Linear independence of exponential functions

I'm honestly a bit baffled by this problem...

Let a_1, ... , a_n = distinct numbers that do not equal 0. show that e^(a_1*t), ... , e^(a_n*t) are linearly independent over the numbers.

It says hint: suppose c_1*e^(a_1*t) + ... + c_n*e^(a_n*t) = 0 for constants c_1, ..., c_n, differentiate n-1 times. The determinant of the coefficients of the system of linear equations should be 0 (Why?)

I tried the hint, but I don't see how I can prove that the determinant of the coefficients have to equal 0.. Help!

Re: Linear independence of exponential functions

Hi and how are you

I have the following function:

Let z be a complex number. Is the functions (n^z) are linearly independent for all n natural. If this is true can you indicate a reference for that.

Thank you in advance.

Re: Linear independence of exponential functions

This follows from the problem at hand. If

Here . For a countable set of functions to be independent, every finite subset of functions must be independent (definition). By the problem above we see that this is true, so the are all independent for natural .

Re: Linear independence of exponential functions

Thank you very much for your answer. Can you suggest a reference such as book or a paper about this topic. It is very imporatnt to me to see the refernce cited in my reaserch.