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.

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 the 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

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