So I was reading about change of basis matrices and I have trouble understanding the following result in my book. It says that given a vector space $V$ with bases $B$ and $C$, $P$ the transition matrix from $B$ to $C$, for any $x\in V$ we have $X=\text{Mat}_B(x)$ and $X'=\text{Mat}_{C}(x).$ Then show that $$X'=PX.$$
Now according to me I would first write the matrix $P$ with subscripts $P_{CB}$ which reminds me that $$P_{CB}=[[b_1]_C,[b_2]_C,...,[b_n]_C]$$ where $B=\{b_1,b_2,...,b_n\}.$ Then it seems natural to me to say that "P takes a vector in basis $B$ and converts it into a vector in basis $C$." In fact, this is what happens in this video tutorial at this time. So why does this formula work and what is natural way to remember it?
Furthermore, writing the change of basis matrix like this also helps me to keep track of basis when working with matrices so for instance if you want a matrix of a linear transformation $T$ with respect to basis $P$ and $Q$ then I would write something like this $$[T]_{QP}=P_{QC}[T]_{CB}P_{BP}.$$ This usually works on some of the problems I have been solving but why is this not working here? Any insights/advice will be much appreciated.
I don't understand quite well your second part; but for the first, I can say this:
You should denote the change of basis matrix from basis $\mathcal B$ to basis $\mathcal C$ as $P_{\mathcal C,\,\mathcal B}$ as it is the matrix of the identity map ($V$ is the vector space): $$\operatorname{id}_{V}:(V,\mathcal B)\longrightarrow (V,\mathcal C),$$ so that, if you denote $X_{\mathcal B}$ and $X_{\mathcal C}$ the coordinates of the same vector in each basis, and similarly $A_{\mathcal B}$, $A_{\mathcal C}$ the matrices of the linear transformation $T$, we have the relations $$X_{\mathcal B}=P_{\mathcal B,\,\mathcal C}X_{\mathcal C},\qquad A_{\mathcal C}= P_{\mathcal C,\,\mathcal B}\,A_{\mathcal B}\,P_{\mathcal B,\,\mathcal C}.$$