QUESTION I
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
Let $N$ be the fixed matrix
Consider the linear map $T : M_2(\mathbb{R}) \to M_2(\mathbb{R})$ given by:
$T(A) = AN - NA$.
(a) Describe bases for the two spaces $\ker(T)$, and $\text{im}(T)$
(b) Let $X^* = \{e_{11}, e_{12}, e_{21}, e_{22}\}$ be the dual basis to the standard basis $X = \{e_{11}, e_{12}, e_{21}, e_{22}\}$ for $M_2(\mathbb{R})$.
Describe bases for the two spaces $\ker(T^*)$, and $\text{im}(T^*)$.
(c) Determine the rank and nullity of both operators $T$ and $T^*$.
QUESTION 2
(a) Count the number of matrices in $M_3(\mathbb{F}_p)$ of each rank. Explain.
(b) Count the number of conjugacy classes in $M_3(\mathbb{F}_2)$. Explain.
