Calculate the transformationmatrix $D^B_C(\phi)$ given two Vectorspaces $V$ and $W$

Question

$V:= \begin{pmatrix} a&b&c&d\\b&a&d&c\\c&d&a&b\\d&c&b&a\\\end{pmatrix}$ $W:= \begin{pmatrix} u&v&w\\x&y&0\\z&0&0\\\end{pmatrix}$ $\phi: V \rightarrow W, \begin{pmatrix} a&b&c&d\\b&a&d&c\\c&d&a&b\\d&c&b&a\\\end{pmatrix} \mapsto \begin{pmatrix} a-b&a+c&a-d\\b+c&b-d&0\\c+d&0&0\\\end{pmatrix}$

So $D^B_C(\phi)$ is bascially a transformation of the spanvectors of V to W. Now the basis of $V$ are obviously $B: \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}, \begin{pmatrix} 0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{pmatrix}, \begin{pmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\\\end{pmatrix}, \begin{pmatrix} 0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\\end{pmatrix}$ and the basis of $W$ $C: \begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\\\end{pmatrix}, \begin{pmatrix} 0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}, \begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\\\end{pmatrix}, \begin{pmatrix} 0&0&0\\1&0&0\\0&0&0\\\end{pmatrix}, \begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\\\end{pmatrix}, \begin{pmatrix} 0&0&0\\0&0&0\\1&0&0\\\end{pmatrix}$

Now I have no idea how I get the transformation matrix, help is very appreciated

Answer 1

We have $$\phi\begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}=\begin{pmatrix} 1&1&1\\0&0&0\\0&0&0\\\end{pmatrix}$$

$$=1\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\\\end{pmatrix}+1 \begin{pmatrix} 0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}+1 \begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\\\end{pmatrix}$$

$$ +0\begin{pmatrix} 0&0&0\\1&0&0\\0&0&0\\\end{pmatrix}+0 \begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\\\end{pmatrix}+ 0\begin{pmatrix} 0&0&0\\0&0&0\\1&0&0\\\end{pmatrix}.$$

$$\ldots$$ Transposing,

$$D^B_C(\phi)=\begin{bmatrix}{1}&{*}&{*}&*\\{1}&{*}&{*}&*\\{1}&{*}&{*}&*\\{0}&{*}&{*}&*\\{0}&{*}&{*}&*\\{0}&{*}&{*}&*\end{bmatrix}.$$

Answer 2

$\phi(v_1)=1w_1+1w_2+1w_3$ where $v_i$ is the $i$-th basisvector of $B$ and $w_j$ is the $j$-th basisvector of $C$. Hence the first column of the desired matrix is $\begin{pmatrix} 1\\1\\1\\0\\0\\0 \end{pmatrix}$.

Similarly, find $\phi(v_i)$ and write it with respect to $C$ to get the coefficients that make up the $i$-th column of the desired matrix.