Given three exponential functions e^x , e^2x , e^3x , show they are linearly independent.
Say that for all .
Then taking the derivative, .
Take the derivative again, .
For any value of this gives us a system of equation.
The determinant of this system (for this value of ) is:
Show this determinat is not the zero function. Therefore, it would imply that for some we have a homogenous system with a non-vanishing determinant and hence the trivial solution is the only such solution.
You could, of course, do this without differentiating at all (although that is easier!).
You want to find . Take x= 0 so you have . Take x= 1 so you have . Take x= 2 so you have . You now have three linear equations to solve for a, b, and c. Do that and show that a= b= c= 0 is the only solution.