On finding matrices of linear transformations I'm trying to understand the meaning behind matrix of a linear transformation. To illustrate what I don't get consider the example below.
Suppose $t: \mathbb{R}^2 \rightarrow \mathbb{R}^3 $ is a linear transformation given by
$$(x,y) \mapsto (2x, 3x+y, y)$$
Suppose also that the domain basis is $E = \{(1,1), (1,0) \}$ and the codomain basis is $F= \{(1,1,1), (0,1,1),(0,0,1) \}$. Then in order to find the matrix of this transformation all we have to do is to find the transformation of the basis vectors, so in this case:
$$t(1,1) = (2,4,1) $$ $$t(1,0) = (2,3,0)$$
But then we need to convert these transformations in the codomain basis and we have:
$$(2,4,1) = (2,2,-3)_F$$
$$(2,3,0) = (2,1,-3)_F$$
From here we deduce that the matrix of the linear transformation is given by
$$\begin{bmatrix}
2 & 2 \\ 2 & 1 \\ -3 & -3 
\end{bmatrix}$$
Now I understand how to do computations like the one above. However, I don't understand what does the intermediate  transformation "basis vectors" represent (i.e. what is the meaning of $t(1,1) = (2,4,1)$)? What do these vectors represent in the codomain? And how does converting these vectors to their representation in the codomain basis give the correct columns for the matrix we are looking for? Why do we need to convert in the first place?
More concisely given $\vec{v} \in V$ the matrix $A$ of the transformation satisifies $[T(\vec{v})]_F = A[\vec{v}]_E$, but what does $T(\vec{v})$ represent (without conversion to basis $F$)?
 A: As a general case, where you have different bases in both the domain and the codomain, there has to be something in between the raw vectors and the transformation matrix in order to change the vectors you want to transform from their original basis to the basis in which you have defined your linear transformation. Your linear transformation will only correctly transform vectors that are expressed in its same basis.
In order to do this in a simple manner, what we use is a change of basis matrix. The columns of the change of basis matrices are the vectors of the original basis expressed in coordinates of the basis of the transformation. These change of basis matrices are sometimes denoted as $M(B_2,B_1)$, where $B_1$ is the original basis in which the vectors are expressed (input), and $B_2$ is the basis of the linear transformation (output). This means that the "new" change of basis matrix $A(B_2,B_1)$ will be:
$$A(B_2,B_1)=A(B_2,B_2)\cdot M(B_2,B_1)$$
Where $A(B_2,B_2)$ is your original change of basis matrix that takes vectors from the basis $B_2$ and outputs transformed vectors also in $B_2$.
Note that here the output vectors will be in the basis $B_2$. If you want them back in the basis $B_1$, then you need another change of basis matrix to take those vectors back into $B_1$:
$$A(B_1,B_1)=M(B_1,B_2)\cdot A(B_2,B_2)\cdot M(B_2,B_1)$$
$M(B_1,B_2)$ can be computed easily as the inverse of $M(B_2,B_1)$, as it is doing the inverse operation.
I hope this helps to understand the use of change of basis matrices for linear transformations.
