Matroid Induced by Linear Independence in Abelian Group is Matroid

Theorem
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 $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $G$ on $S$.

That is, $\mathscr I$ is the set of linearly independent subsets of $S$.

Then $\struct{S, \mathscr I}$ is a matroid.

Proof
It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$.