Representation of Linear Transformation with respect to basis please helppp

Question

Let $A = (1,3) (2,5)$
be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ with respect to the basis $E = (1,0) (0,1)$.that is for a vector (x,y under the basis E, $T(x,y) =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$) What is the representation of this linear transformation with respect to the basis $A$?

Answer 1

You want the matrix $M\prime$ w.r.t. the basis $A$. What you want to do here is chain together the matrices, first applying change of basis from $A$ to $E$, then apply the transformation, and lastly change the basis from $E$ to $A$. You can denote this as $M\prime = C_{A\leftarrow E}MC_{E\leftarrow A}$. $C_{E\leftarrow A}$ is simply the basis vectors of $A$ as column vectors, and $C_{A\leftarrow E} = C_{E\leftarrow A}^{-1}$. So $M\prime = C_{E\leftarrow A}^{-1}MC_{E\leftarrow A}$, found through the multiplication of these matrices. $$ \begin{align*} M\prime &= C_{E\leftarrow A}^{-1}MC_{E\leftarrow A}\\ &= \left[\begin{array}{rr}1 & 2\\3 & 5\end{array}\right]^{-1} \left[\begin{array}{rr}1 & -2\\3 & 0\end{array}\right] \left[\begin{array}{rr}1 & 2\\3 & 5\end{array}\right] \end{align*} $$