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 $2×2$ (not vectors) and the transformation matrix $P$ is a $3×3$ 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.
Best Answer
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.