Exercise 1.3.7

Question

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

Jing Huang · Accepted Answer

\begin{proof}
Suppose a linear transformation $T: \mathbb{C} \xRightarrow{} \mathbb{C}$. $T(1)=a+ib$ and then $T(-1)=-T(1)=-a-ib$. Note that $i^2=-1$. Then $T(-1)=T(i^2)=iT(i)$. Thus 
$$ T(i)=\frac{-a-ib}{i}=i(a+ib).$$
So for any $\omega=x+iy \in \mathbb{C}$,
\begin{equation*}
\begin{split}
    T(\omega) & = T(x+iy) = xT(1)+yT(i) \
              & = x(a+ib)+yi(a+ib) \
              & = (x+iy)(a+ib) \
              & = \omega T(1) \
              & = \omega \alpha,
\end{split}
\end{equation*}
and $\alpha = T(1)$.
\end{proof}

Exercise 1.3.7

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Exercise 1.3.7

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