Synopsis of Linear Associative Algebra: A Report on Its Natural Development and Results Reached Up to the Present Time (Classic Reprint) - Softcover

Shaw, James Byrnie

Softcover

ISBN 10: 1333421451 ISBN 13: 9781333421458

Verlag: Forgotten Books, 2018

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Excerpt from Synopsis of Linear Associative Algebra: A Report on Its Natural Development and Results Reached Up to the Present Time

The memoir is divided into three parts: General Theory, Particular Sys tems, Applications. Under the General Theory is given the development of the subject from fundamental principles, no use being made of other mathematical disciplines, such as bilinear forms, matrices, continuous groups, and the like.

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Excerpt from Synopsis of Linear Associative Algebra: A Report on Its Natural Development and Results Reached Up to the Present Time

The last chapter of the general theory gives a sketch of the theory of general algebra, placing linear associative algebra in its genetic relations to general linear algebra. Some scant work has been done in this development, particularly along the line of symbolic logic.1 On the philosophical side, which this general treatment leads up to, there have always been two views of complex algebra. The one regards a number in such an algebra as in reality a duplex, triplex, or multiplex of arithmetical numbers or expressions. The socalled units become mere umbrae serving to distinguish the different coordinates. This seems to have been cayley's2 view. It is in essence the view of most writers on the subject. The other regards the number in a linear algebra as a single entity, and multiplex only in that an equality between two such numbers implies n equalities between certain coordinates or functions of the numbers. This was hamilton's3 view, and to a certain extent grass mann's.' The first view seeks to derive all properties from a multiplication table. The second seeks to derive these properties from definitions applying to all numbers of an algebra. The attempt to base all mathematics on arith metic leads to the first view. The attempt to base all mathematics on algebra, or the theory of entities defined by relational identities, leads to the second view. It would seem that the latter would be the more profitable from the standpoint of utility. This has been the case notably in all developments along this line, for example, quaternions and space - analysis in general. Hamilton, and those who have caught his idea since, have endeavored to form expressions for other algebras which will serve the purpose which the scalar, vector, conjugate, etc., do in quaternions, in relieving the system of reference to any unit-system. Such definition of algebra, or of an algebra, is a develop ment in terms of what may be called the fundamental invariant forms ofthe algebra. The characteristic equation of the algebra and its derived equations are of this character, since they are true for all numbers irrespective of the units which define the algebra; or, in other words, these relations are identically the same for all equivalent algebras. The present memoir undertakes to add to the development of this view of the subject.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

Details zur Ausgabe

• Verlag: Forgotten Books
• Erscheinungsdatum: 2018
• Sprache: Englisch
• ISBN 10: 1333421451
• ISBN 13: 9781333421458
• Einband: Tapa blanda
• Anzahl der Seiten: 150
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