Range of linear maps #
The range LinearMap.range
of a (semi)linear map f : M → M₂
is a submodule of M₂
.
More specifically, LinearMap.range
applies to any SemilinearMapClass
over a RingHomSurjective
ring homomorphism.
Note that this also means that dot notation (i.e. f.range
for a linear map f
) does not work.
Notations #
- We continue to use the notations
M →ₛₗ[σ] M₂
andM →ₗ[R] M₂
for the type of semilinear (resp. linear) maps fromM
toM₂
over the ring homomorphismσ
(resp. over the ringR
).
Tags #
linear algebra, vector space, module, range
The range of a linear map f : M → M₂
is a submodule of M₂
.
See Note [range copy pattern].
Equations
Instances For
Restrict the codomain of a linear map f
to f.range
.
This is the bundled version of Set.rangeFactorization
.
Equations
Instances For
The range of a linear map is finite if the domain is finite.
Note: this instance can form a diamond with Subtype.fintype
in the
presence of Fintype M₂
.
Equations
If N ⊆ M
then submodules of N
are the same as submodules of M
contained in N
.
See also Submodule.mapIic
.
Equations
Instances For
A monomorphism is injective.
If O
is a submodule of M
, and Φ : O →ₗ M'
is a linear map,
then (ϕ : O →ₗ M').submoduleImage N
is ϕ(N)
as a submodule of M'