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

$$c_1 f_1 + \cdots + c_n f_n = 0?$$