The Linear Algebra a Beginning Graduate Student Ought to Know
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Beschreibung
Linear algebra is a living, active branch of mathematics which is central to almost all other areas of mathematics, both pure and applied, as well as computer science, the physical and social sciences, and engineering. It entails an extensive corpus of theoretical results as well as a large body of computational techniques. Unfortunately, in recent years the content of the linear algebra courses required to complete an undergraduate degree in mathematics has been depleted to the extent that they fail to provide a sufficient theoretical or computational background. Students are not only less able to formulate or even follow mathematical proofs, they are also less able to understand the mathematics of the numerical algorithms they need for applications. Certainly, the material presented in the average undergraduate linear algebra course is insufficient for graduate study. This book is intended to fill this gap by providing enough material "theoretical and computational" to allow the student to work independently or in advanced courses.
The book is intended to be used in one of several possible ways:
(1) as a self-study guide ;
(2) as a textbook for a course in advanced linear algebra, either at the upper-class undergraduate level or at the first-year graduate level ; or
(3) as a reference book.
It is also designed to prepare a student for the linear algebra portion of prelim exams or PhD qualifying exams.
The volume is self-contained to the extent that it does not assume any previous formal knowledge of linear algebra, though the reader is assumed to have been exposed, at least informally, to some basic ideas and techniques, such as the solution of a small system of linear equations over the real numbers. More importantly, it does assume a seriousness of purpose and a modicum of mathematical sophistication on the part of the reader. The book also contains over 1000 exercises, many of which are very challenging.
Deals rigorously with the abstract theory and, at the same time, devotes considerable space to the numerical and computational aspects of linear algebra
Contains a large number of thumbnail portraits of researchers who have contributed to the development of linear algebra as we know it today
The problem set is far more extensive than that usually found in linear algebra books, and the problems are more varied and challenging
Linear algebra is a living, active branch of mathematics. This book deals rigorously with the abstract theory and, at the same time, devotes considerable space to the numerical and computational aspects of linear algebra. It contains a large number of thumbnail portraits of researchers who have contributed to the development of linear algebra as we know it today. The volume is self-contained to the extent that it does not assume any previous formal knowledge of linear algebra, though the reader is assumed to have been exposed, at least informally, to some basic ideas and techniques. More importantly, it does assume a seriousness of purpose and a modicum of mathematical sophistication. The book also contains over 1000 exercises, many of which are very challenging.
Autor*in
Jonathan S. Golan
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From the reviews:
"The book is written in a formal mathematical style, but concepts and theorems are carefully introduced and explained … . the motivated reader will be well rewarded for his perseverance by finding a very complete coverage of linear algebra … . A novel and welcome feature is the inclusion of nearly a hundred small pictures of mathematicians … . This book is a very useful addition to the … set of serious mathematical introductions to linear algebra and can be warmly recommended … ." (Rabe von Randow, Zentralblatt MATH, Vol. 1048 (23), 2004)
"This is an unusual book … . Golan ‘assumes no previous knowledge of linear algebra’. A particular strength of this text is its treatment of linear algebra over finite fields, an important component in cryptography. Another strength is its generality. The exercises, which are interesting, numerous, and of widely differing difficulty, contain examples, counterexamples, and useful insights. They are an important part of the book … . There is something new for everyone here. … is an interesting book to book to leaf through." (Roy Mathias, SIAM Review, Vol. 48 (1), 2006)
"A text that presents the abstract theory of vector spaces over a field and the linear transformations on them. The book is written in the classical pure math style … . There are … many examples and almost a thousand delightfully quirky exercises … . Both the examples and the exercises provide a concreteness … . any good student could pick up a lot of useful knowledge by reading through this book and working as many of the exercises as possible." (David Scott Watkins, Mathematical Reviews, Issue 2006 c)
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