Vol. 15, No. 1, 1965

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Unimodular solutions of infinite systems of linear equations

Donald Charles Benson

Vol. 15 (1965), No. 1, 1–11

It is well known that if a series of real numbers n=1an converges, but not absolutely, then for any b, there exists a sequence {xi},xi = ±1, such that n=1anxn = b. In §1, a criterion is given on a system of denumerably many equations of this type, with real coefficients, so that solutions xi = ±1 exist for arbitrary right hand sides. A sequence {xi} such that xi = ±1 will be called unimodular. In §2, there results are extended to finite systems, and it is shown that an infinite system has unimodular solutions for arbitrary right hand sides if and only if every finite subsystem has this property. §3 shows that if a system satisfies the criterion of §1, then, in a certain sense, “almost any” sequence {xi},xi = ±1, “satisfies” the system for any choice of right hand sides. In §4, conditions are given whereby infinite systems can be constructed which satisfy the criterion of §2. It follows, for example, that the system

j=1(1)[j2i] jαx j = bi,i = 1,2,;0 < α 1

has solutions (xi = ±1) for any bi(i = 1,2,). The bi are allowed to be real numbers or ±.

Mathematical Subject Classification
Primary: 40.99
Received: 22 January 1964
Published: 1 March 1965
Donald Charles Benson