I am having some confusion regarding change of basis, and am unsure how to execute the following question:
If $T\in \text{End}(V)$ such that $T(x_1)=2x_1+x_2$ and $T(x_2)=x_1$, and $y_1=4x_1+2x_2$ and $y_2=x_1-x_2$, determine the matrix $T$ with respect to the basis $\{x_1,x_2\}$ and with respect to the new basis $\{y_1,y_2\}$. Furthermore, it is possible to find an invertible matrix $P$ such that $P^{-1}AP=B$, where $A$ is the matrix transformation with respect to the basis $\{x_1,x_2\}$ and $B$ is the matrix transformation with respect to the basis $\{y_1,y_2\}$.
For the first part, I know I need to find some matrix $D=C^{-1}AC$, such that $A$ is the transformation matrix with respect to the standard basis, and $C$ is the change of basis matrix, but I am unsure how to construct $C$ and thus $C^{-1}$. The transformation matrix for $T$ is:
$$A=\begin{bmatrix} 2 & 1\\ 1 & 0\\ \end{bmatrix} $$
I apologize regarding the simplicity of the question and would appreciate any help.