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refactor(linear_algebra/basic): extract content about linear_map.gene…
…ral_linear_group This makes the file a little shorter.
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/- | ||
Copyright (c) 2019 Johan Commelin. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Johan Commelin | ||
-/ | ||
import algebra.module.equiv | ||
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/-! | ||
# The general linear group of linear maps | ||
The general linear group is defined to be the group of invertible linear maps from `M` to itself. | ||
See also `matrix.general_linear_group` | ||
## Main definitions | ||
* `linear_map.general_linear_group` | ||
-/ | ||
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variables (R M : Type*) | ||
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namespace linear_map | ||
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variables [semiring R] [add_comm_monoid M] [module R M] | ||
variables (R M) | ||
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/-- The group of invertible linear maps from `M` to itself -/ | ||
@[reducible] def general_linear_group := (M →ₗ[R] M)ˣ | ||
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namespace general_linear_group | ||
variables {R M} | ||
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instance : has_coe_to_fun (general_linear_group R M) (λ _, M → M) := by apply_instance | ||
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/-- An invertible linear map `f` determines an equivalence from `M` to itself. -/ | ||
def to_linear_equiv (f : general_linear_group R M) : (M ≃ₗ[R] M) := | ||
{ inv_fun := f.inv.to_fun, | ||
left_inv := λ m, show (f.inv * f.val) m = m, | ||
by erw f.inv_val; simp, | ||
right_inv := λ m, show (f.val * f.inv) m = m, | ||
by erw f.val_inv; simp, | ||
..f.val } | ||
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/-- An equivalence from `M` to itself determines an invertible linear map. -/ | ||
def of_linear_equiv (f : (M ≃ₗ[R] M)) : general_linear_group R M := | ||
{ val := f, | ||
inv := (f.symm : M →ₗ[R] M), | ||
val_inv := linear_map.ext $ λ _, f.apply_symm_apply _, | ||
inv_val := linear_map.ext $ λ _, f.symm_apply_apply _ } | ||
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variables (R M) | ||
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/-- The general linear group on `R` and `M` is multiplicatively equivalent to the type of linear | ||
equivalences between `M` and itself. -/ | ||
def general_linear_equiv : general_linear_group R M ≃* (M ≃ₗ[R] M) := | ||
{ to_fun := to_linear_equiv, | ||
inv_fun := of_linear_equiv, | ||
left_inv := λ f, by { ext, refl }, | ||
right_inv := λ f, by { ext, refl }, | ||
map_mul' := λ x y, by {ext, refl} } | ||
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@[simp] lemma general_linear_equiv_to_linear_map (f : general_linear_group R M) : | ||
(general_linear_equiv R M f : M →ₗ[R] M) = f := | ||
by {ext, refl} | ||
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@[simp] lemma coe_fn_general_linear_equiv (f : general_linear_group R M) : | ||
⇑(general_linear_equiv R M f) = (f : M → M) := | ||
rfl | ||
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end general_linear_group | ||
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end linear_map |
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