I'm given a vector space $V = M_{2\times2}(\mathbb{R})$ and a function $T(A) = 2A^t - A$ and told to find the determinant and characteristic polynomial for T.
I found that $T\begin{pmatrix}a&b \\ c&d\end{pmatrix} = \begin{pmatrix}a&2c-b \\ 2b-c & d \end{pmatrix}$, but I'm having a hard time finding a matrix to represent T. Is it possible for $T$ to be a $2\times2$ matrix or would I have to represent $A$ as a $4\times1$ vector $(a,b,c,d)^t$ and make T a $4\times4$ matrix?