Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra
About this Title
Jiří Matoušek, Charles University, Prague, Czech Republic
Publication: The Student Mathematical Library
Publication Year 2010: Volume 53
ISBNs: 978-0-8218-4977-4 (print); 978-1-4704-1636-2 (online)
MathSciNet review: MR2656313
MSC: Primary 15-01; Secondary 05A10, 05C70
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.
Table of Contents
- 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