Thirty-three Miniatures

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 Lovasz 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: 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. (STML/53)

Cover 1 Title page 2 Contents 4 Preface 6 Notation 10 Fibonacci numbers, quickly 12 Fibonacci numbers, the formula 14 The clubs of Oddtown 16 Same-size intersections 18 Error-correcting codes 22 Odd distances 28 Are these distances Euclidean? 30 Packing complete bipartite graphs 34 Equiangular lines 38 Where is the triangle? 42 Checking matrix multiplication 46 Tiling a rectangle by squares 50 Three Petersens are not enough 52 Petersen, Hoffman–Singleton, and maybe 57 56 Only two distances 62 Covering a cube minus one vertex 66 Medium-size intersection is hard to avoid 68 On the difficulty of reducing the diameter 72 The end of the small coins 78 Walking in the yard 82 Counting spanning trees 88 In how many ways can a man tile a board? 96 More bricks—more walls? 108 Perfect matchings and determinants 118 Turning a ladder over a finite field 124 Counting compositions 130 Is it associative? 136 The secret agent and umbrella 142 Shannon capacity of the union: a tale of two fields 150 Equilateral sets 158 Cutting cheaply using eigenvectors 164 Rotating the cube 174 Set pairs and exterior products 182 Index 190 Back Cover 194

Jiří Matoušek: Charles University, Prague, Czech Republic

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