Show that the set consisting of the functions $x, e^x, e^{-x}$ on $\mathbb R$ is linearly independent.
So I have the equation $ax + be^x + ce^{-x} = 0$ and I want to show that this is only satisfied when $a = b = c = 0$
Letting x = 0, $b + c = 0$
Letting x = 1, $a + be + ce^{-1} = 0$
Letting x = -1, $-a + be^{-1} + ce = 0$
Using these equations as columns of a matrix I have
$\begin{bmatrix} 0 & 1 & 1 \\ 1 & e & e^{-1} \\ -1 & e^{-1} & e \end{bmatrix}$
$\begin{bmatrix} 1 & e & e^{-1} \\ 0 & 1 & 1 \\ -1 & e^{-1} & e \end{bmatrix}$
$\begin{bmatrix} 1 & e & e^{-1} \\ 0 & 1 & 1 \\ 0 & e^{-1}+e & e^{-1}+e \end{bmatrix}$
But if I now subtract (e^{-1}+e) times row 2 from row 3 I will get all zeros in the last row meaning linear dependence. So what have I done wrong?