Message from discussion Infinite system of linear equations

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From: "Jules" <julianro...@gmail.com>
Newsgroups: sci.math
Subject: Infinite system of linear equations
Date: 12 Nov 2006 22:12:39 -0800
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Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers,
with i, j positive integers.  Suppose also that we have a sequence
{c_i} of reals.  We wish to find a sequence {a_j} of reals so that the
sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i.
This is, in some sense, a collection of countably-many linear equations
in countably-many variables.  Are there any conditions on the
coefficients b_(i, j) that would guarantee existence and/or uniqueness
of a solution {a_j}?  If there were only finitely-many equations and
variables (the same number of each), then one could simply check that
the determinant of the coefficient matrix is non-zero.  Is there any
analog of determinant for an "omega-by-omega" matrix?