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1、线性代数发展简介线性代数发展简介中文版 英文版ONON THETHE HISTORYHISTORY OFOF DETERMINANTSDETERMINANTS By G. A. MILLER, University of Illinois The history of determinants is an unusually interesting part of the history of elementary mathematics in view of the fact that it illustrates very clearly some of the difficulties
2、in this history which result from the use of technical terms therein without exhibiting the definite meaning which is to be given to these terms. Many modern writers have based their definitions of a determi- nant on the existence of a square matrix. This was done, for instance, in the widely used W
3、eber-Wellstein Encyklopadie der Elementarmathematik, volume 1, 4th edition, 1922, page 304. From this point of view a determinant does not exist without its square matrix, and, judging from many of the textbooks on elementary mathematics, it is likely that many students consider the square matrix as
4、 an essential part of a determinant, so that the term determinant conveys to them a dual concept composed of a square matrix and a certain polynomial associated therewith. When they speak of the rows and columns of a determinant they naturally are thinking of its matrix and when they speak of the va
5、lue thereof they are naturally thinking of the polynomial implied by the term determinant. When a student who is familiar with no definition of the term determinant except the dual one noted in the preceding paragraph meets with the common statement that the discovery of determinants is usually ascr
6、ibed to G. W. Leibniz, he naturally concludes that a square matrix and a polynomial were associated by G. W. Leibniz in about the same way as they are associated at the present time. This is, however, not the case. In fact G. W. Leibniz associ- ated a polynomial with two square matrices and he deriv
7、ed this polynomial therefrom in a way which differs widely from the one now followed in expanding a determinant. Hence the question arises whether it is desirable to associate the name of G. W. Leibniz with the discovery of determinants. To throw some light on this question it may be desirable to co
8、nsider here the motives which led to some of the early developments which are now commonly as- sociated with the beginnings of the theory of determinants. The three subjects which are commonly associated with the early history of determinants are : The solution of a system of n linear equations in n
9、 unknowns, the elimination of the unknowns from a system of n +1 linear equations in n unknowns, and linear transformations. The first of these subjects is naturally one of the oldest in the history of mathematics and when general methods the polynomials which are now used to exhibit the general sol
10、ution of such a system of equations. Hence some mathematical historians have been inclined to trace the history of determinants to methods used by the ancient Chinese and the ancient Japanese in regard to the solution of a system of linear equa- tions. In view of the great disparity between their me
11、thods and those now com- monly employed it is difficult to exhibit any definite contact between their methods and our modern ideas of determinants. Much closer approaches to such contacts are exhibited by the work of G. W. Leibniz since he employed a notation which is almost equivalent to our double
12、 subscript notation and thereby was able to write the general formulas for the eliminant of a system of linear equations as well as for the solution of such a system. There is, however, a wide difference between the motives involved in finding such general formulas and those relating to the study of
13、 the advantages gained by associating a square matrix and a polynomial as is now being commonly done in the use of determinants. Hence it would appear to be entirely justifiable to say that the work of G. W. Leibniz had no direct connection with determinants, and thus to distinguish sharply between
14、work motivated by the desire to find general rules for obtaining the polynomials involved in solving a system of general linear equations or in eliminating the unknowns from such a system, and the work relating directly to studying the advantages resulting from the association of a square matrix and
15、 a well defined polynomial related thereto. We do not mean to imply that the dual definition of the term determinant, according to which this term implies both a square matrix and a certain poly- nomial associated therewith, is the only one for which there is good authority. In fact, some of the bes
16、t authorities define the term determinant without any reference to the existence of a square matrix. Our object is only to exhibit here some of the advantages resulting from this definition as regards the history of elementary mathematics, and especially as regards the simplification of the history of the subject of determinants itself. In particular, it may be noted here that if this definition had been adopted in the very useful work by Thomas Muir which appeared in four volumes under the
