The Transition Matrix

Suppose we have a vector space  
\[V\]
  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  
\[ \left( \begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array} \right) \]