I was reading this article on the change of basis, and when I finally thought I understood the concept, it comes out another inconsistency, just because people write differently from place to place, and everything, even simple things like this, becomes confusing.
In that article, it's roughly written the following (with some my own additions of details and comments).
Suppose we have two arbitrary basis $f = \{f_1, f_2 \}$ and $e = \{ e_1, e_2 \}$ for $\mathbb{R}^2$, and an arbitrary vector $v$.
We know that $$v = c_1 e_1 + c_2 e_2$$ so $$[v]_e = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$$
That is $c_1$ and $c_2$ are the coordinates of $v$ with respect to the basis $e$. (I think therefore that $v$ is represented with respect to the standard basis). $v$ is a linear combinations of the two column vectors of $e$, and that previous expression can be written in matrix-vector product
$$v = E \cdot [v]_e = \begin{pmatrix} e_1 & e_2 \end{pmatrix} [v]_e$$
Similarly, $v$ can also be represented with respect to the basis $f$ as follows
$$v = F \cdot [v]_f = \begin{pmatrix} f_1 & f_2 \end{pmatrix} [v]_f$$
Conclusion:
$$v = E \cdot [v]_e = F \cdot [v]_f$$
(For now, nothing strange for me, assuming that $v$ is in practice the coordinates of a vector with respect to the standard basis).
Since $e$ is a basis, we can represent $f_1$ and $f_2$ as a linear combination of $e_1$ and $e_2$ as follows:
$$f_1 = a e_1 + b e_2 \\ f_2 = c e_1 + d e_2$$
(Well this makes sense for me).
The change of basis matrix from $E$ to $F$ is $$P = \begin{pmatrix} a & c \\ b & d\end{pmatrix}$$ (Note yet sure why, but I'm showing you now).
Note that $$E = \begin{pmatrix} e_1 & e_2 \end{pmatrix} \\ F = \begin{pmatrix} f_1 & f_2\end{pmatrix}$$
Assume $e_1 = \begin{pmatrix} e_{11} \\ e_{21} \end{pmatrix}$, $e_2 = \begin{pmatrix} e_{12} \\ e_{22} \end{pmatrix}$, $f_1 = \begin{pmatrix} f_{11} \\ f_{21} \end{pmatrix}$ and $f_2 = \begin{pmatrix} f_{12} \\ f_{22} \end{pmatrix}$. So, lets re-write $E$ and $F$
$$E = \begin{pmatrix} e_{11} & e_{12} \\ e_{21} & e_{22}\end{pmatrix} \\ F = \begin{pmatrix} f_{11} & f_{12} \\ f_{21} & f_{22}\end{pmatrix}$$
Now, take the following $$F = EP$$ or $$\begin{pmatrix} f_{11} & f_{12} \\ f_{21} & f_{22}\end{pmatrix} = \begin{pmatrix} e_{11} & e_{12} \\ e_{21} & e_{22}\end{pmatrix} * \begin{pmatrix} a & c \\ b & d\end{pmatrix}$$
If we multiply the two matrices on the right we obtain
$$\begin{pmatrix} f_{11} & f_{12} \\ f_{21} & f_{22}\end{pmatrix} = \begin{pmatrix} ae_{11} + be_{12} & ce_{11} + d e_{12} \\ ae_{21} + be_{22} & ce_{21} + d e_{22}\end{pmatrix}$$
Note that $$f_1 = \begin{pmatrix} f_{11} \\ f_{21}\end{pmatrix} = \begin{pmatrix} ae_{11} + be_{12} \\ ae_{21} + be_{22}\end{pmatrix} = \begin{pmatrix} e_{11} & e_{12} \\ e_{21} & e_{22} \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix}$$
which is the same thing as $$f_1 = a e_1 + b e_2$$
The same reasoning can be done for the second column of both matrices.
In other words, I've just shown that $$f_1 = a e_1 + b e_2 \\ f_2 = c e_1 + d e_2$$ is equivalent to $$F = E P$$
Now suppose $v$ has the known coordinates $[v]_e$ in the basis $e$, and $F = EP$, then $$v = E [v]_e = \left(F P^{-1} \right) \cdot [v]_e = F [v]_f$$
Take this part:
$$F P^{-1} \cdot [v]_e = F [v]_f$$
And remove from both sides $F$, we obtain $$P^{-1} \cdot [v]_e = [v]_f$$
In other words, if $P$ changes the basis from $E$ to $F$, then $P^{−1}$ changes the coordinates from $v_e$ to $v_f$, which was my doubt.
Apparently, change of basis is, as the name suggests, really a change of basis, i.e. it changes one basis into the other, i.e. it converts the vectors in one basis to the vectors in the other.
I've just realised that I solved my original problem related to this question, but I still have a few doubts.
In the literature, people usually say that $E$ and $F$ are actually the change of basis and not $P$, but it isn't true.
- Why people keep saying that $E$ and $F$ are the change of basis? They should be called the change of coordinates of a vector represented in a basis $E$ or $F$ (respectively) to the standard basis's representation of that same vector, given of course the representation of the vector with respect to $E$ and respectively $F$ (of course if I understood correctly everything).
