Show that a set of functions is linearly independent

Hello everybody

I have to show that this set of vectors a = (e^-t, e^-it, e^t, e^it ) is linearly independent.

My attempt :

f(x) = k1 * e^-t + k2 * e^-it + k3 * e^t + k4 * ei^t

f '(x) = k1 * -e^{-t +} k2 * -ie^-it + k3 * e^t + k4 * ie^it

f ''(x) = k1 * e^{-t +} k2 * -e^-it + k3 * e^t + k4 * -e^it

f(0) = k1 * 1 + k2 * 1 + k3 * 1+ k4 * 1 = 0

<sup>f '(0) = k1 * -1 + k2 * -i + k3 * 1+ k4 * i = 0

f ''(0) = k1 * 1 + k2 * -1 + k3 * 1+ k4 * -1 = 0

But when I plot this in maple and reduce it I get this :

Attachment 25614

They're linearly dependent but this can't be correct. So I guess I'm doing something wrong but what ?</sup>

I would really appreciate it if someone could help me.

You have four functions so you need to look at the third derivative also so that you will have four equations in four unknowns.

I have solved this but thank you

But I'm wondering if you could help me with something else á la this ?