\[V\]
with bases \[\{ \mathbf{e}_i \}\]
and \[\{ \mathbf{f}_i \}\]
.The transition matrix from
\[\{ \mathbf{e}_i \}\]
to \[\{ \mathbf{f}_i \}\]
.is the matrix
\[B\]
satisfying \[ \mathbf{e}_i = B \mathbf{f}_j\]
where \[B= \left( \begin{array}{cc} b_{11} & b_{21} \\ b_{12} & b_{22} \end{array} \right) \]
Solving to find this matrix is equivalent to expressing the
\[\{ \mathbf{e}_i \}\]
in terms of the \[\{ \mathbf{f}_i \}\]
.Suppose
\[\{ \mathbf{e}_i \} = \left\{ \begin{pmatrix}1\\0\end{pmatrix} , \begin{pmatrix}0\\1\end{pmatrix} \right\} \: \{ \mathbf{f}_i \} = \left\{ \begin{pmatrix}1\\1\end{pmatrix} , \begin{pmatrix}-1\\0\end{pmatrix} \right\}\]
We solve
\[ \begin{pmatrix}1\\0\end{pmatrix} = b_{11} \begin{pmatrix}1\\1\end{pmatrix} +b_{12} \begin{pmatrix}-1\\0\end{pmatrix}\]
and
\[ \begin{pmatrix}0\\1\end{pmatrix} = b_{21} \begin{pmatrix}1\\1\end{pmatrix} +b_{22} \begin{pmatrix}-1\\0\end{pmatrix}\]
The first of these gives us the simultaneous equations
\[1=b_{11}-b_{12}\]
and \[0=b_{11}\]
Hence
\[b_{11}=0, \: b_{12}=-1\]
The second returns the simultaneous equations\[0=b_{21}-b_{22}, \: 1=b_{21}\]
Hence
\[b_{21}=b_{22}=1\]
.The transition matrix is
\[ \left( \begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array} \right) \]