Linear Algebras

Linear and Associative Algebras.

It was left to Benjamin Peirce to open up this region. The novelty of quaternions, and their great adaptability to three-dimensional geometry occupied Hamilton for most of his too brief life. Peirce, however, under took the construction of algebras in general which were linear and associative. He studied systems that could be expressed in terms of one, two, three, four, five, and to some extent six independent units. These were such that their squares were not necessarily — I, in fact usually they were not — 1. And he introduced the new type of number called "nilpotent," which is a number such that some power of it is zero. A. L. Cauchy had previously used "clefs" which, though not quite the same in fundamental concept as hyper numbers, were practically so, and such that their squares were zero. It is, however, in the studies of Peirce that we find the explicit development of the idea of hypernumber in general. Since Peirce's time non-associative hypernumbers have been studied occasionally, but this very wide region is as yet practically unexplored.

Non-associative Algebras.

The first example of a non-asso ciative algebra that was studied, and one that is interesting for its applications, is the "octave" algebra of Cayley. This algebra depends essentially upon eight independent hypernumbers, and these may be easily selected in infinitely many ways. The simplest definition is that of Dickson, as follows : Let w be a unit hyper number such that —1, and let the real quaternions be repre sented by Q, R, S, etc. Then in the Cayley octave any number is given by the form 0=Q-1-Rw where Q and R run through the domain of all real quaternions. The definition of the algebra is then given by the defining equations (in which the accent indicates the conjugate of the quaternion) This algebra is not associative, the letters representing qua ternions cannot be transferred without change from one to the other side of w . The essential features are shown better in the multiplication table This algebra has unique division except by zero, on either right hand or left-hand. Any number, as 4, satisfies an equation where Q0--=Q-1-Q' . The norm, or last term of this equation, (QQ'+RR'), is the sum of eight squares. Since the norm of a product of two octave hypernumbers is the product of the norms, it follows that the product of two sums of eight squares each, may be written as the sum of eight squares. It was known in Euler's time that there was a theorem of this form for four squares, and complex numbers furnish such a theorem for two squares. There can be no extension of the theorem, however, to larger numbers of squares. However, there are many algebras in which the product of the norms of two hyper numbers is the norm of the product. This leads to a real general ization of the theorem on the squares to forms other than sums of squares, but such that the product of two such forms is a form of the same kind.

It is shown in the study of linear associative algebras in general that their classification can be made to depend upon that of the nilpotent algebras, and hence the classification of nilpotent alge bras furnishes one of the important problems of the study of algebras. A nilpotent algebra is such that some power of every number vanishes, and there is a maximum power for which any number vanishes. A simple class of nilpotent algebras is furnished by the algebras generated by E„, N units, such that for each such unit we have and for any two different ones An algebra of this kind will have units, such as E1,- .E1E2, • Such an algebra is nilpotent because at most there can be but n different factors in any term, hence every hypernum ber raised to the power n+--i will vanish. This algebra was utilized

by H. Grassmann for the study of geometrical configurations, and the multiplication of this kind he called "outer." The generating units can be affixed to n points, the products of pairs then being affixed to segments of lines, the triples to parallelograms, the quad ruples to parallelepipeds, etc. In fact, Grassmann was led to study hypernumbers and different kinds of multiplication by attempting to make a purely geometric symbolism similar to algebra for repre senting geometric facts. This led to the Ausdehnungslehre, a study of projective geometry by a suitable algebra (see QUATERNIONS). These algebras are not exactly linear, however, inasmuch as the products become new units, and while those products may be called units, they are not homogeneous, some units then repre senting points, some lines, some areas, etc. From the formal side this is not a great obstacle if the number of units is finite. The "geometric algebras," as one might expect, have done little to advance the general theory of linear associative algebra.

Algebras with an infinity of linearly independent units, and con sequently a much greater infinity of hypernumbers, occur in the general study of linear operation. For instance, in connection with the development of functions in series of orthogonal func tions, such as Fourier series, Legendre polynomials, and the like, there is an underlying algebra upon whose structure depends the whole theory of the linear operator and its inversion. These operators occur in differential equations, integral equations, and functional equations in general. The operator in the simplest case can be broken up into a sum of idempotent operators with proper coefficients, as in the integral equations of the Fredholm type. This gives an infinite algebra defined by j,=I, 2 There are other forms of infinite algebras, however, where the units are neither idempotent nor nilpo tent, but all powers of the units exist. Such algebras occur in the underlying structure of linear operational equations that are of the Volterra type. For instance, the operation of integration from o to x, is a Volterra operator, and it has the simplest kind of algebra, the units being all the integral powers of a single unit E. The fundamental functions for such an operator are all powers of x, rational or irrational. Infinite algebras, their structure and their properties, as well as their applications, furnish a very large and almost unexplored region of the subject, of immense im portance for the advance of mathematics.

As mathematics has progressed it has become evident that the world of mathematical objects is an ideal world, of singular beauty, and of intense fascination. And ultimately when these objects have reached their greatest sublimations they become universals of the most subtle character. The study of their com binations and operations upon them may be properly said to be the object of algebra. Such universals have a right to be called hypernumbers, and if their combinations have the property of linearity (that is, the combinations are linear in the coefficients of the factors), they furnish linear algebras. Some of these have been rather extensively studied. For instance, all finite groups, as Galois, linear, or homogeneous, may be looked upon as algebras. From this point of view their elements, frequently called operators, become hypernumbers. The same thing may be said of continuous groups and their elements. The near future will see an extensive development of this part of mathematics.