# Basis - Complex Vector Space and Real Vector Space

## 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?

## The Attempt at a Solution

I'm really not sure where to begin with this question.
Are the ej's something like (i, 0, 0,...), (0,i,0,...)?
If so, could we take {ie1,....,ien} 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.

## Answers and Replies

Office_Shredder
Staff Emeritus
Gold Member
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.

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

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