I've taken courses on linear algebra before, so I know enough where I can usually get by, but I saw the following question on my homework, and have no idea how to interpret it:
For the next two questions, let:
$$ B_2 = \Biggl\{ \begin{bmatrix}1 & 1\\1 & 3\end{bmatrix}, \begin{bmatrix}1 & 1\\3 & 1\end{bmatrix}, \begin{bmatrix}1 & 3\\1 & 1\end{bmatrix}, \begin{bmatrix}3 & 1\\1 & 1\end{bmatrix} \Biggl\} $$ $$ B_3 = \Biggl\{ \begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix} \Biggl\} $$
Find: $P_{B_3 \leftarrow B_2}$
I have done multiple searches (although I honestly have no idea what to search for at this point) trying to figure out what on Earth "$P_{B_3 \leftarrow B_2}$" could possibly mean in this context, and I have come up with nothing. It looks like it may be referring to polynomial space with "$P$", but that doesn't make any sense given the provided matrices.
Does anyone know what "$P_{B_3 \leftarrow B_2}$" means?
It means "the transition matrix from the (ordered) basis $B_2$ to the (ordered) basis $B_3$". Thus, for this question, it is $$P_{B_3\leftarrow B_2}=\begin{bmatrix} 1&1&1&3\\ 1&1&3&1\\ 1&3&1&1\\ 3&1&1&1\end{bmatrix}$$ It takes a vector in $M_{2×2}(\mathbb R)$ expressed in the basis $B_2$ and returns that same vector in the basis $B_3$.