matrix representation of operator

Vector $\vec v\$ in basis E = $[\vec e_1 \vec e_2 \ldots \vec e_n]$

$$\vec v = E \ \begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$

Now, operator acts upon it

$$A(\vec v) = v_1 A(\vec e_1) + v_2 A(\vec e_2) + \ldots + v_n A(\vec e_n) = [A(\vec e_1) A(\vec e_2) \ldots A(\vec e_n)] \ \begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$$

Now, when v is a first standard vector, we get the first column, $A(\vec e_1)$. Everybody says that it is decomposable into sum, like I did for $\vec v$,

$$A(e_1) = a_{11} e_1 + a_{21} e_2 + \dots + a_{n1} e_n = \sum a_{i1} e_i = E \vec a_1$$

This is a matrix E by vector (column) $a_1$ multiplication. It produces another vector $\vec {A(\vec e_1)}$). I just wonder what is the column $a_1$? Which matrix it belongs to? Everybody says that the resulting A($e_1$) column must be the first col of the wanted matrix A. So, $a_1$ cannot be part of A also. What is the matrix of columns $[a_1 a_2 ... a_n]$? How do yo call it? Dear, is it the real operator matrix whereas notorious $[A(\vec e_1) A(\vec e_2) \ldots A(\vec e_n)]$ is just A mixed up with the basis vectors? Is it a wrong matrix A then?

Elaboration My purpose is to understand the matrix, which represents the operator. I follow the Miami tutorial and it seems that they assume that we must be able to determine the elements $a_{ij}$ by acting with operator A on every basis vector $|i\rangle$ and know the coordinates (or components?)

$$\begin{bmatrix}a_{1i}\\a_{2i}\\a_{3i}\end{bmatrix}$$

of the response column vector $|a_i\rangle = A\,|i\rangle$ in the same basis $\begin{bmatrix}|1\rangle |2\rangle |3\rangle \end{bmatrix} = E$:

$$A\,|i\rangle =\begin{bmatrix}|1\rangle |2\rangle |3\rangle \end{bmatrix} \ \begin{bmatrix}a_{1i}\\a_{2i}\\a_{3i}\end{bmatrix} = E\ \begin{bmatrix}a_{1i}\\a_{2i}\\a_{3i}\end{bmatrix}.$$

If we apply A to all basis vectors in series, that is if we apply A to matrix E, we get

$$AE = A \begin{bmatrix}|1\rangle |2\rangle |3\rangle \end{bmatrix} = \begin{bmatrix}|1\rangle |2\rangle |3\rangle \end{bmatrix} \begin{bmatrix}a_{11}&a_{12}&a_{33} \\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{bmatrix} = E \ \begin{bmatrix}a_{11}&a_{12}&a_{33}\\a_{21}&a_{22}&a_{23} \\a_{31}&a_{32}&a_{33} \end{bmatrix}.$$

Multiplying by $E^{-1}$, we get

$$E^{-1}AE = \begin{bmatrix}a_{11}&a_{12}&a_{33}\\a_{21}&a_{22}&a_{23} \\a_{31}&a_{32}&a_{33} \end{bmatrix} = [A]$$

I have just introduced the shortcut $[A] = \begin{bmatrix}a_{11}&a_{12}&a_{33}\\a_{21}&a_{22}&a_{23} \\a_{31}&a_{32}&a_{33} \end{bmatrix}$

Now, any vector $|v\rangle = E\ \begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix} = E\ [v]$ where $[v] = \begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix}$ is the vector of coordinates (or components?) is translated to $|u\rangle = E\,[u] = A \,|v\rangle = A\,E\,[v]$ by the operator A. Therefore, the coordinates of new vector,

$$[u] = E^{-1}AE[v] = [A]\,[v]$$

I have got $E^{-1}AE = [A]$ instead of get $A = [A]$ expected. It seems that Miami tutorial treats $[A]$ as the matrix representation of the operator. My question is about the relationship between matrix of A and obtained matrix $[A]$.

I see that $E^{-1}AE = A$ in case of standard basis, E = I. Should I compute the matrix A in standard basis and then translate it into the sought matrix of the operator A? Which of the matrices, $A$ or $[A] = E^{-1}AE$ is the operator matrix and which is the operator in the standard basis? Why I cannot find tutorial which says that I must use standard basis to look for the matrix of my operator?

Quantum mechanics

How is this approach related with Quantum mechanics, who use orthonormal bases, so that operator O elements are $o_{ij} = \langle i | O | j \rangle\$? I see how similar this expression is to our $E^{-1}AE$. The inner product with $\langle i|$ gives i-th component of $O|j\rangle$ in case of orthonormal $|i\rangle$ and $|j\rangle$, as it is in quantum mechanics. This is not true in the general case of $E^{-1}AE$. I do not understand what is computed with $E^{-1}AE$. If $\langle i | O | j \rangle\$ computes i-th component of O-transformed j-th basis vector then it is a number, the element of sought matrix. But what is $E^{-1}AE$? How do I find the elements of this matrix?