Define what it means for two $n \times n$ matrices to be similar to each other. Show that if two $n \times n$ matrices are similar, then the linear transformations they define have isomorphic kernels and images.

If $A$ and $B$ are $n \times n$ real matrices, we define $[A, B]=A B-B A$. Let

$$\begin{aligned} K_{A} &=\left\{X \in M_{n \times n}(\mathbb{R}) \mid[A, X]=0\right\} \\ L_{A} &=\left\{[A, X] \mid X \in M_{n \times n}(\mathbb{R})\right\} \end{aligned}$$

Show that $K_{A}$ and $L_{A}$ are linear subspaces of $M_{n \times n}(\mathbb{R})$. If $A$ and $B$ are similar, show that $K_{A} \cong K_{B}$ and $L_{A} \cong L_{B}$.

Suppose that $A$ is diagonalizable and has characteristic polynomial

$$\left(x-\lambda_{1}\right)^{m_{1}}\left(x-\lambda_{2}\right)^{m_{2}}$$

where $\lambda_{1} \neq \lambda_{2}$. What are $\operatorname{dim} K_{A}$ and $\operatorname{dim} L_{A} ?$