# What does "complex linear" mean?

• Oct 7th 2011, 09:11 PM
Sheld
What does "complex linear" mean?
This question says to regard the set of complex numbers as a vector space $V$ over the reals

and to find a linear transformation $V$ to $V$ which is not complex linear.

What does complex linear mean?
• Oct 7th 2011, 09:27 PM
Deveno
Re: What does "complex linear" mean?
the complex numbers are also a field, so we have 2 choices for regarding them as a vector space:

1) as a vector space over the complex field, of dimension 1, with basis {1}
2) as a vector space over the real field, of dimension 2, with basis {1, i}.

you are being asked to find a mapping which is linear in the sense of (2), but not in the sense of (1).
• Oct 8th 2011, 10:41 AM
Sheld
Re: What does "complex linear" mean?

so the transformation that sends

$(x,y)$ to $(x,y)$ doesn't not work because that's the same thing as

$(x+iy)*1$ or $x*1+y*i$

Why does complex conjugation work though?

isn't $\bar{z}$ both

$\bar{z}*1$

$x*1+-y*i$