Generalize the method of Problem to prove directly that the functions
F0(x) = 1, f1(x) = x, f2(x) = x2, ..., fn(x) = x"
are linearly independent on the real line.
Problem
Prove directly that the functions
F1(x) = 1, f2(x) = x, and f3(x) = x2
are linearly independent on the whole real line. (Suggestion. Assume that c1 + c2a + c3a2 = 0. Differentiate this equation twice, and conclude from the equations you get that c2 = c2 = c3 = 0.)
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