Linear independence of the set of exponential functions

[Monocles]

Homework Statement

For each n \in \mathbb{N}, let f_n(x) = e^{nx} for x \in \mathbb{R}. Prove that f_1, ... , f_n are linearly independent vectors in {\cal F}(\mathbb{R}, \mathbb{R})

Homework Equations

The Attempt at a Solution

I know that the simple way to prove this for n=2 would be by setting x to 0 and 1 and showing that c_1 and c_2 must be 0 with two simultaneous equations. However I don't know how to generalize that to an arbitrary n. I considered making a generalized Wronskian, but I think that would get sloppy and confusing very quickly.

 

[morphism]

What happens when you differentiate both sides of

c_1 f_1 + \cdots + c_n f_n = 0?