"Let $P$ be the change-of-basis matrix from a basis $S$ to a basis S' in a vector space $V$. Then, for any vector $v \in V$, we have P[v]_{S'}=[v]_{S} \text{ and hence, } P^{-1}[v]_{S} = [v]_{S'}
Namely, if we multiply the coordinates of $v$ in the original basis $S$ by $P^{-1}$, we get the coordinates of $v$ in the new basis S'." - Schaum's Outlines: Linear Algebra. 4th Ed.
I am having a lot of difficulty keeping these matrices straight. Could someone please help me understand the reasoning behind (what appears to me as) the counter-intuitive naming of $P$ as the change of basis matrix from $S$ to S'? It seems like $P^{-1}$ is the matrix which actually changes a coordinate vector in terms of the 'old' basis $S$ to a coordinate vector in terms of the 'new' basis S'...
Added:
"Consider a basis $S = \{u_1,u_2,...,u_n\}$ of a vector space $V$ over a field $K$. For any vector $v\in V$, suppose $v = a_1u_1 +a_2u_2+...+a_nu_n$
Then the coordinate vector of $v$ relative to the basis $S$, which we assume to be a column vector (unless otherwise stated or implied), is denoted and defined by $[v]_S = [a_1,a_2,...,a_n]^{T}$. "
"Let $S = \{ u_1,u_2,...,u_n\}$ be a basis of a vector space $V$, and let S'=\{v_1,v_2,...,v_n\} be another basis. (For reference, we will call $S$ the 'old' basis and S' the 'new' basis.) Because $S$ is a basis, each vector in the 'new' basis S' can be written uniquely as a linear combination of the vectors in S; say,
$\begin{array}{c} v_1 = a_{11}u_1 + a_{12}u_2 + \cdots +a_{1n}u_n \\ v_2 = a_{21}u_1 + a_{22}u_2 + \cdots +a_{2n}u_n \\ \cdots \cdots \cdots \\ v_n = a_{n1}u_1 + a_{n2}u_2 + \cdots +a_{nn}u_n \end{array}$
Let $P$ be the transpose of the above matrix of coefficients; that is, let $P = [p_{ij}]$, where $p_{ij} = a_{ij}$. Then $P$ is called the \textit{change-of-basis matrix} from the 'old' basis $S$ to the 'new' basis S'." - Schaum's Outline: Linear Algebra 4th Ed.
I am trying to understand the above definitions with this example:
Basis vectors of $\mathbb{R}^{2}: S= \{u_1,u_2\}=\{(1,-2),(3,-4)\}$ and S' = \{v_1,v_2\}= \{(1,3), (3,8)\} the change of basis matrix from $S$ to S' is $P = \left( \begin{array}{cc} -\frac{13}{2} & -18 \\ \frac{5}{2} & 7 \end{array} \right)$.
My current understanding is the following: normally vectors such as $u_1, u_2$ are written under the assumption of the usual basis that is $u_1 = (1,-2) = e_1 - 2e_2 = [u_1]_E$. So actually $[u_1]_S = (1,0)$ and I guess this would be true in general... But I am not really understanding what effect if any $P$ is supposed to have on the basis vectors themselves (I think I understand the effect on the coordinates relative to a basis). I guess I could calculate a matrix P' which has the effect P'u_1, P'u_2,...,P'u_n = v_1, v_2,..., v_n but would this be anything?