Is $T(M)=PMP^{-1}$, where $P=\begin{bmatrix}2&3\\5&7\end{bmatrix}$ linear? If so, how to prove?

Question

If I define $\vec{v}=\begin{bmatrix}a\\b\end{bmatrix}\text{and }\vec{w}=\begin{bmatrix}c\\d\end{bmatrix}$, I end up getting T$({\vec{v}})=\begin{bmatrix}2+3d\\5b+7d\end{bmatrix}\cdot\begin{bmatrix}-7&3\\5&-2\end{bmatrix}$ which is the product of a $2\times1$ vector and $2\times2$ matrix...unless I'm missing something, the product is undefined;

It's easy to show that $T(T^{-1}M)=M$, which would imply that $T$ is an isomorphism if I could prove linearity.

Answer 1

$T(aM+N)=P(aM+N)P^{-1}=aPMP^{-1}+PNP^{-1}=aT(M)+T(N)$