Linear equivalences involving submodules #
A linear equivalence which maps a submodule of one module onto another, restricts to a linear equivalence of the two submodules.
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A linear equivalence of two modules restricts to a linear equivalence from the preimage of any submodule to that submodule.
This is LinearEquiv.ofSubmodule
but with comap
on the left instead of map
on the right.
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A linear map f : M →ₗ[R] M₂
with a left-inverse g : M₂ →ₗ[R] M
defines a linear
equivalence between M
and f.range
.
This is a computable alternative to LinearEquiv.ofInjective
, and a bidirectional version of
LinearMap.rangeRestrict
.
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An Injective
linear map f : M →ₗ[R] M₂
defines a linear equivalence
between M
and f.range
. See also LinearMap.ofLeftInverse
.
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A bijective linear map is a linear equivalence.
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Given p
a submodule of the module M
and q
a submodule of p
, p.equivSubtypeMap q
is the natural LinearEquiv
between q
and q.map p.subtype
.
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A linear injection M ↪ N
restricts to an equivalence f⁻¹ p ≃ p
for any submodule p
contained in its range.
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The restriction of a linear map on the target to a submodule of the target given by an inclusion.
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The restriction of a bilinear map to a submodule in which it takes values.