## Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?

## Problem 528

Let $V$ denote the vector space of all real $2\times 2$ matrices.

Suppose that the linear transformation from $V$ to $V$ is given as below.

\[T(A)=\begin{bmatrix}

2 & 3\\

5 & 7

\end{bmatrix}A-A\begin{bmatrix}

2 & 3\\

5 & 7

\end{bmatrix}.\]
Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.