[ 2 posts ] 
 Exercise [18.01] 
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Joined: 04 Feb 2009, 15:52
Posts: 8
Post Exercise [18.01]
Hi again.

I don't know if i understood this excersise. Here is my solution:

C_0:\mathbb{CE}^4 \longrightarrow \mathbb{E}^4
(\omega,\xi,\eta,\zeta) \longrightarrow \frac{1}{2}(\omega+\overline{\omega},\xi+\overline{\xi},\eta+\overline{\eta},\zeta+\overline{\zeta})

C_1:\mathbb{CE}^4 \longrightarrow \mathbb{M}
(\omega,\xi,\eta,\zeta) \longrightarrow \frac{1}{2}(\omega+\overline{\omega},\xi-\overline{\xi},\eta-\overline{\eta},\zeta-\overline{\zeta})

C_2:\mathbb{CE}^4 \longrightarrow \mathbb{\widetilde{M}}
(\omega,\xi,\eta,\zeta) \longrightarrow \frac{1}{2}(\omega-\overline{\omega},\xi+\overline{\xi},\eta+\overline{\eta},\zeta+\overline{\zeta})

Are these mappings the answer to the excersise?
I think these mappings aren't involutory...but I didn't find another answer...
Can anybody clarify this question?

See you soon!!

06 Jul 2009, 20:15

Joined: 26 Mar 2010, 04:39
Posts: 109
Post Re: Exercise [18.01]
I'm afraid your answer is incorrect. Your answer does leave the subspaces invariant, but it is not involutary as you noticed. Instead of isolating the real and imaginary parts as you have done, notice that the complex conjugation operation leaves real numbers unchanged and the negative of complex conjugation operation leaves imaginary numbers unchanged.

You can find the solution here: Exercise 18.1.

Please let me know if you need any more clarification.

08 Jun 2010, 12:35
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