Hi, Everyone:(adsbygoogle = window.adsbygoogle || []).push({});

Just curious about two things:

1) if we are given the complexes as a vector space V over R , so that

z1,..,zn are a basis, I heard there is a "natural way" of turning this

into a vector space of R^{2*n} over R; IIRC , is this how it is done:

{ z1,iz1,z2iz2,...,zn,izn}is a new basis.

Does this mean that , for, say, n=2 , if v1=a+ib and v2=c+id are

a basis, then

{ (a,b,0,0), (-b,a,0,0), (0,0,c,d), (0,0,-d,c)}

Is the associated basis for R^4:=R^{2*2} over R (or over any other

field)?

2) How do we define orientability/orientation of C^n over R (over F):

In the case of a vector space of R over F , we say that two bases

B1 and B2 have the same orientation if the matrix M taking

(the rows/columns of ) B1 to B2 has positive determinant. BUT**

if the basis vectors are complex, M is a complex matrix, and the

determinant may not be real.

3)Given 2: what do we mean when we say every complex vector space

is positively-oriented?

Thanks (sorry if post is too long).

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# Complexes C and C^n as Vector Spaces.

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