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Description of  Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Student Mathematical Library)

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)