change of base for $M_{22}$ I have been stuck on this question for a while now. I can easily do the change of basis matrix if the entries in the basis are vectors as opposed to a matrix. 
Let
$$B_1 = \{\begin{bmatrix} 
1 & 1 \\
1 & -1 
\end{bmatrix}
,\begin{bmatrix} 
0 & 1 \\
1 & 0 
\end{bmatrix}
,\begin{bmatrix} 
0 & -1 \\
1 & 0 
\end{bmatrix}\}
\quad and \quad B_2 = \{\begin{bmatrix} 
1 & 1 \\
0 & -1 
\end{bmatrix}
,\begin{bmatrix} 
1 & 0 \\
1 & -1 
\end{bmatrix}
,\begin{bmatrix} 
0 & 1 \\
0 & 0 
\end{bmatrix}\}$$
be two bases for $span(B_1)$ in $M_{22}$, where the usual left to right ordering is assumed. Find the transition matrix (change of coordinate/change of basis matrix) $P_{B1\rightarrow B2}$

Take the first vector of $B_1$ (in this case, by vector I mean the matrix) and write it as a linear combination of the elements in $B_2$, like this:
$$\begin{pmatrix} 1&1 \\ 1&-1\end{pmatrix}=0\begin{pmatrix} 1&1 \\ 0&-1\end{pmatrix}+1\begin{pmatrix} 1&0 \\ 1&-1\end{pmatrix}+1\begin{pmatrix} 0&1 \\ 0&0\end{pmatrix}$$
the scalars are those that go in the first column of the desired matrix. Repeat the same process with the rest of the elements of $B_1$
$$\begin{pmatrix} 0&1 \\ 1&0\end{pmatrix}=-1\begin{pmatrix} 1&1 \\ 0&-1\end{pmatrix}+1\begin{pmatrix} 1&0 \\ 1&-1\end{pmatrix}+2\begin{pmatrix} 0&1 \\ 0&0\end{pmatrix}$$
$$\begin{pmatrix} 0&-1 \\ 1&0\end{pmatrix}=-1\begin{pmatrix} 1&1 \\ 0&-1\end{pmatrix}+1\begin{pmatrix} 1&0 \\ 1&-1\end{pmatrix}+0\begin{pmatrix} 0&1 \\ 0&0\end{pmatrix}$$
and put the scalars in their respective columns. Thus
$$P=\begin{pmatrix} 0&-1&-1 \\ 1&1&1 \\ 1&2&0\end{pmatrix}$$
