Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Student Mathematical Library) by Jiri Matousek (PDF)

Description

This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lovász bound on the Shannon capacity, and a counterexample to Borsuk’s conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for seminar presentations.

User’s Reviews

Editorial Reviews: Review Finding examples of “linear algebra in action” that are both accessible and convincing is difficult. Thirty-three Miniatures is an attempt to present some usable examples. . . . For me, the biggest impact of the book came from noticing the tools that are used. Many linear algebra textbooks, including the one I use, delay discussion of inner products and transpose matrices till later in the course, which sometimes means they don’t get discussed at all. Seeing how often the transpose matrix shows up in Matousek’s miniatures made me realize space must be made for it. Similarly, the theorem relating the rank of the product of two matrices to the ranks of the factors plays a big role here. Most linear algebra instructors would benefit from this kind of insight. . . . Thirty-three Miniatures would be an excellent book for an informal seminar offered to students after their first linear algebra course. It may also be the germ of many interesting undergraduate talks. And it’s fun as well. –Fernando Q. Gouvêa, MAA Reviews[This book] is an excellent collection of clever applications of linear algebra to various areas of (primarily) discrete/combinatiorial mathematics. … The style of exposition is very lively, with fairly standard usage of terminologies and notations. … Highly recommended. –Choice About the Author Ji í Matou ek: Charles University, Prague, Czech Republic

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Jiri Matousek is an accomplished mathematician working in Computational Geometry and he has written several books. I expected some powerful stuff from him in this book and in my opinion he more than delivered. My mathematical background is nothing I can boast about, but from whatever understanding I have of Linear Algebra and Combinatorics, the text is a wonderful treat of mathematical capsules each containing powerful idea(s). I have yet to go through the text completely and while I do not do that for most of the books, I will most certainly complete this one. I fell in love with the text as soon as I finished the 3rd miniature called “the clubs of oddtown”. The author attributes it to Babai and Frankl and discusses a related problem (Fisher’s inequality) in miniature 4. I feel I must stress here that you (or more appropriately, people who do not have much experience deploying an algebraic argument for a seemingly combinatorial problem) should not read a miniature like you are reading a novel. Most likely a pure combinatorial solution will evade the readers or it can be found only after too much effort. Therefore, you should try to think through the problem the miniature discusses and then plan your own attack on the problem. If I may, I would suggest that deploy a combinatorial, geometric whatever kind of attack that seems natural to you. If your methods work, great. If they do not, then I believe you will share the same hair raising experience I had when I read through the first few chapters and to me it seemed that a linear algebraic solution just jumped out of the blue – it was kind of unexpected. I believe this book will certainly develop more facility in deploying algebraic attacks on combinatorial problems by giving a flavour of (what I consider) unusual.The only other thing I wish is that Professor Matousek writes a sequel and yes, I have a different opinion than his; I am most willing to see his four A4 sheets rule bend (Prof Matousek has rarely included items which exceed four A4 sheets, he does not call them a miniature).

⭐I just liked it! Reading fiction or popular science is not that entertaining if you are a theoretical physicist.This is a kind of entertaining math book about algebraic combinatorics for a physicist or mathematician from a different field.Starting from miniature 22, chapters become a bit complex for me, so I had no time to go through them yet.In any case, recommend to my colleagues, theoretical physicists, if they are curious about this topic.

⭐This is a beautiful book, very well-written, as other books of Jiri Matousek. It was a pity that this wonderful author passed away recently. Each chapter is short and can be read in one setting. The book is short yet contains a great number of nice and deep ideas. It demonstrates how thinking linear algebraically can help in solving various problems in different areas. It was a pleasure to read it.

⭐This book suffers from a lack of clarity in many parts. (I’ve confirmed this with a few others…)I rounded my rating up to 4.0 because there are some remarkably interesting things in here.

⭐This book is fantastically clear, concise, and just lovely. I wish Matousek would write one of these for every introductory mathematical subfield.

⭐Enjoying every bit of it. Lots of super cool ideas – great read ..

⭐Excellent

⭐Some math book require our full attention (and then some) You must take notes and read and then reread to understand the proofs. In many math books a good stopping place is hard to find, one thing leads to the next and to the next… and were it not for the sheer density of the material one would feel compelled to read it all at one sitting! This is an enjoyable enough activity for anyone who likes math, but it isn’t really relaxing. I took this lovely book with me on a vacation to Hong Kong and with it discovered a wholly different math-book experience.The book has 33 “chapters” of 1-4 pages each (most are two pages) and in each a proof is suggested first (so you can try to work it out) and then reveled. Basic theorems from undergraduate linear algebra come in handy, but very little advanced knowledge is needed. A pen a paper will help on some of the proofs but most can be done just by thinking.What makes it all especially exciting is that the proofs are useful! many suggest fun ideas for applications.Since it’s presented in little bite-sized chunks you can put it down at any moment.I think any undergrad who has taken linear algebra, grad students and mathematics instructors and professors would enjoy this book. I couldn’t help but share one or two of these little gems with my students!

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