Find transformation matrix in terms of basis b and c

Question

Linear transformation: $[x,y]\to[4y-x, 2x, 8x-2y]$ with respect to bases:

$B=[e_1,e_1+e_2]$

$C=[e_1,e_2,e_3]$

Answer 1

Hint

$e_1=[1,0]\mapsto [-1,2,8]=-1e_1+2e_2+8e_3$

$e_2=[1,1]\mapsto [3,2,6]=3e_1+2e_2+6e_3$

Let's call the transformation $f$ then $(f:B,C)=\begin{pmatrix} -1 & 3 \\ 2 & 2\\ 8 & 6 \end{pmatrix}$

(I wrote the coefficients vertically)

Answer 2

Let $E = \{e_1, e_2\}$ be the canonical basis for $\mathbb{R}^2$. Your map, call it $A$, has this matrix with respect to canonical bases for $\mathbb{R}^2$ and $\mathbb{R}^3$:

$$A_{(C,E)} = \begin{bmatrix} -1 & 4 \\ 2 & 0 \\ 8 & -2\end{bmatrix}$$

To get $A_{(C,B)}$, the transformation matrix is the matrix of the identity map with respect to bases $B$ and $E$, namely $I_{(E, B)} = \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$. Now we have

$$A_{(C,B)} = A_{(C,E)}I_{(E,B)} = \begin{bmatrix} -1 & 4 \\ 2 & 0 \\ 8 & -2\end{bmatrix}\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} = \begin{bmatrix} -1 & 3 \\ 2 & 2 \\ 8 & 6\end{bmatrix}$$