Can exists an endomorphism $f:\Bbb C \rightarrow \Bbb C$ , $f(z)=\Re(z)+\Im(z)$?

Solution 1:

It depends on whether you want it to be an endomorphism over $\mathbb R$ or $\mathbb C$. As you observed it is indeed an endomorphism over $\mathbb R$ (i.e. if you consider $\mathbb C$ as a 2-dimensional real vector space), and can be represented by the matrix you gave. However it is not an endomorphism over $\mathbb C$ because for example $$f(iz) = \Re(iz)+\Im(iz) = -\Im(z)+\Re(z)$$ but this is not the same as $$i f(z) = i(\Re(z)+\Im(z)). $$ Another way to see this immediately is to note that the image of $f$ is $\mathbb R$, which is not a complex vector space. But the image of a linear transformation of vector spaces is always a vector space over the base field.