Q:

How to determin the matrix of a linear map

let $U = \mathbb{C}^2$ and $V = \mathbb{C}^2$, viewed as vector spaces over $\mathbb{C}$. Let f : U → V be the linear map represented by the matrix
$(2i,1+4i)$ 
with respect to the standard bases of $\mathbb{C}^2$ and $\mathbb{C}$.
(The standard bases are $\{(1,0), (0,1)\}$ and $\{1\}$ respectively.)
Now view U and V as vector spaces over $\Bbb{R}$. Determine the matrix of the same map f : U →V as above with respect to the ordered bases B = $\{(1,0),(i,0),(0,1),(0,i)\}$ and C = $\{1,i\}$.
I've already proved that that B and C are bases of U and V, i'm not sure yet how to do the rest. 

A:

Let $\beta=\{(1,0),(0,1)\}$ and $\gamma=\{1\}$, and by the assumption that 
$[f]_{\beta}^\gamma=\begin{pmatrix}2i&1+4i\end{pmatrix}$. Then
\begin{align*}
f\begin{pmatrix}1\\0\end{pmatrix}&=\begin{pmatrix}2i&1+4i\end{pmatrix}
\begin{pmatrix}1\\0\end{pmatrix}=2i=0\cdot1+2\cdot i\\
f\begin{pmatrix}i\\0\end{pmatrix}&=\begin{pmatrix}2i&1+4i\end{pmatrix}
\begin{pmatrix}i\\0\end{pmatrix}=-2=-2\cdot1+0\cdot i\\
f\begin{pmatrix}0\\1\end{pmatrix}&=\begin{pmatrix}2i&1+4i\end{pmatrix}
\begin{pmatrix}0\\1\end{pmatrix}=1+4i=1\cdot1+4\cdot i\\
f\begin{pmatrix}0\\i\end{pmatrix}&=\begin{pmatrix}2i&1+4i\end{pmatrix}
\begin{pmatrix}0\\i\end{pmatrix}=-4+i=-4\cdot1+1\cdot i
\end{align*}
So the matrix representation should be
\begin{align*}
[f]_B^C=\begin{pmatrix}0&-2&1&-4\\2&0&4&1\end{pmatrix}.
\end{align*}

