A Group Homomorphism is Injective if and only if the Kernel is Trivial
Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.
Definitions/Hint.
We recall several […]

Diagonalize a 2 by 2 Symmetric Matrix
Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
Solution.
The characteristic polynomial $p(t)$ of the matrix $A$ […]

A Ring is Commutative if Whenever $ab=ca$, then $b=c$
Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$.
Then prove that $R$ is a commutative ring.
Proof.
Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$.
Consider the […]

If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix
Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.
Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.
Proof.
Because $A$ has rank $n$, we know that the $n \times n$ […]

If Quotient $G/H$ is Abelian Group and $H < K \triangleleft G$, then $G/K$ is Abelian
Let $H$ and $K$ be normal subgroups of a group $G$.
Suppose that $H < K$ and the quotient group $G/H$ is abelian.
Then prove that $G/K$ is also an abelian group.
Solution.
We will give two proofs.
Hint (The third isomorphism theorem)
Recall the third […]

Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$
Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)
For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of […]

Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?
If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.
Can this action be extended to make $M$ into a $\Q$-module?
Proof.
In general, we cannot extend a $\Z$-module into a $\Q$-module.
We give a counterexample. Let $M=\Zmod{2}$ be the order […]

Find the Formula for the Power of a Matrix Using Linear Recurrence Relation
Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.
Then for each positive integer $n$ find $a_n$ and $b_n$ such that
\[A^{n+1}=a_nA+b_nI,\]
where $I$ is the $2\times 2$ identity matrix.
Solution.
Since $-1, 3$ are eigenvalues of the […]