# Linear algebra - finding a basis knowing transition matrix

I have come across a problem in vector spaces dealing with transition matrices and I am not sure in which direction to go.
So far, everytime I've encountered some transition matrix problem, I've been trying to find one, knowing the two bases.

This time I know the transformation matrix $P$ from basis $A$ to $B$ and I know the basis $A$. My task is to find the basis $B$. How do I proceed when the basis $A$ is defined by three matrices $2x2$ (not vectors) and the transformation matrix $P$ is a $3x3$ matrix?

I was thinking by computing the matrix $Q=P^{-1}_{A\rightarrow B}$, then taking its columns which should be the coordinates of vectors of $B$ with respect to $A$. Knowing the vectors in $A$ I thought I could now compute the ones in $B$ but I can't seem to find a way to do it. Especially since I am dealing with matricec and not vectors.

Than you.

You're wrong: the coordinates in basis $\mathcal A$ of the vectors of basis $\mathcal B$ are the column vectors of $P_{\mathcal A\to\mathcal B}$, not of its inverse.

If $P=\begin{pmatrix}2&1&0\\0&1&2\\3&2&4 \end{pmatrix}$ and $\mathcal B=\{w_1,w_2,w_3\}$, we have, for instance: $$w_1=2v_1+3v_3=2\begin{pmatrix}2&4\\0&2\end{pmatrix}+3\begin{pmatrix}0&1\\3&0\end{pmatrix}=\begin{pmatrix}4&11\\9&4\end{pmatrix}$$ and so on.

• Hmm, okay. So how do I work with it when those vectors are $3x1$ and the matrices forming our basis $A$ are $2x2$?
– tawi
Dec 8, 2015 at 18:29
• I don't understand this point, as you didn't say what is the vector space you work with. Dec 8, 2015 at 18:33
• yes, I completely forgot. The vector space is $V \leq \mathbb Z^{2x2}_5$.
– tawi
Dec 8, 2015 at 18:43
• Do you mean $V$ is a subspace of the space of $2\times 2$ matrices with coefficients in the field $\mathbf Z/5\mathbf Z$? How is $V$ defined? Dec 8, 2015 at 18:48