There are two basic ways to get "sub-matroids", one of which is rather obvious, the other of which is less so. Given an element x of the ground set E, we can either delete x, or contract it.

To delete x, we just think of knocking x right out of the ground set E; the new independent sets are {T\x | T in I}. The new matroid we get is denoted M\x. If we continually delete elements so that we've deleted an entire set X from the matroid, we denote the new matroid M\X. To delete an element/edge x from a graphic matroid M(G), we just erase that edge from the graph (but we leave its vertices in place).

Contraction isn't so easy. The contraction of M by x, denoted M/x, is defined (not very pleasantly) as M/x = (M*\x)*, but if x is a loop of M, we just delete it normally. To contract a non-loop element x of a graph G, think of shrinking the edge down to nothing, pulling the edge's end-vertices together, so that the edge itself vanishes and its endpoints have been welded together into a single vertex. (To contract a loop, we just delete it.) M/X is defined in the obvious way.

A minor of a matroid M is a matroid gotten from M by contracting some (or none) of M's elements and deleting others. In obtaining a minor, it doesn't matter which order you delete or contract in, or whether you do deleting or contracting first. Thus, we'll always be able to write minors as M\S/T (where we've deleted all the elements in the set S, and contracted out all the elements in the set T).