The definition I have been giving of a spanning set is the following;
If $V = span(S)$, then $S$is called a spanning set for $V$ and we
say $V$ is spanned by $S$.
My confusion here is what if all the vectors in $V$ can be written as linear combinations of the ones in $S$, but there are some linear combinations of the elements in $S$ that are not in $V$, is it still a spanning set, because it still spans the whole space so it seems like it'd be true but the definition implies otherwise.
Best Answer
If all the vectors in $V$ can be written as a linear combination of the ones in $S$ then $S$ spans $V$. Your concern about there being some linear combinations of elements in $S$ that are not in $V$ is trivial. Since each element of $S$ is an element of $V$ and $V$ is a vector space, all linear combinations of elements in $S$ will be elements of $V$. This is because of the axioms of a vector space, namely that it is closed under addition and scalar multiplication.