Use the result of Problem 1 and the definition of linear independence to prove directly th...

Use the result of Problem 1 and the definition of linear independence to prove directly that, for any constant r, the functions

f0(x) = erx, f1(x) = xerx, ..., fn(x) = xnerx

are linearly independent on the whole real line.

Problem 1

Generalize the method of Problem 2 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 2

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.)