Matroids – a quick prehistory
Matroid theory is an active area of mathematics that uses ideas from abstract and linear algebra, geometry, combinatorics and graph theory. The study of matroids thus offers students a unique opportunity to synthesize several different areas within mathematics typically studied at the undergraduate level. Furthermore, matroids are an active area of research; Mathematical Reviews lists some 2000 publications with the word “matroid” in the title, with more than a third of these appearing in the last decade.
Why have we written this book? Our motivation is direct: There is no comprehensive text written for undergraduates on this topic. There are several more advanced treatments of the subject, suitable for graduate students or researchers, butmost of these are difficult for undergraduates to read. To paraphrase an old joke, this text seeks to fill this “muchneeded gap.”
This text introduces matroids by emphasizing geometry, focusing especially on geometric (affine) dependence. Interpreting this approach for finite subsets of a vector space, points in Euclidean space or the edges of a graph gives a matroid spin to linear algebra, discrete geometry and graph theory. We believe the geometric approach, which both authors learned from their common Ph.D. advisor, Thomas Brylawski, to be the most natural, useful and powerful in understanding the subject.
The common thread that ties the various classes of matroids together is the abstract notion of independence. This unifying idea is due to Hassler Whitney, who defined matroids in his foundational paper [42] in 1935.