Basis for a free abelian group of finite rank

Question

My question is as follows:

Given a free abelian group $G$ of finite rank $n$, is it true that every linearly independent set of $n$ members of $G$ form a basis for $G$ (when $G$ is viewed as a $\mathbb Z$-module)? I know this is true in the case of vector spaces, but I was wondering if this is true more generally for free modules.

Thanks!

Answer 1

As you pointed out in the comments, this is not correct. However, you may prove that if $\{m_1,\dots,m_n\}$ are independent in $G$ and $H=\langle m_1,\dots,m_n\rangle$, then $G/H$ is a finite abelian group.