On the Teaching of Linear Algebra
To a large extent, it lies, no doubt, in what is presented in this work under the title of ‘meta lever‘, a method which it is certainly interesting to develop and further refine. There exists in mathematics courses a strange prudery which forbids one to a
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Mathematics Education Library VOLUME 23
Managing Editor A.J. Bishop, Monash University, Melbourne, Australia
Editorial Board H. Bauersfeld, Bielefeld, Germany J.P. Becker, Illinois, U.S.A. G. Leder, Melbourne, Australia O.Skovsmose, Aalborg, Denmark A. Sfard, Haifa, Israel S. Turnau, Krakow, Poland
The titles published in this series are listed at the end of this volume.
ON THE TEACHING OF LINEAR ALGEBRA
Edited by
JEAN-LUC DORIER Laboratoire Leibniz, Grenoble, France
KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW
eBook ISBN: Print ISBN:
0-306-47224-4 0-792-36539-9
©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2000 Kluwer Academic Publishers Dordrecht All rights reserved
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TABLE OF CONTENTS
FOREWORD TO THE ENGLISH EDITION...........................................xiii PREFACE ............................................................................................................ xv INTRODUCTION............................................................................................... xix
PART I. EPISTEMOLOGICAL ANALYSIS OF THE GENESIS OF THE THEORY OF VECTOR SPACES ..................................... 1 J.-L. Dorier 1. INTRODUCTION........... ................................................................................ 3 2. ANALYTICAL AND GEOMETRICAL ORIGINS.............................................. 6 2.1. Systems of Numerical Linear Equations: First Local Theory of Linearity ...... 6 2.1.1. Euler and the Dependence of Equations....................................................... 6 2.1.2. Cramer and the Birth of the Theory of Determinants.................................... 8 2.1.3. The Concept of Rank ....................................................................................... 9 2.1.4. Georg Ferdinand Frobenius......................................................................... 10 2.1.5. Further Development.................................................................................... 11 2.2. Geometry and Linear Algebra: a Two-Century-Long Process of Complex Reciprocal Exchanges.............................................................................. 11 2.2.1. Analytic Geometry.................................................................................... .13 2.2.2. Leibniz’s Criticism..................................................................................... .13 2.2.3. Geometrical Representation of Imaginary Quantities ............................... 14 2.2.4. Möbius’s Barycentric Calculus ......................................................................... 15 2.2.5. Bellavitis’s Calulus of Equipollences...............................................
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