Linear Associative Algebra

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Էջ 119 - A very general form of a vid of inversion is (A:A)± (B:B)± (C:C) ± &c., in which each doubtful sign corresponds to two cases, except that at least one of the signs must be negative. The negative of unity might also be regarded as a symbol of inversion, but cannot take the place of an independent vid. Besides the above vids of inversion, others may be formed by adding to either of them a vid consisting of two different letters, which correspond to two of the one-lettered vids of different signs...
Էջ 119 - Thus (A:A) + (B.:B)-(C:C)+x(A:C)+y(B:C) is a vid of inversion. The new vid which Professor Clifford has introduced into his biquaternions is a vid of inversion. SEMI-INVERSION. A vid of which the square is a vid of inversion, is a vid of semiinversion. A very general form of a vid of semi-inversion is (A: A) ± (B:B) ±^/-l(C:C)± &c.
Էջ 1 - BOOK I* THE LANGUAGE OF ALGEBRA 5. The language of algebra has its alphabet, vocabulary, and grammar. 6. The symbols of algebra are of two kinds: one class represent its fundamental conceptions and may be called its letters, and the other represent the relations or modes of combination of the letters and are called the signs. 7. The alphabet of an algebra consists of its letters; the vocabulary defines its signs and the elementary combinations of its letters; and the grammar gives the rules of composition...
Էջ 117 - But there is no difficulty in reducing them to a linear form, and, indeed, my algebra (e3) is the simplest case of Hankel's alternate numbers ; and in any other case, in which n is the number of the Hankel elements employed, the complete number of vids of the corresponding linear algebra is 2" — 1 . The limited character of the algebras which I have investigated may be regarded as an accident of the mode of discussion. There is, however, a large number of unlimited algebras suggested by the investigations,...
Էջ 4 - В . 20. The sign + is called plus in common algebra and denotes addition. It may be retained with the same name, and the process which it indicates may be called addition. In the simplest cases it expresses a mere mixture, in which * This, of course, supposes that С does not vanish.

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