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We obtain the following results. 1) Suppose that and its first derivatives , are continuous functions with values in a normed linear vector space. We define a class of linear functionals and show that if a functional in the class is applied to and vanishes for but does not vanish for , then the vectors are linearly independent for each in the domain of . 2) If now are mean-square continuous random processes such that has a nonvanishing white-noise component, then the random variables , are linearly independent. These results are shown to be related both in formulation and method of solution.