Basis - Complex Vector Space and Real Vector Space

[azdang]

Homework Statement

Let {e₁,...,eⁿ} be a basis for a complex vector space X. Find a basis for X regarded as a real vector space. What is the dimension of X in either case?

Homework Equations

The Attempt at a Solution

I'm really not sure where to begin with this question.
Are the e_j's something like (i, 0, 0,...), (0,i,0,...)?
If so, could we take {ie₁,...,ie_n} as the basis for X regarded as a real vector space?

Also, I think that the dimension in either case is n. Does anyone know if that is correct?

Thank you very much.

 

[Office_Shredder]

Consider the case when n=1 and try to generalize. Notice that over C:

a*e₁ = (x+iy)e₁ = xe₁ + yie₁

And this is zero if and only if x and y are, so we immediately get that e₁ and ie₁ are linearly independent.

 

[azdang]

Thank you for your response.

I definitely see why that is true, but I am confused as to what role that fact plays into this problem. Do you think you could explain it? Thanks so much.

 

[Dick]

The point is that over C, e1 and i*e1 are not linearly independent. Over R, they are.

 

[Bacle]

A blast from the past:
Sorry, I did a search, and got this post; thought it would be better to followup on
it instead of doing a new post:


Is this considered to be the canonical way of turning an n- complex vector space into
a 2n-real vector space.?. I mean, there are many ways of getting a real basis once
we are given a complex basis, but this one seems to be nice in that the original basis
seems to be somehow "embedded" in the real basis {e1,ie1,...,en,ien}.

Also: is there a construction that allows us to go in the opposite direction, i.e.,
we are given an n-real vector space V_r , and we want to construct a complex
vector space in which V_r is " embedded" , in the sense that if we were to forget/drop
the complex part, we would get V_r back, i.e., if we took the basis {e1,ie1,..,en,ien}
as above, and we ignored the vectors iej , to get the vector space V_r with basis:
{e1,e2,...,en}. Is this the complexification of V_r.?

Thanks.

 

[Dick]

You are really just talking about choices of basis here. {e1,(1+i)e1...en,(1+i)en} is also a 2n vector real basis for the complex space. The 'real' part is still embedded in there. It's all pretty arbitrary, as far as I know.