A Polynomial Approach to Linear Algebra
A Polynomial Approach to Linear Algebra is a text which is heavily biased towards functional methods. In using the shift operator as a central object, it makes linear algebra a perfect introduction to other areas of mathematics, operator theory in particu
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Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Universit`a degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah ´ CNRS, Ecole Polytechnique Endre S¨uli University of Oxford Wojbor A. Woyczynski Case Western Reserve University
Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext.
For further volumes: http://www.springer.com/series/223
Paul A. Fuhrmann
A Polynomial Approach to Linear Algebra Second Edition
123
Paul A. Fuhrmann Ben-Gurion University of the Negev Beer Sheva Israel
ISSN 0172-5939 e-ISSN 2191-6675 ISBN 978-1-4614-0337-1 e-ISBN 978-1-4614-0338-8 DOI 10.1007/978-1-4614-0338-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941877 Mathematics Subject Classification (2010): 15-02, 93-02 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Nilly
Preface
Linear algebra is a well-entrenched mathematical subject that is taught in virtually every undergraduate program, both in the sciences and in engineering. Over the years, many texts have been written on linear algebra, and therefore it is up to the author to justify the presentation of another book in this area to the public. I feel that my jusification for the writing of this book is based on a different choice of material and a different approach to the classical core of linear algebra. The main innovation in it is the emphasis placed on functional models and polynomial algebra as the best vehicle for the analysis of linear transformations and quadratic forms. In pursuing this innova
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