Further results on (semi)linear equivalences. #
If M
and M₂
are both R
-semimodules and S
-semimodules and R
-semimodule structures
are defined by an action of R
on S
(formally, we have two scalar towers), then any S
-linear
equivalence from M
to M₂
is also an R
-linear equivalence.
See also LinearMap.restrictScalars
.
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Alias of Module.End.isUnit_iff
.
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The tautological action by M ≃ₗ[R] M
on M
.
This generalizes Function.End.applyMulAction
.
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LinearEquiv.applyDistribMulAction
is faithful.
Any two modules that are subsingletons are isomorphic.
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g : R ≃+* S
is R
-linear when the module structure on S
is Module.compHom S g
.
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Each element of the group defines a linear equivalence.
This is a stronger version of DistribMulAction.toAddEquiv
.
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Each element of the group defines a module automorphism.
This is a stronger version of DistribMulAction.toAddAut
.
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An additive equivalence whose underlying function preserves smul
is a linear equivalence.
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An additive equivalence between commutative additive monoids is a linear equivalence between ℕ-modules
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An additive equivalence between commutative additive groups is a linear equivalence between ℤ-modules
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The equivalence between R-linear maps from R
to M
, and points of M
itself.
This says that the forgetful functor from R
-modules to types is representable, by R
.
This is an S
-linear equivalence, under the assumption that S
acts on M
commuting with R
.
When R
is commutative, we can take this to be the usual action with S = R
.
Otherwise, S = ℕ
shows that the equivalence is additive.
See note [bundled maps over different rings].
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The R
-linear equivalence between additive morphisms A →+ B
and ℕ
-linear morphisms A →ₗ[ℕ] B
.
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The R
-linear equivalence between additive morphisms A →+ B
and ℤ
-linear morphisms A →ₗ[ℤ] B
.
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Ring equivalence between additive group endomorphisms of an AddCommGroup
A
and
ℤ
-module endomorphisms of A.
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Between two zero modules, the zero map is an equivalence.
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Between two zero modules, the zero map is the only equivalence.
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If a linear map has an inverse, it is a linear equivalence.
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x ↦ -x
as a LinearEquiv
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A linear isomorphism between the domains and codomains of two spaces of linear maps gives an additive isomorphism between the two function spaces.
See also LinearEquiv.arrowCongr
for the linear version of this isomorphism.
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A linear isomorphism between the domains an codomains of two spaces of linear maps gives a linear isomorphism with respect to an action on the domains.
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If M
and M₂
are linearly isomorphic then the endomorphism rings of M
and M₂
are isomorphic.
See LinearEquiv.conj
for the linear version of this isomorphism.
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Multiplying by a unit a
of the ring R
is a linear equivalence.
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A linear isomorphism between the domains and codomains of two spaces of linear maps gives a linear isomorphism between the two function spaces.
See LinearEquiv.arrowCongrAddEquiv
for the additive version of this isomorphism that works
over a not necessarily commutative semiring.
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If M₂
and M₃
are linearly isomorphic then the two spaces of linear maps from M
into M₂
and M
into M₃
are linearly isomorphic.
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If M
and M₂
are linearly isomorphic then the two spaces of linear maps from M
and M₂
to
themselves are linearly isomorphic.
See LinearEquiv.conjRingEquiv
for the isomorphism between endomorphism rings,
which works over a not necessarily commutative semiring.
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An R
-linear isomorphism between two R
-modules M₂
and M₃
induces an S
-linear
isomorphism between M₂ →ₗ[R] M
and M₃ →ₗ[R] M
, if M
is both an R
-module and an
S
-module and their actions commute.
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Multiplying by a nonzero element a
of the field K
is a linear equivalence.
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An equivalence whose underlying function is linear is a linear equivalence.
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Given an R
-module M
and an equivalence m ≃ n
between arbitrary types,
construct a linear equivalence (n → M) ≃ₗ[R] (m → M)
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The product over S ⊕ T
of a family of modules is isomorphic to the product of
(the product over S
) and (the product over T
).
This is Equiv.sumPiEquivProdPi
as a LinearEquiv
.
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The product Π t : α, f t
of a family of modules is linearly isomorphic to the module
f ⬝
when α
only contains ⬝
.
This is Equiv.piUnique
as a LinearEquiv
.