Question

prove if the set {e^ax, xe^ax} is linearly independent

Answer 1

To show this is linearly independent, you need to let \[a_1e^{ax}+a_2xe^{ax}=0\]and show that \(a_1\) and \(a_2\) are 0.

Answer 2

In other words, show that \(a_1e^{ax}\neq a_2 xe^{ax}\) for all \(x\).

To begin, you might want to multiply by \(e^{-ax}\).

Answer 3

You could also use the Wronskian determinant: http://en.wikipedia.org/wiki/Wronskian

Answer 4

I highly recommend using the Wronskian.

Answer 5

And Abel's theorem allows you to simplify it even further because if it vanishes at a point t_0 then it is identically zero for all t.

Answer 6

If I remember DiffEQ correctly haha