I guys, I'm having a hard time working on the following exercise:
$\begin{array}{l}Let\;A:=\begin{pmatrix}1&1\\2&3\end{pmatrix},\;and\;consider\;the\;linear\;map\\\end{array}$
$$L_A:\mathbb{R}^2\rightarrow\mathbb{R}^2,\;\mathcal x\mapsto A\cdot\mathcal x$$
$\begin{array}{l}with\;respect\;to\;the\;stan dard\;basis\;\begin{pmatrix}1\\0\end{pmatrix},\begin{pmatrix}0\\1\end{pmatrix}\;for\;\mathbb{R}^2.\;Find\;the\;transformed\;matrix\;T\;such\;that\;the\;same\;map\;L_A\;can\;be\;expressed\;as\\\\\end{array}$
$$L_A:\mathbb{R}^2\rightarrow\mathbb{R}^2,\;\mathcal x\mapsto T\cdot\mathcal x$$
$\begin{array}{l}with\;respect\;to\;the\;non-stan dard\;basis\;\begin{pmatrix}1\\1\end{pmatrix},\begin{pmatrix}0\\1\end{pmatrix}\;for\;\mathbb{R}^2.\\\end{array}$
So far, I have I have multiplied the matrix A times each basis, and then I tried to find a matrix T, s.t. T times the non-standard basis is equal to the first linear map. I am not sure if that's the good way to do it.
Can someone help me? Thank you!
EDIT: I have found the matrix $T=\begin{pmatrix}0&1\\-1&3\end{pmatrix}$. With this T if I multiply by the non-standard basis, the result is the same as in the first linear map.