Systems of Linear Inequalities (Little Mathematics Library) by A. S Solodovnikov
By A. S Solodovnikov
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Extra info for Systems of Linear Inequalities (Little Mathematics Library)
TqBq, where f t - t2~ ... , tq are arbitrary nonnegative numbers. The theorem can be proved in a. few words. ff. ) or (0) (the origin of coordinates), we find that in the case where the system is normal our statement is valid. ). Notice that if all points A 1, A 2 , ... , A 2 ,... , A p> also coincides with 0; then only the addend is left of the sum (14). , Bq coincide with 0, the set (B 1 , B 2 , ... , Bq ) also coincides with 0 and only the augend is left of the sum (14). A2 , t iB, 43 The converse theorem also holds, though with some reservation.
Ff r' the intersection of $' with the j-axis, is a segment with the ends C 1 (0, 1) and C 2(0, 2). :ff is a set of points of the form (Fig. 42) (0, y) + (x, - x) == (x, y - x) where x is arbitrary and y is any number in the interval from I to 2. * Notice that the system (12) (viewed as a system of inequalities in one unknown) is now normal. Indeed. fl. 42 In conclusion we shall briefly discuss a theorem which follows from the results obtained above. e. where everything takes place in the plane) this theorem is not particularly striking and it would be right to regard it as a starting point of the extension to the "n-dimcnsional" case to be studied in Section 7.
As to the first case there is as a matter of fact nothing to be proved in it, for then (B 1 , .. /' Bq + I) is' also the whole of space, Let the second case occur and (8 1 , ... , B) be a convex polyhedral ~one. K with each point of the ray (B q + I)~ and, as shown earlier, any convex 29 polyhedral cone g is either an infinite convex pyramid or one of the sets 1 to 8. Having considered the above union of segments for each of these cases, it is not difficult to make sure (check it for yourself) that either it coincides with the whole of space or is again a convex polyhedral cone.