The converse could read: “Suppose that $V$ is a vector space and $S=\set{\vectorlist{v}{m}}$ is a set of vectors in $V$. If, for each $\vect{w}\in V$, there are unique scalars $a_1,\,a_2,\,a_3,\,\ldots,\,a_m$ such that
\begin{equation*}
\vect{w}=\lincombo{a}{v}{m}
\end{equation*}
then $S$ is a linearly independent set that spans $V$.”
Since every vector $\vect{w}\in V$ is assumed to be a linear combination of the elements of $S$, it is easy to see that $S$ is a spanning set for $V$ (Definition SSVS).
To establish linear independence, begin with an arbitrary relation of linear dependence on the vectors in $S$ (Definition RLD). One way to form such a relation is the trivial way, where each scalar is zero. But our hypothesis of uniqueness then implies that that the only way to form this relation of linear dependence is the trivial way. But this establishes the linear independence of $S$ (Definition LI).