# Hundred - A First Look at Vectors T W Krner o Trinity Hall...

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A First Look at Vectors T. W. K¨orner Trinity Hall Cambridge If you find this document useful, please send an e-mail to [email protected] with heading ‘Vectors thanks’ saying ‘Thank you. This message needs no re- ply’. Please note that I am still at a very early stage of checking for misprints. I would very much appreciate lists of corrections etc. Since a change in one place may produce a line change somewhere else in the document, it would be helpful if you quoted the date January 5, 2012 when this version was pro- duced and you gave corrections in a form like ‘Page 73 near sesquipedelian replace log 10 by log 20’. I would also like to hear about misplaced exercises or non-standard notation. More general comments including suggestions for further exercises, improved proofs and so on are extremely welcome but I reserve the right to disagree with you. The present documents hangs half way between extended notes and a book. I assert copyright.

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i In general the position as regards all such new calculi is this. — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided that such a cal- culus corresponds to the inmost nature of frequent needs, anyone who mas- ters it thoroughly is able — without the unconscious inspiration which no one can command — to solve the associated problems, even to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. . . . Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less of inventive genius. Gauss Werke, Bd. 8, p. 298. (Quoted by Moritz [12].) For many purposes of physical reasoning, as distinguished from calcula- tion, it is desirable to avoid explicitly introducing . . . Cartesian coordinates, and to fix the mind at once on a point of space instead of its three coor- dinates, and on the magnitude and direction of a force instead of its three components . . . . I am convinced that the introduction of the idea [of vectors] will be of great use to us in the study of all parts of our subject, and espe- cially in electrodynamics where we have to deal with a number of physical quantities, the relations of which to each other can be expressed much more simply by [vectorial equations rather] than by the ordinary equations. Maxwell Treatise on Electricity and Magnetism [9] We [Halmos and Kaplansky] share a love of linear algebra. . . . And we share a philosophy about linear algebra: we think basis-free, we write basis- free, but when the chips are down we close the office door and compute with matrices like fury. Kaplansky in Paul Halmos: Celebrating Fifty Years of Mathematics [5] Marco Polo describes a bridge, stone by stone.

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