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).
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!