Linear Algebra (Classic Reprint)

Straiglit Xjines in A leliraio Expressions 1. The author here desires to say to the reader, that though the present chapter contains nothing new it is yet of unquestioned importance for a clear understanding of the chapters following. 2. If the lines A B, NO, for example, of a geometric figure are in different directions, and if not only their absolute lengths are considered, but their respective directions as well, it is evident that, though the lengths of these lines are equal, it cannot be said AB=NO. 3. By the expression AB =NO, in Linear Algebra and in the science of Quaternions also, it is understood that the length of AB is equal to that of NO, and also that the direction of the line AB is the same as that of NO, that is to say they are either on the same straight line or are parallel to each other in the same direction. But inN umerical Algebra it is the absolute equality of the lengths only of these lines which is understood. 4. In describing a line AB for example, if we say the line AB or simply AB we mean the special line AB which has a determined direction and length. If we write the line AB, or simply AJB, we mean that the lenght alone is considered. Sometimes I shall write N(AB) for AB and N( ), for .I shall also write N(AB), N( )for N(AB)xN(AB), N(a)xN(a), or for AB xAB,ax .I nthese cases by the letterN prefixed to a line will be meant the number of the abstract length of that line. 5. To represent the different lines of a figure with regard to their directions as well as lengths, Greek letters are often employed. For example, if pis put for the line AB, so long as the problem is not changed, by this pis understood the line AB which by supposition has a determined length and direction. 6. It is obvious that the lines AB, BD, having a determined direction and length, the line AD will also, necessarily have a determined direction and length; and(Typographical errors above are due to OCR software and don't occur in the book.)

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