# 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.