Consider the matrix $\partial_1=\begin{bmatrix}-1 & 0&-1&-1&0&0\\1 & -1&0&0&-1&0\\0&1&1&0&0&-1\\0&0&0&1&1&1\end{bmatrix}$
Which is the incidence matrix of the graph $K_4$.
I want to check if $Ker \partial_1 \oplus Im \partial_1 \cong \mathbb{Z}^6$ as $\mathbb{Z}$-modules but I have no idea how to do this type of calculation. Is it possible to do row operations while preserving the associated spaces?