Description
Book SynopsisContains a collection of mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof, and assumes only a modest background in linear algebra. The topics include Hamming codes, the matrix-tree theorem, the Lovász bound on the Shannon capacity, and a counterexample to Borsuk's conjecture.
Trade ReviewFinding 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." -
ChoiceTable of Contents
- Preface
- Notation
- Fibonacci numbers, quickly
- Fibonacci numbers, the formula
- The clubs of Oddtown
- Same-size intersections
- Error-correcting codes
- Odd distances
- Are these distances Euclidean?
- Packing complete bipartite graphs
- Equiangular lines
- Where is the triangle?
- Checking matrix multiplication
- Tiling a rectangle by squares
- Three Petersens are not enough
- Petersen, Hoffman–Singleton, and maybe 57
- Only two distances
- Covering a cube minus one vertex
- Medium-size intersection is hard to avoid
- On the difficulty of reducing the diameter
- The end of the small coins
- Walking in the yard
- Counting spanning trees
- In how many ways can a man tile a board?
- More bricks—more walls?
- Perfect matchings and determinants
- Turning a ladder over a finite field
- Counting compositions
- Is it associative?
- The secret agent and umbrella
- Shannon capacity of the union: a tale of two fields
- Equilateral sets
- Cutting cheaply using eigenvectors
- Rotating the cube
- Set pairs and exterior products
- Index
