I am having a hard time in getting the notation right for problems involving a change of basis. So suppose we have a linear operator $T: U\to V$ with $B$ being the basis for $U$ and $D$ being the basis for $V$ then we in my notation I would write $A=[T]_{DB}$ as the matrix of transformation. You take something from the $B$ basis and send it $D$ basis.
Then there is a notation of change of base matrix. Suppose you consider a matrix which represents the change of basis from a basis $X=\{x_1,x_2,...,x_k\}$ to a basis $Y.$ I would denote it as $P_{YX}=[[x_1]_Y [x_2]_Y...[x_{k}]_Y]$ with each vector $x_i$ expressed in terms of $Y.$ So if we want the linear transformation expresses in a new basis say $B'$ and $D'$ then I would write $$[T]_{D'B'}=[Q]_{D'B}[T]_{DB}[P]_{BB'}.$$
The problem comes with the formula representing coordinate change. If $P$ is a matrix representing a change of basis from $B$ to $B'$ then we in my notation we have $$(v)_{B}=[P]_{B'B}(v)_{B'}$$ which is wrong. How do I resolve this conflict?
Let consider $v=T(u)$ with $u\in U$ and $v\in V$, then we can write
$$(v)_{D}=[T]_{DB}(u)_{B}$$
and
$$(u)_{B'}=[P]_{B'B}(u)_{B}\iff (u)_{B}=([P]_{B'B})^{-1}(u)_{B'}\iff (u)_{B}=[P]_{BB'}(u)_{B'}$$
$$(v)_{D'}=[Q]_{D'D}(v)_{D} \iff (v)_{D}=([Q]_{D'D})^{-1}(v)_{D'}\iff (v)_{D}=[Q]_{DD'}(v)_{D'}$$
thus
$$(v)_{D}=[T]_{DB}(u)_{B} \iff [Q]_{DD'}(v)_{D'}=[T]_{DB}[P]_{BB'}(u)_{B'}$$
$$\iff (v)_{D'}= [Q]_{D'D}[T]_{DB}[P]_{BB'} (u)_{B'} \iff (v)_{D'}=[T]_{D'B'}(u)_{B'} $$
with