Basis of a real vector space with complex vectors

1. Sep 29, 2008

_Andreas

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution

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

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

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

Last edited: Sep 29, 2008
2. Sep 29, 2008

_Andreas

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

Fixed.

Last edited: Sep 29, 2008
3. Sep 29, 2008

HallsofIvy

Staff Emeritus
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. Sep 30, 2008

_Andreas

Yeah, I fixed the code.

Thanks for your help!

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