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In this paper a new method is developed for obtaining necessary and sufficient conditions which, when satisfied, guarantee that a linear non-autonomous system will be uniformly asymptotically stable. The method is essentially an extension of two earlier theorems on the asymptotic stability of linear autonomous systems; these theorems, developed by the same author, are here extended to non-autonomous systems. Basically, the method consists of ascertaining whether a certain matrix is positive definite. In general, it is shown that this knowledge is necessary and sufficient to determine whether a linear non-autonomous system is exponentially stable and hence uniformly asymptotically stable.