\begin{definition}[Definition:Matroid Induced by Linear Independence/Abelian Group]
Let $\struct{G, +}$ be a torsion-free Abelian group.
Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.
Let $S$ be a finite subset of $G$.
Let $\mathscr I$ be the set of linearly independent subsets of $S$.
Then the ordered pair $\struct{S, \mathscr I}$ is called the '''matroid induced by linear independence in $G$ on $S$'''.
\end{definition}