All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8
Determine all the conjugacy classes of the dihedral group
\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.
Hint.
You may directly compute the conjugates of each element
but we are going to use the following theorem to simplify the […]

If $ab=1$ in a Ring, then $ba=1$ when $a$ or $b$ is Not a Zero Divisor
Let $R$ be a ring with $1\neq 0$. Let $a, b\in R$ such that $ab=1$.
(a) Prove that if $a$ is not a zero divisor, then $ba=1$.
(b) Prove that if $b$ is not a zero divisor, then $ba=1$.
Definition.
An element $x\in R$ is called a zero divisor if there exists a […]

If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable
Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix.
Show that the matrix $A$ is diagonalizable.
Hint.
Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]

Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation
Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\]
for each $X\in V$.
(a) Prove that $T:V\to V$ is a linear transformation.
(b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]

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 […]

Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic
Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.
Hint.
Note that any homomorphism between fields over $\Q$ fixes $\Q$ pointwise.
Proof.
Assume that there is an isomorphism $\phi:\Q(\sqrt{2}) \to \Q(\sqrt{3})$.
Let […]

Find the Formula for the Power of a Matrix
Let
\[A=\begin{bmatrix}
1 & 1 & 1 \\
0 &0 &1 \\
0 & 0 & 1
\end{bmatrix}\]
be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.
Proof.
We first compute several powers of $A$ and guess the general formula.
We […]