How to use Linear-equations in a sentence
The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.
Linear Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of linear equations.
Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.
Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m.
By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.Advertisement
Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.
By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.
Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.
The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.
Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.Advertisement
The set of linear equations is then solved by means of a matrix inversion.
The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.