Difficulties understanding these statements about change of basis I understood more or less what a change of basis matrix is and how I can use it to pass to one coordinate system to another. Basically, a change of basis matrix is a matrix whose columns are the entries of the vectors in the new basis.
For example, if we have the basis $B = \{ \vec{b_1} ... \vec{b_n} \}$, and we have our nice vector $\vec{v}$ represented as a linear combination of these vectors, lets denote this representation as $[\vec{v}]_B$. We want now to represent $\vec{v}$ with respect to a new basis $D = \{ \vec{d_1} ... \vec{d_n} \}$. We can construct our change of basis matrix for $D$ like this: $$C = \left[ \begin{matrix} \vec{d_1} & \vec{d_2} & ... & \vec{d_n}\end{matrix}\right]$$
Now, if we want to find the coordinates of $\vec{v}$ with respect to $D$, we can simply multiply $C^{-1}$ times $[\vec{v}]_B$ (coordinates of vector $\vec{v}$ with respect to the basis $B$): $$C^{-1} \cdot [\vec{v}]_B = [\vec{v}]_D$$ 
If this all correct, I cannot understand what's going on some notes, which I need to fix

Assume we have a vector space $V$. Let us also assume we have basis $B$ for $V$ consisting of the vectors $\vec{b}_1, \vec{b}_2, \dots , \vec{b}_n$ and another basis $D = \{ \vec{d}_1, \vec{d}_2, \dots, \vec{d}_n \}$.

Now basically there is a vector $\vec{v}$ that has some coordinates with respect to $B$ and $D$. And then I also have that the vectors of $D$ represented as a linear combination of the vectors in $B$:
$$
\vec{d}_1 = S_{11}\vec{b}_1+S_{12}\vec{b}_2+\dots +S_{1n}\vec{b}_n\\
. \\
. \\
. \\
\vec{d}_n = S_{n1}\vec{b}_1+S_{n2}\vec{b}_2+\dots +S_{nn}\vec{b}_n
$$
And then I have the following:
$$\left(  \begin{matrix} \vec{d}_1 \\ . \\ . \\ . \\ \vec{d}_n \end{matrix} \right) = S \cdot \left(  \begin{matrix} \vec{b}_1 \\ . \\ . \\ . \\ \vec{b}_n \end{matrix} \right)$$
The problem is that I am not quite understanding what's going on here, and what's the relation between what I know about change of basis. What is $S$ and why do we want this? 
I think I am not fully understanding all the concepts...
Ok, there are the 3 pages about Changes of basis that I need to fix and understand:
Page 1:

Page 2:

Page 3

Sincerely, there are some steps that I really don't get what's going on, mostly in the second and third pages...
 A: I think I understand the root of your uncertainty. The thing you mention at the start of your post only applies to changing the basis on $\mathbb{R}^n$. Only in that context can we just directly take the basis and form a basis matrix.
In the case of abstract vector spaces, we have to study the linear dependence of one basis on another. In the notes you post this is accomplished through the $S^T$ or $S$ matrix. Your notes appear correct to me. Different folks face the coordinate change problems differently, if you look at the video I posted or my notes you'll see my approach looks quite different, but, it's really the same. Since you're where you are I'd recommend you try to understand the $S$ and $S^T$ matrix.
Exercise: take $V = P_2(\mathbb{R})$ that is quadratic polynomials with real coefficients. Take basis $B = \{ x^2,x,1 \}$ and basis $D = \{ 1,x,x^2 \}$. It ought to be fairly easy to find $S$ in this case. Then, take a specific vector $f(x)$ in $V$ to see how $S^T$ changes the coordinate vector for $f(x)$. Then, finally, study some $L: P_2 \rightarrow  P_2$ say $L(f(x)) = f'(x)$. To see what the coordinate change for operators is about. I think this is what you'll need to do. Pick an example and trace through these notes. Once you do that, it'll be easy to understand.
