TITLE: How can I find a basis for the kernel of a homomorphism on a free abelian group?
QUESTION [6 upvotes]: Let $\phi\colon F\to G$ be a homomorphism of finitely generated abelian groups. If $F$ is free, then $\ker(\phi)$ is also free and thus admits a basis.
Question: Is there a general procedure to find a basis for $\ker(\phi)$?

As an example, consider the homomorphism $\phi\colon\mathbb Z^3\to (\mathbb Z/2)^2$ given by the matrix
$$
\begin{pmatrix}
1&1&0\\0&1&1
\end{pmatrix}.
$$
After playing around, I guess that the elements $\begin{pmatrix}0\\2\\0\end{pmatrix}$, $\begin{pmatrix}1\\1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\\1\end{pmatrix}$ form a basis. Is this true? How could I see this in a systematic way?

REPLY [2 votes]: Try also the Smith normal form.