examine the following sets for linear independence: u1=eix, u2=e-ix, u3=sinx

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examine the following sets for linear independence: u1=eix, u2=e-ix, u3=sinx

Mathematics
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sorry, but those are no sets.
\[u_{1}=e ^{ix}, u _{2}=e ^{-ix}, u _{3}=sinx\]
still those are only terms with the free variable x

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do you mean \[\{e^{ix}, e^{-ix}, \sin{x}\}\] for an arbitrary x or do you mean \[\{e^{ix} | x \in ℂ \}\] and so on?
an arbitrary
\[c _{1}u _{1}+c _{2}u _{2}+c _{3}u _{3}\] if linear combination is zero c1, c2, c3=0 then its called linear independence.
they dependent: \[\sin x = \frac{e^{ix} - e^{x}}{2i}\]
could you explain how this combination come?
It arises naturally if you define sinus and the exponential function by their power series. But also if you take geometric intuition and define \[e^{a+bi} = e^a(\cos b + i \sin b)\]

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