By I. S. Berezin and N. P. Zhidkov (Auth.)

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**Example text**

X^) we find x^. Then using (x(^\ x£\ . . , χ^) we find x^ and so on. After finding all THE SOLUTION OF SETS OF LINEAR ALGEBRAIC EQUATIONS 57 the x{p we find χ{*\ xf\ . . , in t h e same way until we attain the necessary degree of accuracy. We apply this method t o t h e same set as in t h e last section. The notation of t h e coefficients is not repeated. 367308 Note t h a t we could shorten t h e calculations a n d notation as in t h e case of simple iteration. Above all, several first approximations could be made with fewer decimal places.

The norm of an element χζΗ is a function of its components \\Χ\\ = Ψ(Χ1, X2, . . , xn). (21) The functions φ will differ depending on the definition of the norm, but they all possess certain common properties. || = \c\ \\x\\, (p(cxl9 cx29 . . , cxn)=\c\ φ(χν x2, . . , xn). (22) The function

E. operator A converts each element χζΗ having a single norm into a zero element. This is also true of t h e rest of t h e elements H b y virtue of t h e additivity of operator A. This proves t h a t all t h e properties required of a norm in linear normalized space are fulfilled in respect of norms of linear operators. Thus, the totality of all the linear operators defined in linear normalized space in its turn forms a linear normalized space. 40 COMPUTING METHODS But the operation of multiplication by an operator has still to be defined for sets of linear operators.