u ⨯₃ v ⬝ᵥ w ⨯₃ x = u ⬝ᵥ w * v ⬝ᵥ x - u ⬝ᵥ x * v ⬝ᵥ w := by
simp_rw [cross_apply, vec3_dotProduct]
dsimp only [Matrix.cons_val]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | cross_dot_cross | The scalar quadruple product identity, related to the Binet-Cauchy identity. |
leibniz_cross (u v w : Fin 3 → R) : u ⨯₃ (v ⨯₃ w) = u ⨯₃ v ⨯₃ w + v ⨯₃ (u ⨯₃ w) := by
simp_rw [cross_apply, vec3_add]
apply vec3_eq <;> dsimp <;> ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | leibniz_cross | The cross product satisfies the Leibniz lie property. |
Cross.lieRing : LieRing (Fin 3 → R) :=
{ Pi.addCommGroup with
bracket := fun u v => u ⨯₃ v
add_lie := LinearMap.map_add₂ _
lie_add := fun _ => LinearMap.map_add _
lie_self := cross_self
leibniz_lie := leibniz_cross }
attribute [local instance] Cross.lieRing | def | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | Cross.lieRing | The three-dimensional vectors together with the operations + and ⨯₃ form a Lie ring.
Note we do not make this an instance as a conflicting one already exists
via `LieRing.ofAssociativeRing`. |
cross_cross (u v w : Fin 3 → R) : u ⨯₃ v ⨯₃ w = u ⨯₃ (v ⨯₃ w) - v ⨯₃ (u ⨯₃ w) :=
lie_lie u v w | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | cross_cross | null |
jacobi_cross (u v w : Fin 3 → R) : u ⨯₃ (v ⨯₃ w) + v ⨯₃ (w ⨯₃ u) + w ⨯₃ (u ⨯₃ v) = 0 :=
lie_jacobi u v w | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | jacobi_cross | **Jacobi identity**: For a cross product of three vectors,
their sum over the three even permutations is equal to the zero vector. |
crossProduct_ne_zero_iff_linearIndependent {F : Type*} [Field F] {v w : Fin 3 → F} :
crossProduct v w ≠ 0 ↔ LinearIndependent F ![v, w] := by
rw [not_iff_comm]
by_cases hv : v = 0
· rw [hv, map_zero, LinearMap.zero_apply, eq_self, iff_true]
exact fun h ↦ h.ne_zero 0 rfl
constructor
· rw [LinearIndependent.pair_iff' hv, not_forall_not]
rintro ⟨a, rfl⟩
rw [LinearMap.map_smul, cross_self, smul_zero]
have hv' : v = ![v 0, v 1, v 2] := by simp [← List.ofFn_inj]
have hw' : w = ![w 0, w 1, w 2] := by simp [← List.ofFn_inj]
intro h1 h2
simp_rw [cross_apply, cons_eq_zero_iff, zero_empty, and_true, sub_eq_zero] at h1
have h20 := LinearIndependent.pair_iff.mp h2 (- w 0) (v 0)
have h21 := LinearIndependent.pair_iff.mp h2 (- w 1) (v 1)
have h22 := LinearIndependent.pair_iff.mp h2 (- w 2) (v 2)
rw [neg_smul, neg_add_eq_zero, hv', hw', smul_vec3, smul_vec3, ← hv', ← hw'] at h20 h21 h22
simp only [smul_eq_mul, mul_comm (w 0), mul_comm (w 1), mul_comm (w 2), h1] at h20 h21 h22
rw [hv', cons_eq_zero_iff, cons_eq_zero_iff, cons_eq_zero_iff, zero_empty] at hv
exact hv ⟨(h20 trivial).2, (h21 trivial).2, (h22 trivial).2, rfl⟩ | lemma | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | crossProduct_ne_zero_iff_linearIndependent | null |
cross_cross_eq_smul_sub_smul (u v w : Fin 3 → R) :
u ⨯₃ v ⨯₃ w = (u ⬝ᵥ w) • v - (v ⬝ᵥ w) • u := by
simp_rw [cross_apply, vec3_dotProduct]
ext i
fin_cases i <;>
· simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.reduceFinMk, cons_val,
Pi.sub_apply, Pi.smul_apply, smul_eq_mul]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | cross_cross_eq_smul_sub_smul | The scalar triple product expansion of the vector triple product. |
cross_cross_eq_smul_sub_smul' (u v w : Fin 3 → R) :
u ⨯₃ (v ⨯₃ w) = (u ⬝ᵥ w) • v - (v ⬝ᵥ u) • w := by
simp_rw [cross_apply, vec3_dotProduct]
ext i
fin_cases i <;>
· simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val, cons_val_one,
cons_val_zero, Fin.reduceFinMk, Pi.sub_apply, Pi.smul_apply, smul_eq_mul]
ring | theorem | LinearAlgebra | [
"Mathlib.Algebra.Lie.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas",
"Mathlib.LinearAlgebra.Matrix.Determinant.Basic",
"Mathlib.LinearAlgebra.Matrix.Notation"
] | Mathlib/LinearAlgebra/CrossProduct.lean | cross_cross_eq_smul_sub_smul' | Alternative form of the scalar triple product expansion of the vector triple product. |
equivOfPiLEquivPi {R : Type*} [Finite m] [Finite n] [CommRing R] [Nontrivial R]
(e : (m → R) ≃ₗ[R] n → R) : m ≃ n :=
Basis.indexEquiv (Basis.ofEquivFun e.symm) (Pi.basisFun _ _) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | equivOfPiLEquivPi | If `R^m` and `R^n` are linearly equivalent, then `m` and `n` are also equivalent. |
indexEquivOfInv [Nontrivial A] [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : m ≃ n :=
equivOfPiLEquivPi (toLin'OfInv hMM' hM'M) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | indexEquivOfInv | If `M` and `M'` are each other's inverse matrices, they are square matrices up to
equivalence of types. |
det_comm [DecidableEq n] (M N : Matrix n n A) : det (M * N) = det (N * M) := by
rw [det_mul, det_mul, mul_comm] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_comm | null |
det_comm' [DecidableEq m] [DecidableEq n] {M : Matrix n m A} {N : Matrix m n A}
{M' : Matrix m n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : det (M * N) = det (N * M) := by
nontriviality A
let e := indexEquivOfInv hMM' hM'M
rw [← det_submatrix_equiv_self e, ← submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm,
submatrix_mul_equiv, Equiv.coe_refl, submatrix_id_id] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_comm' | If there exists a two-sided inverse `M'` for `M` (indexed differently),
then `det (N * M) = det (M * N)`. |
det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by
rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_conj_of_mul_eq_one | If `M'` is a two-sided inverse for `M` (indexed differently), `det (M * N * M') = det N`.
See `Matrix.det_conj` and `Matrix.det_conj'` for the case when `M' = M⁻¹` or vice versa. |
det_toMatrix_eq_det_toMatrix [DecidableEq κ] (b : Basis ι A M) (c : Basis κ A M)
(f : M →ₗ[A] M) : det (LinearMap.toMatrix b b f) = det (LinearMap.toMatrix c c f) := by
rw [← linearMap_toMatrix_mul_basis_toMatrix c b c, ← basis_toMatrix_mul_linearMap_toMatrix b c b,
Matrix.det_conj_of_mul_eq_one] <;>
rw [Basis.toMatrix_mul_toMatrix, Basis.toMatrix_self] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_toMatrix_eq_det_toMatrix | The determinant of `LinearMap.toMatrix` does not depend on the choice of basis. |
detAux_def' (b : Basis ι A M) (f : M →ₗ[A] M) :
LinearMap.detAux (Trunc.mk b) f = Matrix.det (LinearMap.toMatrix b b f) := by
rw [detAux]
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | detAux_def' | The determinant of an endomorphism given a basis.
See `LinearMap.det` for a version that populates the basis non-computably.
Although the `Trunc (Basis ι A M)` parameter makes it slightly more convenient to switch bases,
there is no good way to generalize over universe parameters, so we can't fully state in `detAux`'s
type that it does not depend on the choice of basis. Instead you can use the `detAux_def''` lemma,
or avoid mentioning a basis at all using `LinearMap.det`.
-/
irreducible_def detAux : Trunc (Basis ι A M) → (M →ₗ[A] M) →* A :=
Trunc.lift
(fun b : Basis ι A M => detMonoidHom.comp (toMatrixAlgEquiv b : (M →ₗ[A] M) →* Matrix ι ι A))
fun b c => MonoidHom.ext <| det_toMatrix_eq_det_toMatrix b c
/-- Unfold lemma for `detAux`.
See also `detAux_def''` which allows you to vary the basis. |
detAux_def'' {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (tb : Trunc <| Basis ι A M)
(b' : Basis ι' A M) (f : M →ₗ[A] M) :
LinearMap.detAux tb f = Matrix.det (LinearMap.toMatrix b' b' f) := by
induction tb using Trunc.induction_on with
| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b']
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | detAux_def'' | null |
detAux_id (b : Trunc <| Basis ι A M) : LinearMap.detAux b LinearMap.id = 1 :=
(LinearMap.detAux b).map_one
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | detAux_id | null |
detAux_comp (b : Trunc <| Basis ι A M) (f g : M →ₗ[A] M) :
LinearMap.detAux b (f.comp g) = LinearMap.detAux b f * LinearMap.detAux b g :=
(LinearMap.detAux b).map_mul f g | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | detAux_comp | null |
@[elab_as_elim]
det_cases [DecidableEq M] {P : A → Prop} (f : M →ₗ[A] M)
(hb : ∀ (s : Finset M) (b : Basis s A M), P (Matrix.det (toMatrix b b f))) (h1 : P 1) :
P (LinearMap.det f) := by
classical
if H : ∃ s : Finset M, Nonempty (Basis s A M) then
obtain ⟨s, ⟨b⟩⟩ := H
rw [← det_toMatrix b]
exact hb s b
else
rwa [LinearMap.det_def, dif_neg H]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_cases | The determinant of an endomorphism independent of basis.
If there is no finite basis on `M`, the result is `1` instead.
-/
protected irreducible_def det : (M →ₗ[A] M) →* A :=
if H : ∃ s : Finset M, Nonempty (Basis s A M) then LinearMap.detAux (Trunc.mk H.choose_spec.some)
else 1
open scoped Classical in
theorem coe_det [DecidableEq M] :
⇑(LinearMap.det : (M →ₗ[A] M) →* A) =
if H : ∃ s : Finset M, Nonempty (Basis s A M) then
LinearMap.detAux (Trunc.mk H.choose_spec.some)
else 1 := by
ext
rw [LinearMap.det_def]
split_ifs
· congr -- use the correct `DecidableEq` instance
rfl
end
-- Auxiliary lemma, the `simp` normal form goes in the other direction
-- (using `LinearMap.det_toMatrix`)
theorem det_eq_det_toMatrix_of_finset [DecidableEq M] {s : Finset M} (b : Basis s A M)
(f : M →ₗ[A] M) : LinearMap.det f = Matrix.det (LinearMap.toMatrix b b f) := by
have : ∃ s : Finset M, Nonempty (Basis s A M) := ⟨s, ⟨b⟩⟩
rw [LinearMap.coe_det, dif_pos this, detAux_def'' _ b]
@[simp]
theorem det_toMatrix (b : Basis ι A M) (f : M →ₗ[A] M) :
Matrix.det (toMatrix b b f) = LinearMap.det f := by
haveI := Classical.decEq M
rw [det_eq_det_toMatrix_of_finset b.reindexFinsetRange,
det_toMatrix_eq_det_toMatrix b b.reindexFinsetRange]
@[simp]
theorem det_toMatrix' {ι : Type*} [Fintype ι] [DecidableEq ι] (f : (ι → A) →ₗ[A] ι → A) :
Matrix.det (LinearMap.toMatrix' f) = LinearMap.det f := by simp [← toMatrix_eq_toMatrix']
@[simp]
theorem det_toLin (b : Basis ι R M) (f : Matrix ι ι R) :
LinearMap.det (Matrix.toLin b b f) = f.det := by
rw [← LinearMap.det_toMatrix b, LinearMap.toMatrix_toLin]
@[simp]
theorem det_toLin' (f : Matrix ι ι R) : LinearMap.det (Matrix.toLin' f) = Matrix.det f := by
simp only [← toLin_eq_toLin', det_toLin]
/-- To show `P (LinearMap.det f)` it suffices to consider `P (Matrix.det (toMatrix _ _ f))` and
`P 1`. |
det_comp (f g : M →ₗ[A] M) :
LinearMap.det (f.comp g) = LinearMap.det f * LinearMap.det g :=
LinearMap.det.map_mul f g
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_comp | null |
det_id : LinearMap.det (LinearMap.id : M →ₗ[A] M) = 1 :=
LinearMap.det.map_one | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_id | null |
@[simp]
det_smul [Module.Free A M] (c : A) (f : M →ₗ[A] M) :
LinearMap.det (c • f) = c ^ Module.finrank A M * LinearMap.det f := by
nontriviality A
by_cases H : ∃ s : Finset M, Nonempty (Basis s A M)
· have : Module.Finite A M := by
rcases H with ⟨s, ⟨hs⟩⟩
exact Module.Finite.of_basis hs
simp only [← det_toMatrix (Module.finBasis A M), LinearEquiv.map_smul,
Fintype.card_fin, Matrix.det_smul]
· classical
have : Module.finrank A M = 0 := finrank_eq_zero_of_not_exists_basis H
simp [coe_det, H, this] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_smul | Multiplying a map by a scalar `c` multiplies its determinant by `c ^ dim M`. |
det_zero' {ι : Type*} [Finite ι] [Nonempty ι] (b : Basis ι A M) :
LinearMap.det (0 : M →ₗ[A] M) = 0 := by
haveI := Classical.decEq ι
cases nonempty_fintype ι
rwa [← det_toMatrix b, LinearEquiv.map_zero, det_zero] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_zero' | null |
@[simp]
det_zero [Module.Free A M] :
LinearMap.det (0 : M →ₗ[A] M) = (0 : A) ^ Module.finrank A M := by
simp only [← zero_smul A (1 : M →ₗ[A] M), det_smul, mul_one, MonoidHom.map_one] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_zero | In a finite-dimensional vector space, the zero map has determinant `1` in dimension `0`,
and `0` otherwise. We give a formula that also works in infinite dimension, where we define
the determinant to be `1`. |
det_eq_one_of_not_module_finite (h : ¬Module.Finite R M) (f : M →ₗ[R] M) : f.det = 1 := by
rw [LinearMap.det, dif_neg, MonoidHom.one_apply]
exact fun ⟨_, ⟨b⟩⟩ ↦ h (Module.Finite.of_basis b)
@[nontriviality] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_eq_one_of_not_module_finite | null |
det_eq_one_of_subsingleton [Subsingleton M] (f : M →ₗ[R] M) :
LinearMap.det (f : M →ₗ[R] M) = 1 := by
have b : Basis (Fin 0) R M := Basis.empty M
rw [← f.det_toMatrix b]
exact Matrix.det_isEmpty | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_eq_one_of_subsingleton | null |
det_eq_one_of_finrank_eq_zero {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M]
[Module 𝕜 M] (h : Module.finrank 𝕜 M = 0) (f : M →ₗ[𝕜] M) :
LinearMap.det (f : M →ₗ[𝕜] M) = 1 := by
classical
refine @LinearMap.det_cases M _ 𝕜 _ _ _ (fun t => t = 1) f ?_ rfl
intro s b
have : IsEmpty s := by
rw [← Fintype.card_eq_zero_iff]
exact (Module.finrank_eq_card_basis b).symm.trans h
exact Matrix.det_isEmpty | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_eq_one_of_finrank_eq_zero | null |
@[simp]
det_conj {N : Type*} [AddCommGroup N] [Module A N] (f : M →ₗ[A] M) (e : M ≃ₗ[A] N) :
LinearMap.det ((e : M →ₗ[A] N) ∘ₗ f ∘ₗ (e.symm : N →ₗ[A] M)) = LinearMap.det f := by
classical
by_cases H : ∃ s : Finset M, Nonempty (Basis s A M)
· rcases H with ⟨s, ⟨b⟩⟩
rw [← det_toMatrix b f, ← det_toMatrix (b.map e), toMatrix_comp (b.map e) b (b.map e),
toMatrix_comp (b.map e) b b, ← Matrix.mul_assoc, Matrix.det_conj_of_mul_eq_one]
· rw [← toMatrix_comp, LinearEquiv.comp_coe, e.symm_trans_self, LinearEquiv.refl_toLinearMap,
toMatrix_id]
· rw [← toMatrix_comp, LinearEquiv.comp_coe, e.self_trans_symm, LinearEquiv.refl_toLinearMap,
toMatrix_id]
· have H' : ¬∃ t : Finset N, Nonempty (Basis t A N) := by
contrapose! H
rcases H with ⟨s, ⟨b⟩⟩
exact ⟨_, ⟨(b.map e.symm).reindexFinsetRange⟩⟩
simp only [coe_det, H, H', MonoidHom.one_apply, dif_neg, not_false_eq_true] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_conj | Conjugating a linear map by a linear equiv does not change its determinant. |
isUnit_det {A : Type*} [CommRing A] [Module A M] (f : M →ₗ[A] M) (hf : IsUnit f) :
IsUnit (LinearMap.det f) := IsUnit.map LinearMap.det hf | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | isUnit_det | If a linear map is invertible, so is its determinant. |
isUnit_iff_isUnit_det [Module.Finite R M] [Module.Free R M] (f : M →ₗ[R] M) :
IsUnit f ↔ IsUnit f.det := by
let b := Module.Free.chooseBasis R M
rw [← isUnit_toMatrix_iff b, ← det_toMatrix b, Matrix.isUnit_iff_isUnit_det (toMatrix b b f)] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | isUnit_iff_isUnit_det | null |
finiteDimensional_of_det_ne_one {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] (f : M →ₗ[𝕜] M)
(hf : LinearMap.det f ≠ 1) : FiniteDimensional 𝕜 M := by
by_cases H : ∃ s : Finset M, Nonempty (Basis s 𝕜 M)
· rcases H with ⟨s, ⟨hs⟩⟩
exact FiniteDimensional.of_fintype_basis hs
· classical simp [LinearMap.coe_det, H] at hf | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | finiteDimensional_of_det_ne_one | If a linear map has determinant different from `1`, then the space is finite-dimensional. |
range_lt_top_of_det_eq_zero {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] {f : M →ₗ[𝕜] M}
(hf : LinearMap.det f = 0) : LinearMap.range f < ⊤ := by
have : FiniteDimensional 𝕜 M := by simp [f.finiteDimensional_of_det_ne_one, hf]
contrapose hf
simp only [lt_top_iff_ne_top, Classical.not_not, ← isUnit_iff_range_eq_top] at hf
exact isUnit_iff_ne_zero.1 (f.isUnit_det hf) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | range_lt_top_of_det_eq_zero | If the determinant of a map vanishes, then the map is not onto. |
bot_lt_ker_of_det_eq_zero {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] {f : M →ₗ[𝕜] M}
(hf : LinearMap.det f = 0) : ⊥ < LinearMap.ker f := by
have : FiniteDimensional 𝕜 M := by simp [f.finiteDimensional_of_det_ne_one, hf]
contrapose hf
simp only [bot_lt_iff_ne_bot, Classical.not_not, ← isUnit_iff_ker_eq_bot] at hf
exact isUnit_iff_ne_zero.1 (f.isUnit_det hf) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | bot_lt_ker_of_det_eq_zero | If the determinant of a map vanishes, then the map is not injective. |
@[simp] det_ring (f : R →ₗ[R] R) : f.det = f 1 := by
simp [← det_toMatrix (Basis.singleton Unit R)] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_ring | When the function is over the base ring, the determinant is the evaluation at `1`. |
det_mulLeft (a : R) : (mulLeft R a).det = a := by simp | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_mulLeft | null |
det_mulRight (a : R) : (mulRight R a).det = a := by simp | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_mulRight | null |
det_prodMap [Module.Free R M] [Module.Free R M'] [Module.Finite R M] [Module.Finite R M']
(f : Module.End R M) (f' : Module.End R M') :
(prodMap f f').det = f.det * f'.det := by
let b := Module.Free.chooseBasis R M
let b' := Module.Free.chooseBasis R M'
rw [← det_toMatrix (b.prod b'), ← det_toMatrix b, ← det_toMatrix b', toMatrix_prodMap,
det_fromBlocks_zero₂₁, det_toMatrix]
omit [DecidableEq ι] in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_prodMap | null |
det_pi [Module.Free R M] [Module.Finite R M] (f : ι → M →ₗ[R] M) :
(LinearMap.pi (fun i ↦ (f i).comp (LinearMap.proj i))).det = ∏ i, (f i).det := by
classical
let b := Module.Free.chooseBasis R M
let B := (Pi.basis (fun _ : ι ↦ b)).reindex <|
(Equiv.sigmaEquivProd _ _).trans (Equiv.prodComm _ _)
simp_rw [← LinearMap.det_toMatrix B, ← LinearMap.det_toMatrix b]
have : ((LinearMap.toMatrix B B) (LinearMap.pi fun i ↦ f i ∘ₗ LinearMap.proj i)) =
Matrix.blockDiagonal (fun i ↦ LinearMap.toMatrix b b (f i)) := by
ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
unfold B
simp_rw [LinearMap.toMatrix_apply', Matrix.blockDiagonal_apply, Basis.coe_reindex,
Function.comp_apply, Basis.repr_reindex_apply, Equiv.symm_trans_apply, Equiv.prodComm_symm,
Equiv.prodComm_apply, Equiv.sigmaEquivProd_symm_apply, Prod.swap_prod_mk, Pi.basis_apply,
Pi.basis_repr, LinearMap.pi_apply, LinearMap.coe_comp, Function.comp_apply,
LinearMap.toMatrix_apply', LinearMap.coe_proj, Function.eval, Pi.single_apply]
split_ifs with h
· rw [h]
· simp only [map_zero, Finsupp.coe_zero, Pi.zero_apply]
rw [this, Matrix.det_blockDiagonal] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_pi | null |
protected det : (M ≃ₗ[R] M) →* Rˣ :=
(Units.map (LinearMap.det : (M →ₗ[R] M) →* R)).comp
(LinearMap.GeneralLinearGroup.generalLinearEquiv R M).symm.toMonoidHom
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det | On a `LinearEquiv`, the domain of `LinearMap.det` can be promoted to `Rˣ`. |
coe_det (f : M ≃ₗ[R] M) : ↑(LinearEquiv.det f) = LinearMap.det (f : M →ₗ[R] M) :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | coe_det | null |
coe_inv_det (f : M ≃ₗ[R] M) : ↑(LinearEquiv.det f)⁻¹ = LinearMap.det (f.symm : M →ₗ[R] M) :=
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | coe_inv_det | null |
det_refl : LinearEquiv.det (LinearEquiv.refl R M) = 1 :=
Units.ext <| LinearMap.det_id
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_refl | null |
det_trans (f g : M ≃ₗ[R] M) :
LinearEquiv.det (f.trans g) = LinearEquiv.det g * LinearEquiv.det f :=
map_mul _ g f
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_trans | null |
det_symm (f : M ≃ₗ[R] M) : LinearEquiv.det f.symm = LinearEquiv.det f⁻¹ :=
map_inv _ f | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_symm | null |
@[simp]
det_conj (f : M ≃ₗ[R] M) (e : M ≃ₗ[R] M') :
LinearEquiv.det ((e.symm.trans f).trans e) = LinearEquiv.det f := by
rw [← Units.val_inj, coe_det, coe_det, ← comp_coe, ← comp_coe, LinearMap.det_conj]
attribute [irreducible] LinearEquiv.det | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_conj | Conjugating a linear equiv by a linear equiv does not change its determinant. |
@[simp]
LinearEquiv.det_mul_det_symm {A : Type*} [CommRing A] [Module A M] (f : M ≃ₗ[A] M) :
LinearMap.det (f : M →ₗ[A] M) * LinearMap.det (f.symm : M →ₗ[A] M) = 1 := by
simp [← LinearMap.det_comp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.det_mul_det_symm | The determinants of a `LinearEquiv` and its inverse multiply to 1. |
@[simp]
LinearEquiv.det_symm_mul_det {A : Type*} [CommRing A] [Module A M] (f : M ≃ₗ[A] M) :
LinearMap.det (f.symm : M →ₗ[A] M) * LinearMap.det (f : M →ₗ[A] M) = 1 := by
simp [← LinearMap.det_comp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.det_symm_mul_det | The determinants of a `LinearEquiv` and its inverse multiply to 1. |
LinearEquiv.isUnit_det (f : M ≃ₗ[R] M') (v : Basis ι R M) (v' : Basis ι R M') :
IsUnit (LinearMap.toMatrix v v' f).det := by
apply isUnit_det_of_left_inverse
simpa using (LinearMap.toMatrix_comp v v' v f.symm f).symm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.isUnit_det | null |
LinearEquiv.isUnit_det' {A : Type*} [CommRing A] [Module A M] (f : M ≃ₗ[A] M) :
IsUnit (LinearMap.det (f : M →ₗ[A] M)) :=
isUnit_of_mul_eq_one _ _ f.det_mul_det_symm
set_option linter.unusedSimpArgs false in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.isUnit_det' | Specialization of `LinearEquiv.isUnit_det` |
LinearEquiv.det_coe_symm {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] (f : M ≃ₗ[𝕜] M) :
LinearMap.det (f.symm : M →ₗ[𝕜] M) = (LinearMap.det (f : M →ₗ[𝕜] M))⁻¹ := by
simp [field, IsUnit.ne_zero f.isUnit_det'] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.det_coe_symm | The determinant of `f.symm` is the inverse of that of `f` when `f` is a linear equiv. |
@[simps]
LinearEquiv.ofIsUnitDet {f : M →ₗ[R] M'} {v : Basis ι R M} {v' : Basis ι R M'}
(h : IsUnit (LinearMap.toMatrix v v' f).det) : M ≃ₗ[R] M' where
toFun := f
map_add' := f.map_add
map_smul' := f.map_smul
invFun := toLin v' v (toMatrix v v' f)⁻¹
left_inv x :=
calc toLin v' v (toMatrix v v' f)⁻¹ (f x)
_ = toLin v v ((toMatrix v v' f)⁻¹ * toMatrix v v' f) x := by
rw [toLin_mul v v' v, toLin_toMatrix, LinearMap.comp_apply]
_ = x := by simp [h]
right_inv x :=
calc f (toLin v' v (toMatrix v v' f)⁻¹ x)
_ = toLin v' v' (toMatrix v v' f * (toMatrix v v' f)⁻¹) x := by
rw [toLin_mul v' v v', LinearMap.comp_apply, toLin_toMatrix v v']
_ = x := by simp [h]
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.ofIsUnitDet | Builds a linear equivalence from a linear map whose determinant in some bases is a unit. |
LinearEquiv.coe_ofIsUnitDet {f : M →ₗ[R] M'} {v : Basis ι R M} {v' : Basis ι R M'}
(h : IsUnit (LinearMap.toMatrix v v' f).det) :
(LinearEquiv.ofIsUnitDet h : M →ₗ[R] M') = f := by
ext x
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.coe_ofIsUnitDet | null |
LinearMap.equivOfDetNeZero {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M]
[FiniteDimensional 𝕜 M] (f : M →ₗ[𝕜] M) (hf : LinearMap.det f ≠ 0) : M ≃ₗ[𝕜] M :=
have : IsUnit (LinearMap.toMatrix (Module.finBasis 𝕜 M)
(Module.finBasis 𝕜 M) f).det := by
rw [LinearMap.det_toMatrix]
exact isUnit_iff_ne_zero.2 hf
LinearEquiv.ofIsUnitDet this | abbrev | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearMap.equivOfDetNeZero | Builds a linear equivalence from a linear map on a finite-dimensional vector space whose
determinant is nonzero. |
LinearMap.associated_det_of_eq_comp (e : M ≃ₗ[R] M) (f f' : M →ₗ[R] M)
(h : ∀ x, f x = f' (e x)) : Associated (LinearMap.det f) (LinearMap.det f') := by
suffices Associated (LinearMap.det (f' ∘ₗ ↑e)) (LinearMap.det f') by
convert this using 2
ext x
exact h x
rw [← mul_one (LinearMap.det f'), LinearMap.det_comp]
exact Associated.mul_left _ (associated_one_iff_isUnit.mpr e.isUnit_det') | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearMap.associated_det_of_eq_comp | null |
LinearMap.associated_det_comp_equiv {N : Type*} [AddCommGroup N] [Module R N]
(f : N →ₗ[R] M) (e e' : M ≃ₗ[R] N) :
Associated (LinearMap.det (f ∘ₗ ↑e)) (LinearMap.det (f ∘ₗ ↑e')) := by
refine LinearMap.associated_det_of_eq_comp (e.trans e'.symm) _ _ ?_
intro x
simp only [LinearMap.comp_apply, LinearEquiv.coe_coe, LinearEquiv.trans_apply,
LinearEquiv.apply_symm_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | LinearMap.associated_det_comp_equiv | null |
det_ne_zero [Nontrivial R] : e.det ≠ 0 := fun h => by simpa [h] using e.det_self | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_ne_zero | The determinant of a family of vectors with respect to some basis, as an alternating
multilinear map. -/
nonrec def det : M [⋀^ι]→ₗ[R] R where
toMultilinearMap :=
MultilinearMap.mk' (fun v ↦ det (e.toMatrix v))
(fun v i x y ↦ by
simp only [e.toMatrix_update, map_add, Finsupp.coe_add, det_updateCol_add])
(fun u i c x ↦ by
simp only [e.toMatrix_update, Algebra.id.smul_eq_mul, LinearEquiv.map_smul]
apply det_updateCol_smul)
map_eq_zero_of_eq' := by
intro v i j h hij
dsimp
rw [← Function.update_eq_self i v, h, ← det_transpose, e.toMatrix_update, ← updateRow_transpose,
← e.toMatrix_transpose_apply]
apply det_zero_of_row_eq hij
rw [updateRow_ne hij.symm, updateRow_self]
theorem det_apply (v : ι → M) : e.det v = Matrix.det (e.toMatrix v) :=
rfl
theorem det_self : e.det e = 1 := by simp [e.det_apply]
@[simp]
theorem det_isEmpty [IsEmpty ι] : e.det = AlternatingMap.constOfIsEmpty R M ι 1 := by
ext v
exact Matrix.det_isEmpty
/-- `Basis.det` is not the zero map. |
smul_det {G} [Group G] [DistribMulAction G M] [SMulCommClass G R M]
(g : G) (v : ι → M) :
(g • e).det v = e.det (g⁻¹ • v) := by
simp_rw [det_apply, toMatrix_smul_left] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | smul_det | null |
is_basis_iff_det {v : ι → M} :
LinearIndependent R v ∧ span R (Set.range v) = ⊤ ↔ IsUnit (e.det v) := by
constructor
· rintro ⟨hli, hspan⟩
set v' := Basis.mk hli hspan.ge
rw [e.det_apply]
convert LinearEquiv.isUnit_det (LinearEquiv.refl R M) v' e using 2
ext i j
simp [v']
· intro h
rw [Basis.det_apply, Basis.toMatrix_eq_toMatrix_constr] at h
set v' := Basis.map e (LinearEquiv.ofIsUnitDet h) with v'_def
have : ⇑v' = v := by
ext i
rw [v'_def, Basis.map_apply, LinearEquiv.ofIsUnitDet_apply, e.constr_basis]
rw [← this]
exact ⟨v'.linearIndependent, v'.span_eq⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | is_basis_iff_det | null |
isUnit_det (e' : Basis ι R M) : IsUnit (e.det e') :=
(is_basis_iff_det e).mp ⟨e'.linearIndependent, e'.span_eq⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | isUnit_det | null |
AlternatingMap.eq_smul_basis_det (f : M [⋀^ι]→ₗ[R] R) : f = f e • e.det := by
refine Basis.ext_alternating e fun i h => ?_
let σ : Equiv.Perm ι := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)
change f (e ∘ σ) = (f e • e.det) (e ∘ σ)
simp [AlternatingMap.map_perm, Basis.det_self]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | AlternatingMap.eq_smul_basis_det | Any alternating map to `R` where `ι` has the cardinality of a basis equals the determinant
map with respect to that basis, multiplied by the value of that alternating map on that basis. |
AlternatingMap.map_basis_eq_zero_iff {ι : Type*} [Finite ι] (e : Basis ι R M)
(f : M [⋀^ι]→ₗ[R] R) : f e = 0 ↔ f = 0 :=
⟨fun h => by
cases nonempty_fintype ι
letI := Classical.decEq ι
simpa [h] using f.eq_smul_basis_det e,
fun h => h.symm ▸ AlternatingMap.zero_apply _⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | AlternatingMap.map_basis_eq_zero_iff | null |
AlternatingMap.map_basis_ne_zero_iff {ι : Type*} [Finite ι] (e : Basis ι R M)
(f : M [⋀^ι]→ₗ[R] R) : f e ≠ 0 ↔ f ≠ 0 :=
not_congr <| f.map_basis_eq_zero_iff e
variable {A : Type*} [CommRing A] [Module A M] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | AlternatingMap.map_basis_ne_zero_iff | null |
@[simp]
det_comp (e : Basis ι A M) (f : M →ₗ[A] M) (v : ι → M) :
e.det (f ∘ v) = (LinearMap.det f) * e.det v := by
rw [det_apply, det_apply, ← f.det_toMatrix e, ← Matrix.det_mul,
e.toMatrix_eq_toMatrix_constr (f ∘ v), e.toMatrix_eq_toMatrix_constr v, ← toMatrix_comp,
e.constr_comp]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_comp | null |
det_comp_basis [Module A M'] (b : Basis ι A M) (b' : Basis ι A M') (f : M →ₗ[A] M') :
b'.det (f ∘ b) = LinearMap.det (f ∘ₗ (b'.equiv b (Equiv.refl ι) : M' →ₗ[A] M)) := by
rw [det_apply, ← LinearMap.det_toMatrix b', LinearMap.toMatrix_comp _ b, Matrix.det_mul,
LinearMap.toMatrix_basis_equiv, Matrix.det_one, mul_one]
congr 1; ext i j
rw [toMatrix_apply, LinearMap.toMatrix_apply, Function.comp_apply]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_comp_basis | null |
det_basis (b : Basis ι A M) (b' : Basis ι A M) :
LinearMap.det (b'.equiv b (Equiv.refl ι)).toLinearMap = b'.det b :=
(b.det_comp_basis b' (LinearMap.id)).symm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_basis | null |
det_mul_det (b b' b'' : Basis ι A M) :
b.det b' * b'.det b'' = b.det b'' := by
have : b'' = (b'.equiv b'' (Equiv.refl ι)).toLinearMap ∘ b' := by
ext; simp
conv_rhs =>
rw [this, Basis.det_comp, det_basis, mul_comm] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_mul_det | null |
det_inv (b : Basis ι A M) (b' : Basis ι A M) :
(b.isUnit_det b').unit⁻¹ = b'.det b := by
rw [← Units.mul_eq_one_iff_inv_eq, IsUnit.unit_spec, ← det_basis, ← det_basis]
exact LinearEquiv.det_mul_det_symm _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_inv | null |
det_reindex {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Basis ι R M) (v : ι' → M)
(e : ι ≃ ι') : (b.reindex e).det v = b.det (v ∘ e) := by
rw [det_apply, toMatrix_reindex', det_reindexAlgEquiv, det_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_reindex | null |
det_reindex' {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Basis ι R M)
(e : ι ≃ ι') : (b.reindex e).det = b.det.domDomCongr e :=
AlternatingMap.ext fun _ => det_reindex _ _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_reindex' | null |
det_reindex_symm {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Basis ι R M)
(v : ι → M) (e : ι' ≃ ι) : (b.reindex e.symm).det (v ∘ e) = b.det v := by
rw [det_reindex, Function.comp_assoc, e.self_comp_symm, Function.comp_id]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_reindex_symm | null |
det_map (b : Basis ι R M) (f : M ≃ₗ[R] M') (v : ι → M') :
(b.map f).det v = b.det (f.symm ∘ v) := by
rw [det_apply, toMatrix_map, det_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_map | null |
det_map' (b : Basis ι R M) (f : M ≃ₗ[R] M') :
(b.map f).det = b.det.compLinearMap f.symm :=
AlternatingMap.ext <| b.det_map f | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_map' | null |
@[simp]
Pi.basisFun_det : (Pi.basisFun R ι).det = Matrix.detRowAlternating := by
ext M
rw [Basis.det_apply, Basis.coePiBasisFun.toMatrix_eq_transpose, det_transpose] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | Pi.basisFun_det | null |
Pi.basisFun_det_apply (v : ι → ι → R) :
(Pi.basisFun R ι).det v = (Matrix.of v).det := by
rw [Pi.basisFun_det]
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | Pi.basisFun_det_apply | null |
det_smul_mk_coord_eq_det_update {v : ι → M} (hli : LinearIndependent R v)
(hsp : ⊤ ≤ span R (range v)) (i : ι) :
e.det v • (Basis.mk hli hsp).coord i = e.det.toMultilinearMap.toLinearMap v i := by
apply (Basis.mk hli hsp).ext
intro k
rcases eq_or_ne k i with (rfl | hik) <;>
simp only [Algebra.id.smul_eq_mul, coe_mk, LinearMap.smul_apply,
MultilinearMap.toLinearMap_apply]
· rw [mk_coord_apply_eq, mul_one, update_eq_self]
congr
· rw [mk_coord_apply_ne hik, mul_zero, eq_comm]
exact e.det.map_eq_zero_of_eq _ (by simp [hik]) hik | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_smul_mk_coord_eq_det_update | If we fix a background basis `e`, then for any other basis `v`, we can characterise the
coordinates provided by `v` in terms of determinants relative to `e`. |
det_unitsSMul (e : Basis ι R M) (w : ι → Rˣ) :
(e.unitsSMul w).det = (↑(∏ i, w i)⁻¹ : R) • e.det := by
ext f
change
(Matrix.det fun i j => (e.unitsSMul w).repr (f j) i) =
(↑(∏ i, w i)⁻¹ : R) • Matrix.det fun i j => e.repr (f j) i
simp only [e.repr_unitsSMul]
convert Matrix.det_mul_column (fun i => (↑(w i)⁻¹ : R)) fun i j => e.repr (f j) i
simp [← Finset.prod_inv_distrib] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_unitsSMul | If a basis is multiplied columnwise by scalars `w : ι → Rˣ`, then the determinant with respect
to this basis is multiplied by the product of the inverse of these scalars. |
@[simp]
det_unitsSMul_self (w : ι → Rˣ) : e.det (e.unitsSMul w) = ∏ i, (w i : R) := by
simp [det_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_unitsSMul_self | The determinant of a basis constructed by `unitsSMul` is the product of the given units. |
@[simp]
det_isUnitSMul {w : ι → R} (hw : ∀ i, IsUnit (w i)) :
e.det (e.isUnitSMul hw) = ∏ i, w i :=
e.det_unitsSMul_self _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Tactic.FieldSimp",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.Basis"
] | Mathlib/LinearAlgebra/Determinant.lean | det_isUnitSMul | The determinant of a basis constructed by `isUnitSMul` is the product of the given units. |
lmk (s : Finset ι) : (∀ i : (↑s : Set ι), M i) →ₗ[R] Π₀ i, M i where
toFun := mk s
map_add' _ _ := mk_add
map_smul' c x := mk_smul c x | def | LinearAlgebra | [
"Mathlib.Data.DFinsupp.Submonoid",
"Mathlib.Data.DFinsupp.Sigma",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.LinearAlgebra.Finsupp.SumProd",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/DFinsupp.lean | lmk | `DFinsupp.mk` as a `LinearMap`. |
lsingle (i) : M i →ₗ[R] Π₀ i, M i :=
{ DFinsupp.singleAddHom _ _ with
toFun := single i
map_smul' := single_smul } | def | LinearAlgebra | [
"Mathlib.Data.DFinsupp.Submonoid",
"Mathlib.Data.DFinsupp.Sigma",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.LinearAlgebra.Finsupp.SumProd",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/DFinsupp.lean | lsingle | `DFinsupp.single` as a `LinearMap` |
lhom_ext ⦃φ ψ : (Π₀ i, M i) →ₗ[R] N⦄ (h : ∀ i x, φ (single i x) = ψ (single i x)) : φ = ψ :=
LinearMap.toAddMonoidHom_injective <| addHom_ext h | theorem | LinearAlgebra | [
"Mathlib.Data.DFinsupp.Submonoid",
"Mathlib.Data.DFinsupp.Sigma",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.LinearAlgebra.Finsupp.SumProd",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/DFinsupp.lean | lhom_ext | Two `R`-linear maps from `Π₀ i, M i` which agree on each `single i x` agree everywhere. |
@[ext 1100]
lhom_ext' ⦃φ ψ : (Π₀ i, M i) →ₗ[R] N⦄ (h : ∀ i, φ.comp (lsingle i) = ψ.comp (lsingle i)) :
φ = ψ :=
lhom_ext fun i => LinearMap.congr_fun (h i) | theorem | LinearAlgebra | [
"Mathlib.Data.DFinsupp.Submonoid",
"Mathlib.Data.DFinsupp.Sigma",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.LinearAlgebra.Finsupp.SumProd",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/DFinsupp.lean | lhom_ext' | Two `R`-linear maps from `Π₀ i, M i` which agree on each `single i x` agree everywhere.
See note [partially-applied ext lemmas].
After applying this lemma, if `M = R` then it suffices to verify
`φ (single a 1) = ψ (single a 1)`. |
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