Linear Algebras

LINEAR ALGEBRAS. It might be said with good reason that algebra began with the introduction of negative numbers, and linear algebra with the introduction of the imaginary numbers. These inventions, especially the latter, marked most significant advances of human thought, for through them came the certainty of the independence of the creative ability of the mind from the trammels of natural phenomena. True, one may find this inde pendence also shown in other ways before these, but these are indubitable instances of it. From the fact that natural phenomena did not suggest the negative number nor the imaginary number, we are led to enquire whether any kind of number was so sug gested, and we find that certainly neither fractions nor irrationals were so suggested. Whether we first derived our idea of the in tegers of natural numbers from objects and phenomena of nature or not is a debatable question. However, there is justification for the assertion that all number is a creation of the human mind. This is quite evident for negatives and imaginaries, when we consider the centuries that had to elapse before any use was found for either of these two kinds of number in the study of natural phenomena. Both were for a long time called "fictitious" numbers. They were regarded for the most part as empty symbols, really imaginary, much as the centaur or the dryad were imaginary. They were beautiful fancies. However, the time finally arrived when they took their places along with all other numbers as useful ideas in the study of physics and other sciences. Indeed, the negative number as a debt found a use in accountancy in the 14th and pos sibly 13th century in double-entry book-keeping; and the imag inary is to-day, thanks to the insight of Steinmetz, useful in the practical solution of problems of alternating current transmission lines. In mathematical history the most significant question re garding the imaginary was asked by R. Bombelli, who enquired how it could be possible for the symbolic cube roots of two fictitious expressions to be added together and give the perfectly real number 4. These expressions arise in H. Cardan's solution of

the cubic isx+4, one solution being x=4. This appears in Cardan's formula as : After his time the square root of minus one was treated like any other number.

Progress of Imaginary Numbers.

It is not certain who first created negative numbers, perhaps Diophantos, perhaps Brahma gupta. Cardan in his algebra (Ars. Magna, 1545) mentions imag inary numbers, and Bombelli in his algebra (1572) calls V piI di meno (more than negative). The name "imaginary" was given by Rene Descartes. For a whole century little development occurred, but in the next century the progress was very rapid. By the end of the 18th century the complex number (q.v.) had been so generally recognized that it was thought desirable to furnish the family with a terrain of its own, and Caspar Wessel, of Copen hagen, was the first to attach successfully these numbers to geometry in a plane. His essay (1797) was "On the Analytic Representation of Direction." These investigations were followed by extensive ones by J. R. Argand, C. F. Gauss, A. L. Cauchy and others. The method of affixing complex numbers to directions in a plane is really very simple, as we see it now.

In the figure the complex number a-I-W-1, or as erally written, is attached to the line running from 0 at such an angle that its projection on the initial line is a, and on the per pendicular to the initial line is b. This means that the ratio of such a directed segment OP to a segment of unit length OC on the initial line, is a+bi. From this convention comes a geometric method called "equipollences" which is quite superior for some purposes to ordinary analytic geometry. If a is the resistance of an electric circuit, and b the reactance, ad-bi is the impedance, and the diagram above gives the magnitude of the impedance, and the phase angle. Or, a may be the current in phase with the electro motive force, and b the current at phase with the electro motive force; then a+bi is the complex current ; and by using these complex ex pressions Steinmetz was able to make the laws of Ohm and Kirchoff hold also for alternating currents.