I just wanted to understand the link between set of all linear combinations (span) and allowing all elements from a vector space being written as a linear combination of some vectors. The reason why I want to understand the link is because the definition of span is the set of all linear combinations for any coefficient from a given field, but when someone sais that all elements from a vector space can be written as a linear combination, that kind of points towards the notion of there exists coefficients from a field, rather than all, so I would like to prove the following to show that indeed, if all elements from a vector space can be written as a linear combination of some vectors then these vectors do indeed span the vector space. The proof is straight forward, but in my opinion, the connection is clearer.
What I will prove: If V is a vector space over a field F and all elements of V can be expressed as a linear combination of the elements of S, then the elements of S span V.
Proof:
Assume that the elements of the vector space V can be written as a linear combination of the elements in the set S. Then it follows that the vector space V is a subset of the span(S) and since the span(S) is a subspace of V, it follows that span(S)=V.
Is the proof correct?
So importantly I would like to ask, Since the definition of span is: {a$_1$v$_1$+...+a$_n$v$_n$ : a$_{i}$$\in$ F, v$_i$$\in$V} Does that notation imply for all a$_i$ in F? or for some a$_i$$\in$F ?
There is nothing to prove, it is more a check of the definition of span indeed as you noticed, since any $v\in V$ can be expressed as a linear combination of the elements of $S$, we say that $V$ is spanned by $S$.