Your wording seems a bit off to me, but your idea seems quite right.
how do i prove that every subspace is a spanning set for itself?
sorry im very weak at proofing.
my working:
let S be the subspace.
so we need to show that S = span(S)
we shall show that S⊂span S ( shown in previous post) and span S ⊂S
to show that spanS ⊂ S,
lets assume that S is in n dimension where xi∈ S where 1<i<n
span S =a1x1+...+anxn ⊂ S as S being a subspace obeys the laws of scalar multiplication and addition.
is this done like this?
You want to show that when S is a subspace, we have that span S is contained in S. We don't really need to know anything about the dimension of S.
Take an element y in span S. Since span S consists of finite linear combinations of elements from S, we can write y = a_1x_1+...+a_nx_n for some elements x_i in S, and some scalars a_i (note: this n need not coincide with the dimension of S). And then, as you say, since S is closed under scaling and under vector addition, then a_1x_1+...+a_nx_n will also be in S, whereas y will be in S. So all elements of span S are contained in S - hence span S is a subset of S.