# What does "complex linear" mean?

• Oct 7th 2011, 08:11 PM
Sheld
What does "complex linear" mean?
This question says to regard the set of complex numbers as a vector space $\displaystyle V$ over the reals

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

What does complex linear mean?
• Oct 7th 2011, 08:27 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, 09:41 AM
Sheld
Re: What does "complex linear" mean?

so the transformation that sends

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

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

Why does complex conjugation work though?

isn't $\displaystyle \bar{z}$ both

$\displaystyle \bar{z}*1$

$\displaystyle x*1+-y*i$