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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^{i}^{t}

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

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 : Attachment 25614
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.

Re: Show that a set of functions is linearly independent

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

Re: Show that a set of functions is linearly independent

I have solved this but thank you

Re: Show that a set of functions is linearly independent

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