<s>[INST] Find the matrix representation of a linear map Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a linear map and consider the matrix representation $$M_{B}^{B} =\begin{pmatrix}
    4 & 6\\
    6 & 3
\end{pmatrix}$$
with respect to the basis $B=\bigg\{ \begin{pmatrix}
    2 \\
    2 
\end{pmatrix} , \begin{pmatrix}
    3\\
    2
\end{pmatrix}\bigg\}$.
Find the matrix representation of $f$ with respect to the canonical basis.
MY ATTEMPT:
Let $b_1=\begin{pmatrix}
    2 \\
    2 
\end{pmatrix}$, $b_2=\begin{pmatrix}
    3\\
    2
\end{pmatrix}$ and $e_j$ the j-th vector of the canonical basis.
Then I have $f(b_1)=\begin{pmatrix}
    4\\
    6
\end{pmatrix} \Longrightarrow f(2e_1 +2e_2)=2 f(e_1) +2f(e_2) =4e_1 + 6e_2$
$f(b_2)=\begin{pmatrix}
    6\\
    3
\end{pmatrix} \Longrightarrow f(3e_1 +2e_2)=3 f(e_1) +2f(e_2) =6e_1 + 3e_2$
Solving this system I obtain that $f(e_1)=2e_1 -3e_2=\begin{pmatrix}
    2\\
    -3
\end{pmatrix}$ and $f(e_2)= 6e_2=\begin{pmatrix}
    0\\
    6
\end{pmatrix}$.
So the matrix rappresentation is $M_{E}^{E}=\begin{pmatrix}
    2 & 0\\
    -3 & 6
\end{pmatrix}$.
BUT my theacher says that the solution is $M_{E}^{E}=\begin{pmatrix}
    13 & 10\\
    0 & 0
\end{pmatrix}$. Where I am wrong?
 [/INST] $$f(2e_1 +2e_2)=2 f(e_1) +2f(e_2) \color{red}{=4e_1 + 6e_2}$$
This is where you are wrong, $4$ and $6$ are coordinates in the basis $B$, so there is one more step to get $e_1$ and $e_2$ into the game. Otherwise your approach is nice.
</s>