The Linear Algebra a Beginning Graduate Student Ought to Know / Edition 2

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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.

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. The book also contains over 1000 exercises, many of which are very challenging.

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Editorial Reviews

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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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Product Details

  • ISBN-13: 9781402054945
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 1/19/2007
  • Edition description: 2nd ed.
  • Edition number: 2
  • Pages: 448
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.91 (d)

Table of Contents

1. Notation and terminology.- 2. Fields.- 3. Vector spaces over a field.- 4. Algebras over a field.- 5. Linear Dependence and Dimension.- 6. Linear Transformations.- 7. The endomorphism algebra of a vector space.- 8. Representation of linear transformations by matrices.- 9. The algebra of square matrices.- 10. Systems of linear equations.- 11.Determinants.- 12. Eigenvalues and eigenvectors.- 13. Krylov subspaces.- 14. The dual space.- 15. Inner product spaces.- 16. Orthogonality.- 17. Selfadjoint endomorphisms.- 18. Unitary and normal endomorphisms.- 19. Moore-Penrose pseudoinverses.- 20. Bilinear transformations and forms.

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