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

Prove that the functions

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Re} \,{\left(z\right)},$

and

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Im} \,{\left(z\right)},$

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

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \vert {z}\vert ,$

${}\mathbb {R}$-linear?

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