Change of Basis (Transformation Matrix)

Question

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.

Answer 1

The matrix of $T$ in basis $(x_1,x_2)$ has for columns the coordinates of $T(x_1)$ and $T(x_2)$ in basis $(x_1,x_2)$. Thus $$A=\begin{bmatrix}2&1\\1&0\end{bmatrix}\quad\text{(no standard basis in this)}.$$ For the basis $(y_1,y_2)$, use the change of basis matrix from basis $(x_1,x_2)$ to basis $(y_1,y_2)$: its column vectoes are the coordinates of $y_1$ and $y_2$ in basis $(x_1,x_2)$, i.e. $$P=\begin{bmatrix}4&1\\2&-1\end{bmatrix}.$$ The change of basis formula asserts that the matrix of $T$ in the new basis is $$B=P^{-1}AP,\quad\text{so }\quad A=PBP^{-1}.$$ All that remains to do is calculating the inverse of $P$.