Let f1(x),..., fn(x) be scalar functions. Suppose that every set of sample points x1, . . .

Question:

Let f1(x),..., fn(x) be scalar functions. Suppose that every set of sample points x1, . . . ,xm ∈
R, for all finite m ≥ 1, leads to linearly dependent sample vectors f1.......fn ∈ Rm. Prove
that f1(x),..., fn(x) are linearly dependent functions.