<s>[INST] Find matrix A of linear transformation of two bases. If we have set E = {v1, v2, v3} with 
$v1 = \begin{pmatrix}1\\1\\1\end{pmatrix}$ , 
$v2 = \begin{pmatrix}1\\0\\-2\end{pmatrix}$ , 
$v3 = \begin{pmatrix}1\\1\\0\end{pmatrix}$
and
$L(v1) = \begin{pmatrix}1\\2\\3\\0\end{pmatrix}$ ,
$L(v2) = \begin{pmatrix}3\\0\\1\\0\end{pmatrix}$ ,
$L(v3) = \begin{pmatrix}-1\\-2\\-3\\0\end{pmatrix}$
How do I find a matrix A of L in basis E and standard basis R^4 and a matrix B of L in standard basis R^3 and R^4?
I'm not sure I understand how to find a matrix of L for two bases?
 [/INST] You can write
$LV=U$, where $L_{4 \times 3}$ is the unknowm matrix $V_{3 \times 3}$  is the matricx made from the columns $v_i$ and $U_{4 \times 3}$ is made from the columns $u_i$
$V=\begin{bmatrix}  1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -2 & 0 \end{bmatrix}$ $~~U=\begin{bmatrix} 1 & 3 & -1 \\ 2 & 0 &-2 \\ 3 & 1 & -3 \\  0 & 0 & 0 \end{bmatrix}.$ Then  $L=U V^{-1}= \begin{bmatrix} 7 & -8 & 2 \\ 8 & -10 & 4\\-13 & -16 & 6\\ 0 &  0& 0 \end{bmatrix}$
</s>