In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
Show that the matrix $A=\begin{bmatrix}
1 & \alpha\\
0& 1
\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$.
Comment.
Remark that if $A$ is a square […]

Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?
Let $A$ be an $n \times n$ matrix.
Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.
Solution.
The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]

Questions About the Trace of a Matrix
Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.
(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]

The Intersection of Two Subspaces is also a Subspace
Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$.
Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.
Definition (Intersection).
Recall that the intersection $U\cap V$ is the set of elements that are both elements of $U$ […]

Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]

Basic Properties of Characteristic Groups
Definition (automorphism).
An isomorphism from a group $G$ to itself is called an automorphism of $G$.
The set of all automorphism is denoted by $\Aut(G)$.
Definition (characteristic subgroup).
A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]

Solve a System of Linear Equations by Gauss-Jordan Elimination
Solve the following system of linear equations using Gauss-Jordan elimination.
\begin{align*}
6x+8y+6z+3w &=-3 \\
6x-8y+6z-3w &=3\\
8y \,\,\,\,\,\,\,\,\,\,\,- 6w &=6
\end{align*}
We use the following notation.
Elementary row operations.
The […]

The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]