This work provides an introductory overview of matroid theory, providing examples and investigating base exchanges. The work introduces fundamental concepts, and provides illustrations to visualize and clarify this concepts. It explores base exchanges, presenting some of the actual results and open problems concerning base exchanges. The study focuses on cyclically orderings of bases such that exchanges of elements can be made while preserving bases. We also give a counterexample to an open problem, followed by an easier way to study for which classes of matroids the problem may hold.