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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.

          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.

$QUESTION I < b m a t r i x > Let N be the fixed matrix Consider the linear map T : M2(ℝ) → M2(ℝ) given by: T(A) = AN - NA. (a) Describe bases for the two spaces (T), and im(T) (b) Let X^* = {e11, e12, e21, e22} be the dual basis to the standard basis X = {e11, e12, e21, e22} for M2(ℝ). Describe bases for the two spaces (T^*), and im(T^*). (c) Determine the rank and nullity of both operators T and T^*. QUESTION 2 (a) Count the number of matrices in M3(𝔽p) of each rank. Explain. (b) Count the number of conjugacy classes in M3(𝔽2). Explain.$

Added by Bryan R.

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