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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 :

- Image -

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.

I HAVE SOLVED THIS !

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# Show that a set of functions is linearly independent

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