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I want to find the basis for a vector space expressed as $$K=\left\{\left[\frac{v_1}{|v_1|},\frac{v_2}{|v_2|},\frac{v_3}{|v_3|}\right]^T \quad|\quad v_1,v_2,v_3 \in \Bbb C\right\} $$

$C$ is complex number set. (Sorry I am still learning the set theories, hope this expression is correct.)

Now I need to find the orthogonal basis of this space. My intention is that every component of the basis is a normalized complex value, i.e. the module of the component is 1 and has a specific phase. For example, the vector is kind of similar to this one, $[e^{j\theta_1},e^{j\theta_2},e^{j\theta_3}]^T$.

So, the question is, how to find the the basis set? Thanks a lot.

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why do you think $K$ is a vector space ? –  mercio Mar 4 '13 at 13:28
You talk of "this space", as if we had the slightest idea what you're talking about...*what vector space* (apparently, a complex one) are you talking about?! –  DonAntonio Mar 4 '13 at 13:28
@Donantonio it's not closed under addition, not closed under scalar multiplication, has no additive neutral, that is pretty much something what is no vector space –  Dominic Michaelis Mar 4 '13 at 13:29
@DominicMichaelis, how can you tell that if we haven't yet been told what the objects of that suposed space are?! –  DonAntonio Mar 4 '13 at 13:31
The OP still hasn't specified any vector space: not set, not operations, no nothing. Of course, not inner product in order to be able to talk about orthogonality... –  DonAntonio Mar 4 '13 at 13:41

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