Multilinear Algebra
This book is built around the material on multilinear algebra which in chapters VI to IX of the second edition of Linear Algebra was included but exc1uded from the third edition. It is designed to be a sequel and companion volume to the third edition of L
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Herausgegeben von
]. L. Doob . E. B. Dynkin . E. Heinz . F. Hirzebruch E. Hopf. H. Hopf· W. Maak . S. Mac Lane . W. Magnus D. Mumford· M. M. Postnikov . F. K. Schmidt . K. Stein
Geschäftsführende Herausgeber B. Eckmann und B. L. van der Waerden
Multilinear Algebra W. H. Greub Mathematics Department, University of Toronto
Springer-Verlag Berlin Heidelberg New York 1967
Geschäftsführende Herausgeber:
Prof. Dr. B. Eckmann Eidgenössische Technische Hochschule Zürich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universität Zürich
ISBN 978-3-662-00797-6
ISBN 978-3-662-00795-2 (eBook)
DOI 10.1007/978-3-662-00795-2 All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm andJor microcard) or by other procedure without written permission from Springer-Verlag © by Springer-Verlag Berlin . Heidelberg 1967 Softcover reprint ofthe hardcover 1st edition 1967 Library of Congress Catalog Card Number 67-14017 Title-Nr. 5119
To Rolf Nevanlinna
Preface This book is built around the material on multilinear algebra which was included in chapters VI to IX of the second edition of Linear Algebra but exc1uded from the third edition. It is designed to be a sequel and companion volume to the third edition of Linear Algebra. In fact, the terminology and basic results of that book are frequently used without reference. In particular, the reader should be familiar with chapters I to V and the first part of chapter VI although other sections are occasionally used. The essential difference between the present treatment and that of the second edition lies in the full exploitation of universal properties which eliminates the restrietion to vector spaces of finite dimension. Chapter I contains standard material on multilinear mappings and the tensor product of vector spaces. These results are extended in Chapter 11 to vector spaces with additional structure, such as algebras and differential spaces. The fundamental concept of "tensor product" is used in Chapter 111 to construct the tensor algebra over a given vector space. In the next chapter the link is provided between tensor algebra on the one hand and exterior and symmetrie tensor algebra on the other. Chapter V contains material on exterior algebra which is developed in considerable depth. Exterior algebra techniques are used in the followmg chapter as a powerful tool to obtain matrix-free proofs of many classical theorems on linear transformation. Chapter VII contains the elementary properties of the symmetrie tensor algebra. In particular, the isomorphism between the symmetrie tensor algebra over an n-dimensional vector space and the polynomial algebra in n indeterminates is established. Finally, Chapter VIII treats algebras of multilinear functions over a finite dimensional vector space and shows that th,ey provide examples for tensor, exterior and symmetrie algebras. Many portions of this volume resulted from discussions with
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