Page "Graphic matroid" Paragraph 6

If a set of edges contains a cycle, then the corresponding columns ( multiplied by if necessary to reorient the edges consistently around the cycle ) sum to zero, and is not independent.

Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent.

Therefore, the column matrix is isomorphic to.