components M
Define an equivalence relation ~ on the ground set of M by e ~ f if e = f or $\{e,f\}$ is contained in a circuit. The equivalence classes under ~ are the connected components of M. A matroid is the direct sum of its connected components.






As the examples above show, the connected components of the graphic matroid M(G) need not be the same as the connected components of the graph G (indeed, for any graph G, there exists a connected graph H with M(G) isomorphic to M(H)).