Further Linear Algebra: (Springer Undergraduate Mathematics Series)

Most of the introductory courses on linear algebra develop the basic theory of finite­ dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num­ ber of illustrative and worked examples, as well as many exercises that are strategi­ cally placed throughout the text. Solutions to the exercises are also provided. Many applications of linear algebra require careful, and at times rather tedious, calculations by hand. Very often these are subject to error, so the assistance of a com­ puter is welcome. As far as computation in algebra is concerned, there are several packages available. Here we include, in the spirit of a tutorial, a chapter that gives 1 a brief introduction to the use of MAPLE in dealing with numerical and algebraic problems in linear algebra.

Table of Contents:
The story so far.- 1. Inner Product Spaces.- 2. Direct Sums of Subspaces.- 3. Primary Decomposition.- 4. Reduction to Triangular Form.- 5. Reduction to Jordan Form.- 6. Rational and Classical Forms.- 7. Dual Spaces.- 8. Orthogonal Direct Sums.- 9. Bilinear and Quadratic Forms.- 10. Real Normality.- 11. Computer Assistance.- 12. …. but who were they?.- 13. Solutions to the Exercises.

Review :
From the reviews of the first edition: MAA ONLINE "This book will be of interest to anyone who wishes to have a good grasp of linear llgebra and matrix theory. It can also be used as an advanced undergraduate textbook. Although this book does not treat infinite dimensional linear spaces, it provides the reader with a deep understanding of finite dimensional linear spaces. Many aspects of the theory of finite dimensional linear spaces can easily be generalized to the infinite dimensional case. Therefore, this book will also be helpful to those who intend to study infinite dimensional spaces later. " ZENTRALBLATT MATH "!it embodies a beautiful, concise and precise treatment of the subject, with succinct numerical and algebraic worked examples at the right points, and many exercises!This is an excellent textbook which, together with the earlier book, comprises a very nearly complete introduction to linear algebra which not only the undergraduate but also the advanced reader will enjoy studying." "The present book is a contribution of the volume Basic Linear Algebra of the same authors, which appeared in the same Springer series. ! It is a very interesting, accessible book for undergraduates (or teacher professors), and the many examples and exercises should really help the undergraduate to study its contents." (Koen Thas, Bulletin of the Belgian Mathematical Society, Vol. 11 (4), 2004) "The book is a contribution of the authors' Basic Linear Algebra published in the same series. ! Besides numerous well-chosen examples scattered throughout the text, the reader can also enjoy short biographical profiles of twenty one eminent mathematicians associated with the subject." (European Mathematical Society Newsletter, December, 2003) "The present book is a natural sequel to the same authors' successful SUMS volume 'Basic Linear Algebra'. The most advanced topics here take the reader to the very heart of the subject ! . To sum up, this textbook is well suited for self-study or for a one- or two-semester course. Therefore, we warmly recommend it to undergraduate students studying as well as professors teaching Linear Algebra at any level." (Ferenc Moricz, Acta Scientiarum Mathematicarum, Vol. 69, 2003) "Further Linear Algebra is a natural sequel to the authors' highly acclaimed SUMS volume Basis Linear Algebra. ! An introductory chapter recaps the prerequisites (for those readers unfamiliar with the first volume), and a wide range of worked examples and exercises (with solutions) are strategically placed throughout the text to consolidate understanding." (L'Enseignement Mathematique, Vol. 48 (1-2), 2002)