Question

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

Answer 1

sorry, but those are no sets.

Answer 2

\[u_{1}=e ^{ix}, u _{2}=e ^{-ix}, u _{3}=sinx\]

Answer 3

still those are only terms with the free variable x

Answer 4

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?

Answer 5

an arbitrary

Answer 6

\[c _{1}u _{1}+c _{2}u _{2}+c _{3}u _{3}\] if linear combination is zero c1, c2, c3=0 then its called linear independence.

Answer 7

they dependent: \[\sin x = \frac{e^{ix} - e^{x}}{2i}\]

Answer 8

could you explain how this combination come?

Answer 9

It arises naturally if you define sinus and the exponential function by their power series. But also if you take geometric intuition and define \[e^{a+bi} = e^a(\cos b + i \sin b)\]