Thread: What does "complex linear" mean?

This question says to regard the set of complex numbers as a vector space over the reals

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

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).

Thanks for answering

so the transformation that sends

to doesn't not work because that's the same thing as

or

Why does complex conjugation work though?

isn't both