## anonymous 5 years ago examine the following sets for linear independence: u1=eix, u2=e-ix, u3=sinx

1. nowhereman

sorry, but those are no sets.

2. anonymous

$u_{1}=e ^{ix}, u _{2}=e ^{-ix}, u _{3}=sinx$

3. nowhereman

still those are only terms with the free variable x

4. nowhereman

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?

5. anonymous

an arbitrary

6. anonymous

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

7. nowhereman

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

8. anonymous

could you explain how this combination come?

9. nowhereman

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