End BigSumBasis.
End VectorTheory.
Hint Resolve subvv : core.
Arguments subvP {K vT U V}.
Arguments addv_idPl {K vT U V}.
Arguments addv_idPr {K vT U V}.
Arguments memv_addP {K vT w U V }.
Arguments sumv_sup [K vT I] i0 [P U Vs].
Arguments memv_sumP {K vT I P Us v}.
Arguments subv_sumP {K vT I P Us V}.
Arguments capv_idPl {K vT U V}.
Arguments capv_idPr {K vT U V}.
Arguments memv_capP {K vT w U V}.
Arguments bigcapv_inf [K vT I] i0 [P Us V].
Arguments subv_bigcapP {K vT I P U Vs}.
Arguments directvP {K vT S}.
Arguments directv_addP {K vT U V}.
Arguments directv_add_unique {K vT U V}.
Arguments directv_sumP {K vT I P Us}.
Arguments directv_sumE {K vT I P Ss}.
Arguments directv_sum_independent {K vT I P Us}.
Arguments directv_sum_unique {K vT I P Us}.
Arguments span_subvP {K vT X U}.
Arguments freeP {K vT n X}.
Prenex Implicits coord.
Notation directv S := (directv_def (Phantom _ S%VS)).
(* Linear functions over a vectType *)
Section LfunDefs.
Variable R : ringType.
Implicit Types aT vT rT : vectType R.
Fact lfun_key : unit. Proof. by []. Qed.
Definition fun_of_lfun_def aT rT (f : 'Hom(aT, rT)) :=
r2v \o mulmxr (f2mx f) \o v2r.
Definition fun_of_lfun := locked_with lfun_key fun_of_lfun_def.
Canonical fun_of_lfun_unlockable := [unlockable fun fun_of_lfun].
Definition linfun_def aT rT (f : aT -> rT) :=
Vector.Hom (lin1_mx (v2r \o f \o r2v)).
Definition linfun := locked_with lfun_key linfun_def.
Canonical linfun_unlockable := [unlockable fun linfun].
Definition id_lfun vT := @linfun vT vT idfun.
Definition comp_lfun aT vT rT (f : 'Hom(vT, rT)) (g : 'Hom(aT, vT)) :=
linfun (fun_of_lfun f \o fun_of_lfun g).
End LfunDefs.
Coercion fun_of_lfun : Vector.hom >-> Funclass.
Notation "\1" := (@id_lfun _ _) : lfun_scope.
Notation "f \o g" := (comp_lfun f g) : lfun_scope.
Section LfunVspaceDefs.
Variable K : fieldType.
Implicit Types aT rT : vectType K.
Definition inv_lfun aT rT (f : 'Hom(aT, rT)) := Vector.Hom (pinvmx (f2mx f)).
Definition lker aT rT (f : 'Hom(aT, rT)) := mx2vs (kermx (f2mx f)).
Fact lfun_img_key : unit. Proof. by []. Qed.
Definition lfun_img_def aT rT f (U : {vspace aT}) : {vspace rT} :=
mx2vs (vs2mx U *m f2mx f).
Definition lfun_img := locked_with lfun_img_key lfun_img_def.
Canonical lfun_img_unlockable := [unlockable fun lfun_img].
Definition lfun_preim aT rT (f : 'Hom(aT, rT)) W :=
(lfun_img (inv_lfun f) (W :&: lfun_img f fullv) + lker f)%VS.
End LfunVspaceDefs.
Prenex Implicits linfun lfun_img lker lfun_preim.
Notation "f ^-1" := (inv_lfun f) : lfun_scope.
Notation "f @: U" := (lfun_img f%VF%R U) (at level 24) : vspace_scope.
Notation "f @^-1: W" := (lfun_preim f%VF%R W) (at level 24) : vspace_scope.
Notation limg f := (lfun_img f fullv).
Section LfunZmodType.
Variables (R : ringType) (aT rT : vectType R).
Implicit Types f g h : 'Hom(aT, rT).
Definition lfun_eqMixin := Eval hnf in [eqMixin of 'Hom(aT, rT) by <:].
Canonical lfun_eqType := EqType 'Hom(aT, rT) lfun_eqMixin.
Definition lfun_choiceMixin := [choiceMixin of 'Hom(aT, rT) by <:].
Canonical lfun_choiceType := ChoiceType 'Hom(aT, rT) lfun_choiceMixin.
Fact lfun_is_linear f : linear f.

Lemma lfunP f g : f =1 g <-> f = g.
Definition zero_lfun : 'Hom(aT, rT) := linfun \0.