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 to show that this set of vectors a = (e-t, e-it, et, eit ) is linearly independent.
My attempt :
f(x) = k1 * e-t + k2 * e-it + k3 * et + k4 * eit
f '(x) = k1 * -e-t + k2 * -ie-it + k3 * et + k4 * ieit
f ''(x) = k1 * e-t + k2 * -e-it + k3 * et + k4 * -eit
f(0) = k1 * 1 + k2 * 1 + k3 * 1+ k4 * 1 = 0
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 :
They're linearly dependent but this can't be correct. So I guess I'm doing something wrong but what ?
I would really appreciate it if someone could help me.