Which way to write the transition matrix arrow? On a study guide my professor writes the problem:
Let $B\:=\:\left\{\left(1,-1\right),\left(-2,1\right)\right\}$ and $B'=\left\{\left(-1,1\right),\left(1,2\right)\right\}$ be bases for $\mathbb{R}^2$
And let $A_B=\begin{pmatrix}2 & 1 \\0 & -1\end{pmatrix}$be the matrix for $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$relative to $B$
Find the transition matrix $P_{B\:\rightarrow B'}$.
but in the solutions for the guide she refers to this transition matrix as
$P_{B\:\leftarrow B'}$. Does this mean the same thing?
 A: It is hard to tell without the definitions, but my guess would be that $P_{B\rightarrow B'}$ would be defined to be the inverse matrix of $P_{B\leftarrow B'}$, while on the other hand $P_{B\:\rightarrow B'}=P_{B'\leftarrow B}$. In other words left and right can be interchanged, but it does matter which basis is at the source and which is at the point of the arrow.
Another question is which matrix is meant by $P_{B\rightarrow B'}$ in the first place. Again I have to guess, and I would guess it is $P$ such that $P\cdot B[j]=B'[j]$ where $A[j]$ denotes vector $j$ of the (ordered) basis $A$.
This is an unfortunate convention, which wouldn't make sense in a vector space other than $\Bbb R^n$ (because one cannot multiply abstract vectors by matrices; only their coordinate representations can be operated upon by a matrix), which has the property that $P$ describes the conversion of the coordinates of some$~v$ with respect to the basis $B'$ into the coordinates of$~v$ with respect to$~B$ (notice the reversal of the direction here). But this does seem to be the most common convention anyway.
