Show that the exchange axiom holds for the Partition Matroid defined in class.
The graphic matroid of a graph \(G\) can be represented by the following matrix: we have one row for each vertex, and one column for each edge. The column for edge \(e\) has \(+1\) in the row for one endpoint, \(-1\) in the row for the other endpoint, and \(0\) elsewhere; the choice of which endpoint to give which sign is arbitrary.
Argue that this is a valid representation (i.e, that the forests correspond to linearly independent columns and the subsets of edges that have cycles in them correspond to dependent columns).