Show that any linear transformation in $\mathbb C$ (treated as a complex vector space) is a multiplication by $\alpha\in\mathbb C$.

Solution: Let $T$ be a linear transformation from $\mathbb C$ to $\mathbb C$. Then $T(1)\in \mathbb C$. Set $\alpha= T(1)$.

We show that $T$ is a multiplication by $\alpha\in\mathbb C$.

For any vector in $\mathbb C$, it is represented by a complex number $x$. As a complex vector, it can also be written as $x=x\cdot 1$. Note that $T$ is linear, we have $$T(x)=T(x\cdot 1)=xT(1)=x\alpha=\alpha x.$$ This shows $T$ is a multiplication by $\alpha\in\mathbb C$.