No matter what the elements of the set $S$ are, we can choose the scalars in a linear combination to all be zero. Suppose that $S=\set{\vectorlist{v}{p}}$. Then compute
\begin{align*}
0\vect{v}_1+0\vect{v}_2+0\vect{v}_3+\cdots+0\vect{v}_p
&=\zerovector+\zerovector+\zerovector+\cdots+\zerovector\\
&=\zerovector
\end{align*}
But what if we choose $S$ to be the empty set? The convention is that the empty sum in Definition SSCV evaluates to “zero,” in this case this is the zero vector.