Complex Conjugation is not Linear Mapping

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Theorem

Let $\overline \cdot: \C \to \C: z \mapsto \overline z$ be the complex conjugation over the field of complex numbers.


Then complex conjugation is not a linear mapping.


Proof

\(\ds \overline {i \cdot 1}\) \(=\) \(\ds \overline i \cdot \overline 1\) Product of Complex Conjugates
\(\ds \) \(=\) \(\ds - i \cdot \overline 1\) Definition of Complex Conjugate
\(\ds \) \(\ne\) \(\ds i \cdot \overline 1\)

By definition, it is not a linear mapping.

$\blacksquare$


Sources