Section InvLfun.
Variable (K : fieldType) (aT : FalgType K).
Implicit Types f g : 'End(aT).
Definition lfun_invr f := if lker f == 0%VS then f^-1%VF else f.
Canonical amull_additive := Eval hnf in Additive amull_is_linear.
Canonical amull_linear := Eval hnf in AddLinear amull_is_linear.
Canonical amulr_additive := Eval hnf in Additive amulr_is_lrmorphism.
Canonical amulr_linear := Eval hnf in AddLinear amulr_is_lrmorphism.
Canonical amulr_rmorphism := Eval hnf in AddRMorphism amulr_is_lrmorphism.
Canonical amulr_lrmorphism := Eval hnf in LRMorphism amulr_is_lrmorphism.
Canonical prodv_muloid := Monoid.MulLaw prod0v prodv0.
Canonical addv_addoid := Monoid.AddLaw prodvDl prodvDr.
Canonical prodv_monoid := Monoid.Law prodvA prod1v prodv1.
Definition expv U n := iterop n.+1.-1 prodv U 1%VS.
Local Notation "A ^+ n" := (expv A n) : vspace_scope.
(* Centralisers and centers. *)
Definition centraliser1_vspace u := lker (amulr u - amull u).
Local Notation "'C [ u ]" := (centraliser1_vspace u) : vspace_scope.
Definition centraliser_vspace V := (\bigcap_i 'C[tnth (vbasis V) i])%VS.
Local Notation "'C ( V )" := (centraliser_vspace V) : vspace_scope.
Definition center_vspace V := (V :&: 'C(V))%VS.
Local Notation "'Z ( V )" := (center_vspace V) : vspace_scope.

Lemma cent1vP u v : reflect (u * v = v * u) (u \in 'C[v]%VS).
Lemma centvsP U V : reflect {in U & V, commutative *%R} (U <= 'C(V))%VS.