<s>[INST] Given linear mapping and bases, determine the transformation matrix and the change of basis 
Given is linear mapping $f:\mathbb{R}^3 \rightarrow \mathbb{R}^3$. Matrix of $f$ in terms of the ordered standard basis $B_0^3 = \left\{\vec{e_1}, \vec{e_2}, \vec{e_3}\right\}$ of $\mathbb{R}^3$ is $\,\,\,\,                          
A_{B_0^3 B_0^3}^{f} = \begin{pmatrix}
4 & 0 & -2\\ 
1 & 3 & -2\\ 
1 & 2 & -1
\end{pmatrix} \in \mathbb{R}^{3 \times 3}$, moreover
  $B=\left\{\vec{b_1}, \vec{b_2}, \vec{b_3}\right\} = \left\{\begin{pmatrix}
2\\ 
2\\ 
3
\end{pmatrix};
\begin{pmatrix}
1\\ 
1\\ 
1
\end{pmatrix};
\begin{pmatrix}
2\\ 
1\\ 
1
\end{pmatrix}\right\}$ is an ordered basis of $\mathbb{R}^3$.
Determine the transformation matrix $A_{BB}^f$ and the change of basis $T_{B_0^3}^{B}$.

I try to calculate the transformation matrix $A_{BB}^f$ first. So I think because we have $A_{B_0^3 B_0^3}^{f} = \begin{pmatrix}
4 & 0 & -2\\ 
1 & 3 & -2\\ 
1 & 2 & -1
\end{pmatrix}$, where $B_0^3$ is standard basis, we have that 
$A^f = \begin{pmatrix}
4 & 0 & -2\\ 
1 & 3 & -2\\ 
1 & 2 & -1
\end{pmatrix}$ because standard basis shouldn't have any changes on it.
But for calculating $A^f_{BB}$, I need to know the mapping, but I just know $\mathbb{R}^3 \rightarrow \mathbb{R}^3$.. I don't know the precise mapping from matrix to another matrix. I need that so I can insert the given basis $B$ and determine the transformation matrix like that. But how is this supposed to work here? :S
And about the change of basis I really have no idea because it requires the previous thing I mentioned : /
 [/INST] The change of basis matrix $T_B^{B_0^3}$ is simply$$\begin{pmatrix}2&1&2\\2&1&1\\3&1&1\end{pmatrix}$$and $T_{B_0^3}^B$ is$$\begin{pmatrix}2&1&2\\2&1&1\\3&1&1\end{pmatrix}^{-1}=\begin{pmatrix}0 & -1 & 1 \\ -1 & 4 & -2 \\ 1 & -1 & 0\end{pmatrix}.$$So$$A_{BB}^f=T^B_{B_0^3}A_{B_0^3B_0^3}^fT^{B_0^3}_B=\begin{pmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{pmatrix}.$$
</s>