Basis of a real vector space with complex vectors

  • Thread starter _Andreas
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  • #1
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Homework Statement



Find a basis for [tex]V=\mathbb{C}^1[/tex], where the field is the real numbers.

The Attempt at a Solution



I'd say [tex]\vec{e}_1=(1,0), \vec{e}_2=(i,0)[/tex] is a basis, because it seems to me that [tex] \vec{u}=a+bi \in V[/tex] can be written as

[tex]a(1,0)+b(i,0)=(a,0)+(bi,0)=\mathbf{(a+bi,0)=a+bi+0=a+bi=\vec{u} }[/tex], where a and b are real.

I'm a bit unsure about the bolded part, though. Is it correct?
 
Last edited:

Answers and Replies

  • #2
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Ok, I have no idea why the tex code isn't doing its job.

Fixed.
 
Last edited:
  • #3
HallsofIvy
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tex looks good to me. Since you are doing this over the real numbers, yes, (1, 0) and (i, 0) work fine although I would see no reason to write the "0" and don't think you really need to write this as a pair. "1" and "i" as basis should do.
 
  • #4
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tex looks good to me. Since you are doing this over the real numbers, yes, (1, 0) and (i, 0) work fine although I would see no reason to write the "0" and don't think you really need to write this as a pair. "1" and "i" as basis should do.

Yeah, I fixed the code.

Thanks for your help!
 

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