When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear?
I need to know it for my homework but I can't seem to find the definition anywhere.
2026-04-07 06:27:58.1775543278
Real linear tranformation
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The plane of complex numbers is a two dimensional vector space over the real numbers, its standard basis is $1,\,i$.
A map $T:\Bbb C\to\Bbb C$ is real linear, if additive ($T(x+y)=Tx+Ty$) and preserves multiplication by real scalars: $$T(\lambda x)=\lambda\,T(x)$$ For all real number $\lambda\in\Bbb R$.
We could also say $\Bbb C$-linear, that would require this last property for all complex $\lambda$'s as well.