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 $\displaystyle n > 1$

$\displaystyle n^z = e^{z\log n}$

Here $\displaystyle a_n = \log n$. 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 $\displaystyle n^z$ are all independent for natural $\displaystyle n>1$.

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.