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Linear Algebra Thoroughly Explained

Milan Vujicic

Linear Algebra Thoroughly

Explained

Author

Milan Vujicic

(1931–2005)

Editor

Jeffrey Sanderson

Emeritus Professor,

School of Mathematics & Statistics,

University of St Andrews,

St Andrews,

Scotland

ISBN: 978-3-540-74637-9

e-ISBN: 978-3-540-74639-3

Library of Congress Control Number: 2007936399

2008 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,

reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication

or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,

1965, in its current version, and permission for use must always be obtained from Springer. Violations are

liable to prosecution under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,

even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws

and regulations and therefore free for general use.

Cover Design: eStudio Calamar S.L.

Printed on acid-free paper

987654321

springer.com

Foreword

There are a zillion books on linear algebra, yet this one ﬁnds its own unique place

among them. It starts as an introduction at undergraduate level, covers the essen-

tial results at postgraduate level and reaches the full power of the linear algebraic

methods needed by researchers, especially those in various ﬁelds of contemporary

theoretical physics, applied mathematics and engineering. At ﬁrst sight, the title of

the book may seem somewhat pretentious but it faithfully reﬂects its objective and,

indeed, its achievements.

Milan Vujicic started his scientiﬁc carrier in theoretical nuclear physics in which

he relied heavily in his research problems on linear algebraic and group theoretic

methods. Subsequently, he moved to the ﬁeld of group theory itself and its appli-

cations in various topics in physics. In particular, he achieved, together with Fedor

Herbut, important results in the foundations of and distant correlations in quantum

mechanics, where his understanding and skill in linear algebra was precedent. He

was known as an acute and learned mathematical physicist.

At ﬁrst Vujicic taught group theory at graduate level. However, his teaching ca-

reer blossomed when he moved to the Physics Faculty of the University of Belgrade,

and it continued, even after retirement, at the University of Malta, where he taught

linear algebra at the most basic level to teaching diploma students. He continuously

interested himself in the problems of teaching, and with worthy results. Indeed, his

didactic works were outstanding and he was frequently singled out by students,

in their teaching evaluation questionnaires, as a superb teacher of mathematical

physics.

This book is based on lectures that Vujicic gave to both undergraduate and post-

graduate students over a period of several decades. Its guiding principle is to de-

velop the subject rigorously but economically, with minimal prerequisites and with

plenty of geometric intuition. The book offers a practical system of studies with an

abundance of worked examples coordinated in such a way as to permit the diligent

student to progress continuously from the ﬁrst easy lessons to a real mastery of the

subject. Throughout this book, the author has succeeded in maintaining rigour while

giving the reader an intuitive understanding of the subject. He has imbued the book

with the same good sense and helpfulness that characterized his teaching during

Foreword

his lifetime. Sadly, having just completed the book, Milan Vujicic suddenly died in

December 2005.

Having known Milan well, as my thesis advisor, a colleague and a dear friend, I

am certain that he would wish this book to be dedicated to his wife Radmila and his

sons Boris and Andrej for their patience, support and love.

Belgrade, July 2007

Djordje

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