Section lin_dep_def.
(** %\tocdef{is\_lin\_comb}% *)
Definition is_lin_comb (F : field) (V : vectorspace F)
(x : V) (S : part_set V) :=
exists n : Nat,
(exists a : seq n F,
(exists v : seq n S, x =' sum (mult_by_scalars a (Map_embed v)) in _)).

Lemma is_lin_comb_comp :
forall (F : field) (V : vectorspace F) (x y : V) (S T : part_set V),
S =' T in _ -> x =' y in _ -> is_lin_comb x S -> is_lin_comb y T.