<s>[INST] Finding A Matrix Representation 
Let $T=\begin{pmatrix} 
3x+3y+3z  \\
4x+7y+4z\\
6x+8y+6z 
\end{pmatrix}$ And basis $B=\left\{\begin{pmatrix} 
3  \\
1\\
2
\end{pmatrix},\begin{pmatrix} 
4  \\
2\\
3
\end{pmatrix},\begin{pmatrix} 
2  \\
1\\
2
\end{pmatrix}\right\}$  Find $[T]_B$

Now to find it directly I applied T on the elements of $B$ and wrote the coordinates vectors of the images with respect to basis $B$ and got 
$$\begin{pmatrix} 
-3 & -5 & -3  \\
10 & 15 & 8\\
-4 & -6 & -3
\end{pmatrix}$$
But how can I get the answer using basis change matrices?
I first need to find $[T]_B^{E}$ the transformation form the standard basis to $B$ which is 
$$\begin{pmatrix} 
-5 & -1 & -1  \\
2 & 2 & 1\\
5 & -1 & 0
\end{pmatrix}$$
And multiply it by the matrix of basis change $[I]_B^E$ and $[I]_E^B$?
Meaning $[I]_E^B\cdot[T]^E_B\cdot [I]_B^E$?
 [/INST] Note that 
$$[T]_E=
\begin{pmatrix} 
3 & 3 & 3  \\
4 & 7 & 4\\
6 & 8& 6
\end{pmatrix}$$
and 
$$[M]_{EB}=
\begin{pmatrix} 
3 & 4 & 2  \\
1 & 2 & 1\\
2 & 3& 2
\end{pmatrix}$$
$$v_E=[M]_{EB} \cdot v_B\implies v_B=[M]_{EB}^{-1}\cdot v_E \implies v_B= [M]_{BE}\cdot v_E$$
thus for $w=T(v)$ we have
$$w_E=[T]_{E}\cdot v_E\implies [M]_{EB}\cdot w_B=[T]_{E}\cdot [M]_{EB}\cdot v_B \implies w_B=[M]_{BE}\cdot [T]_{E}\cdot[M]_{EB}\cdot v_B $$
</s>