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Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra

Jiří Matoušek Charles University, Prague, Czech Republic
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Softcover ISBN: 978-0-8218-4977-4
Product Code: STML/53
List Price: $41.00 Individual Price:$32.80
Electronic ISBN: 978-1-4704-1636-2
Product Code: STML/53.E
List Price: $38.00 Individual Price:$30.40
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List Price: $61.50 Click above image for expanded view Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra Jiří Matoušek Charles University, Prague, Czech Republic Available Formats:  Softcover ISBN: 978-0-8218-4977-4 Product Code: STML/53  List Price:$41.00 Individual Price: $32.80  Electronic ISBN: 978-1-4704-1636-2 Product Code: STML/53.E  List Price:$38.00 Individual Price: $30.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$61.50
• Book Details

Student Mathematical Library
Volume: 532010; 182 pp
MSC: Primary 05; 68; 15;
Winner of a CHOICE Outstanding Academic Title Award for 2012!

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.

Undergraduates, graduate students and research mathematicians interested in combinatorics, graph theory, theoretical computer science, and geometry.

• Chapters
• Miniature 1. Fibonacci numbers, quickly
• Miniature 2. Fibonacci numbers, the formula
• Miniature 3. The clubs of Oddtown
• Miniature 4. Same-size intersections
• Miniature 5. Error-correcting codes
• Miniature 6. Odd distances
• Miniature 7. Are these distances Euclidean?
• Miniature 8. Packing complete bipartite graphs
• Miniature 9. Equiangular lines
• Miniature 10. Where is the triangle?
• Miniature 11. Checking matrix multiplication
• Miniature 12. Tiling a rectangle by squares
• Miniature 13. Three Petersens are not enough
• Miniature 14. Petersen, Hoffman–Singleton, and maybe 57
• Miniature 15. Only two distances
• Miniature 16. Covering a cube minus one vertex
• Miniature 17. Medium-size intersection is hard to avoid
• Miniature 18. On the difficulty of reducing the diameter
• Miniature 19. The end of the small coins
• Miniature 20. Walking in the yard
• Miniature 21. Counting spanning trees
• Miniature 22. In how many ways can a man tile a board?
• Miniature 23. More bricks—more walls?
• Miniature 24. Perfect matchings and determinants
• Miniature 25. Turning a ladder over a finite field
• Miniature 26. Counting compositions
• Miniature 27. Is it associative?
• Miniature 28. The secret agent and umbrella
• Miniature 29. Shannon capacity of the union: A tale of two fields
• Miniature 30. Equilateral sets
• Miniature 31. Cutting cheaply using eigenvectors
• Miniature 32. Rotating the cube
• Miniature 33. Set pairs and exterior products

• Reviews

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 532010; 182 pp
MSC: Primary 05; 68; 15;
Winner of a CHOICE Outstanding Academic Title Award for 2012!

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.

Undergraduates, graduate students and research mathematicians interested in combinatorics, graph theory, theoretical computer science, and geometry.

• Chapters
• Miniature 1. Fibonacci numbers, quickly
• Miniature 2. Fibonacci numbers, the formula
• Miniature 3. The clubs of Oddtown
• Miniature 4. Same-size intersections
• Miniature 5. Error-correcting codes
• Miniature 6. Odd distances
• Miniature 7. Are these distances Euclidean?
• Miniature 8. Packing complete bipartite graphs
• Miniature 9. Equiangular lines
• Miniature 10. Where is the triangle?
• Miniature 11. Checking matrix multiplication
• Miniature 12. Tiling a rectangle by squares
• Miniature 13. Three Petersens are not enough
• Miniature 14. Petersen, Hoffman–Singleton, and maybe 57
• Miniature 15. Only two distances
• Miniature 16. Covering a cube minus one vertex
• Miniature 17. Medium-size intersection is hard to avoid
• Miniature 18. On the difficulty of reducing the diameter
• Miniature 19. The end of the small coins
• Miniature 20. Walking in the yard
• Miniature 21. Counting spanning trees
• Miniature 22. In how many ways can a man tile a board?
• Miniature 23. More bricks—more walls?
• Miniature 24. Perfect matchings and determinants
• Miniature 25. Turning a ladder over a finite field
• Miniature 26. Counting compositions
• Miniature 27. Is it associative?
• Miniature 28. The secret agent and umbrella
• Miniature 29. Shannon capacity of the union: A tale of two fields
• Miniature 30. Equilateral sets
• Miniature 31. Cutting cheaply using eigenvectors
• Miniature 32. Rotating the cube
• Miniature 33. Set pairs and exterior products
• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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