Complex numbers/Real and imaginary part/Conjugation/Linearity/Exercise

Prove that the functions

$ℂ⟶ℝ,z⟼\mathrm{Re}\left(z\right),$

and

$ℂ⟶ℝ,z⟼\mathrm{Im}\left(z\right),$

are $ℝ$-linear maps. Prove also that the complex conjugation is $ℝ$-linear, but not $ℂ$-linear. Is the absolute value

$ℂ⟶ℝ,z⟼|z|,$

$ℝ$-linear?

Create a solution