Lorenzo Robbiano

This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is notavailable at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is notavailable at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

O objetivo deste livro é o de fornecer as primeiras ferramentas matemáticas associadas à Álgebra Linear. O texto foi elaborado por um matemático que procurou "sair do seu personagem" para vir ao encontro de um público amplo. O desafio é o de tornar acessível a todos, os primeiros fundamentos e as primeiras técnicas de um saber fundamental para a ciência e a tecnologia. Com esse intuito, o autor escolheu escrevê-lo em um estilo não tradicional e o enriqueceu não somente com exercícios, mas também com frases de auto-referência, aforismos, citações, palíndromos e problemas para serem resolvidos com a ajuda do computador. O livro é para todos, mas tem uma dedicação especial aos estudantes dos cursos de graduação em disciplinas científicas e aos professores cujo texto poderá ser útil também como material de consulta.

This book provides students with the rudiments of Linear Algebra, a fundamental subject for students in all areas of science and technology. The book would also be good for statistics students studying linear algebra. It is the translation of a successful textbook currently being used in Italy. The author is a mathematician sensitive to the needs of a general audience. In addition to introducing fundamental ideas in Linear Algebra through a wide variety of interesting examples, the book also discusses topics not usually covered in an elementary text (e.g. the "cost" of operations, generalized inverses, approximate solutions). The challenge is to show why the "everyone" in the title can find Linear Algebra useful and easy to learn. The translation has been prepared by a native English speaking mathematician, Professor Anthony V. Geramita.

Hofstadter’s Law: It always takes longer than you think it will take, even if you take into account Hofstadter’s Law. (Douglas R. Hofstadter) Dear Reader, what you are holding in your hands now is for youabook. But for us, for our families and friends, it has been known as the book over the last three years. Three years of intense work just to ?ll three centimeters of your bookshelf! This amounts to about one centimeter per year, or roughly two-?fths of an inch per year if you are non-metric. Clearly we had ample opportunity to experience the full force of Hofstadter’s Law. Writing a book about Computational Commutative Algebra is not - like computing a Gr¨ obner basis: you need unshakeable faith to believe that the project will ever end; likewise, you must trust in the Noetherianity of polynomial rings to believe that Buchberger’s Algorithm will ever terminate. Naturally, we hope that the ?nal result proves our e?orts worthwhile. This is a book for learning, teaching, reading, and, most of all, enjoying the topic at hand.

Hofstadter’s Law: It always takes longer than you think it will take, even if you take into account Hofstadter’s Law. (Douglas R. Hofstadter) Dear Reader, why did we begin the foreword of this second volume with the same quote as the ?rst? There we wrote that it took three years of intense work just to ?ll three centimeters of your bookshelf. The completion of this volume took four years and it is about four centimeters thick. Thus we have a con?rmed invariant which governs our writing: our velocity is one centimeter per year, after all e?ects due to Hofstadter’s Law have been taken into account. When westartedthisprojectinthelastmillennium,weplannedabookforlearning, teaching, reading and, most of all, enjoying the topic at hand. Surely there is no law which says that a mathematical book has to be dull, boring, dry, or tedious. But how do you make it enjoyable? Our approach has been to ?ll it with amusing quotes, varied jokes, funny word games, ?owery metaphors and occasional literarye?orts. There are two possible drawbacks of this method. Firstly, not everyone has the same sense of humour and not every metaphor works as intended. For instance, it is easy to joke about certain politicians, but what happens if they read this book? And when we wrote of a small boat sailing slowly into the Brazilian sunset, it was pointed out to us that this entails a geographical problem. Secondly, it is very di?cult to write humorously in a foreign language.

Da dove viene questo strano titolo Algebra lineare per tutti? Tutti in che senso? E` luogo comune il fatto che la matematica non sia materia per tutti ed è vero che uno degli ostacoli alla sua diffusione sono spesso i matematici stessi (per fortuna non tutti). Alcuni di essi tendono a sviluppare un linguaggio astruso, difficile, o addirittura talvolta incomprensibile persino per gli esperti di settori limitrofi. E se provassimo a chiedere ad un matematico di professione di estraniarsi per un poco dal suo linguaggio abitudinario e parlare e scrivere in modo più lineare? E se gli chiedessimo addirittura di essere vivace? E perché non esagerare e chiedergli di essere a tratti persino divertente? Lo scopo di questo libro è quello di fornire i primi strumenti matematici relativi ad un capitolo della scienza che si chiama Algebra Lineare. Il testo è stato scritto da un matematico che ha cercato di uscire dal suo personaggio per venire incontro ad un pubblico ampio. La sfida è quella di rendere accessibili a tutti i primi rudimenti di un sapere fondamentale per la scienza e la tecnologia.

Hofstadter’s Law: It always takes longer than you think it will take, even if you take into account Hofstadter’s Law. (Douglas R. Hofstadter) Dear Reader, why did we begin the foreword of this second volume with the same quote as the ?rst? There we wrote that it took three years of intense work just to ?ll three centimeters of your bookshelf. The completion of this volume took four years and it is about four centimeters thick. Thus we have a con?rmed invariant which governs our writing: our velocity is one centimeter per year, after all e?ects due to Hofstadter’s Law have been taken into account. When westartedthisprojectinthelastmillennium,weplannedabookforlearning, teaching, reading and, most of all, enjoying the topic at hand. Surely there is no law which says that a mathematical book has to be dull, boring, dry, or tedious. But how do you make it enjoyable? Our approach has been to ?ll it with amusing quotes, varied jokes, funny word games, ?owery metaphors and occasional literarye?orts. There are two possible drawbacks of this method. Firstly, not everyone has the same sense of humour and not every metaphor works as intended. For instance, it is easy to joke about certain politicians, but what happens if they read this book? And when we wrote of a small boat sailing slowly into the Brazilian sunset, it was pointed out to us that this entails a geographical problem. Secondly, it is very di?cult to write humorously in a foreign language.

Hofstadter’s Law: It always takes longer than you think it will take, even if you take into account Hofstadter’s Law. (Douglas R. Hofstadter) Dear Reader, what you are holding in your hands now is for youabook. But for us, for our families and friends, it has been known as the book over the last three years. Three years of intense work just to ?ll three centimeters of your bookshelf! This amounts to about one centimeter per year, or roughly two-?fths of an inch per year if you are non-metric. Clearly we had ample opportunity to experience the full force of Hofstadter’s Law. Writing a book about Computational Commutative Algebra is not - like computing a Gr¨ obner basis: you need unshakeable faith to believe that the project will ever end; likewise, you must trust in the Noetherianity of polynomial rings to believe that Buchberger’s Algorithm will ever terminate. Naturally, we hope that the ?nal result proves our e?orts worthwhile. This is a book for learning, teaching, reading, and, most of all, enjoying the topic at hand.