Question

examine the following sets are linear independence or dependence: u1=e^ix, u2=e^-ix, u3= sinx.

Answer 1

u1= e^ix = Cos(x) + i Sin(x)
u2= e^-ix= Cos(-x) + i Sin(-x) = Cos(x) - i Sin(x)
u3= Sin(x)
they are linearly dependent since you are able to write
u1 - u2 =2i u3

theorem: a set of vectors is linearly independent if and only if none of the vectors can be written as a linear combination of the others.

Answer 2

thank you