Equivalent bases for Vector Spaces over Complexes.

[WWGD]

Hi:

Given a fin.dim vector space V over R, and two different bases B_V,B_V'

for V , we say that B_V,B'_V are equivalent as bases ( or have the same

orientation) , if there exists a matrix T with TB=B', and DetT>0.

How do we define equivalent bases for vector spaces over the

Complexes.?. If W is a vector space over C, and we are given

bases B_W, B'_W , then the matrix L with LB=B' may not have

a real-valued determinant.

Is there then a way of defining equivalent bases in the second case.?

Thanks.

 

[lavinia]

I think all bases are equivalent in a complex vector space because the the general linear group over the complex numbers is path connected.