## Quotient Group of Abelian Group is Abelian

## Problem 340

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.

Then prove that the quotient group $G/N$ is also an abelian group.

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.

Then prove that the quotient group $G/N$ is also an abelian group.

Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

1

\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}

1 \\

-1

\end{bmatrix}.\]
The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by

\begin{align*}

T(\mathbf{v}_1)=\begin{bmatrix}

2 \\

4 \\

6

\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}

0 \\

8 \\

10

\end{bmatrix}.

\end{align*}

Find the formula of $T(\mathbf{x})$, where

\[\mathbf{x}=\begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2.\]

Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.

**(1)** \[S_1=\left \{\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in the vector space $\R^3$.

**(2)** \[S_2=\left \{\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}\]
in the vector space $\R^3$.

**(3)** \[S_3=\left \{\, \begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}\]
in the vector space $\R^2$.

**(4)** Let $P_4$ be the vector space of all polynomials of degree $4$ or less with real coefficients.

\[S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}\]
in the vector space $P_4$.

**(5)** \[S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}\]
in the vector space $P_4$.

**(6)** Let $M_{2 \times 2}$ be the vector space of all $2\times 2$ real matrices.

\[S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\} \]
in the vector space $M_{2\times 2}$.

**(7)** \[S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\} \]
in the vector space $M_{2\times 2}$.

(*Linear Algebra Exam Problem, the Ohio State University*)

**(8)** Let $C[-1, 1]$ be the vector space of all real continuous functions defined on the interval $[a, b]$.

\[S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\} \]
in the vector space $C[-2, 2]$.

**(9)** \[S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}\]
in the vector space $C[-1, 1]$.

**(10)** Let $C^2[a, b]$ be the vector space of all real-valued functions $f(x)$ defined on $[a, b]$, where $f(x), f'(x)$, and $f^{\prime\prime}(x)$ are continuous on $[a, b]$. Here $f'(x), f^{\prime\prime}(x)$ are the first and second derivative of $f(x)$.

\[S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}\]
in the vector space $C[-1, 1]$.

**(11)** Let $S_{11}$ be the set of real polynomials of degree exactly $k$, where $k \geq 1$ is an integer, in the vector space $P_k$.

**(12)** Let $V$ be a vector space and $W \subset V$ a vector subspace. Define the subset $S_{12}$ to be the **complement** of $W$,

\[ V \setminus W = \{ \mathbf{v} \in V \mid \mathbf{v} \not\in W \}.\]

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation

\[A=AB-BA.\]

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

Add to solve laterA complex square ($n\times n$) matrix $A$ is called **normal** if

\[A^* A=A A^*,\]
where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.

A matrix $A$ is said to be **nilpotent** if there exists a positive integer $k$ such that $A^k$ is the zero matrix.

**(a)** Prove that if $A$ is both normal and nilpotent, then $A$ is the zero matrix.

You may use the fact that every normal matrix is diagonalizable.

**(b)** Give a proof of (a) without referring to eigenvalues and diagonalization.

**(c)** Let $A, B$ be $n\times n$ complex matrices. Prove that if $A$ is normal and $B$ is nilpotent such that $A+B=I$, then $A=I$, where $I$ is the $n\times n$ identity matrix.

Consider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$.

Let $\alpha$ be any real root of $f(x)$.

Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.

Prove that the polynomial

\[f(x)=x^3+9x+6\]
is irreducible over the field of rational numbers $\Q$.

Let $\theta$ be a root of $f(x)$.

Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.

Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.

Prove that the polynomial

\[f(x)=x^n-t\]
in the ring $S[x]$ is irreducible in $S[x]$.

Let $G=\GL(n, \R)$ be the **general linear group** of degree $n$, that is, the group of all $n\times n$ invertible matrices.

Consider the subset of $G$ defined by

\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\]
Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$.

The subgroup $\SL(n,\R)$ is called **special linear group**

The number $\pi$ is defined a s the ratio of a circle’s circumference $C$ to its diameter $d$:

\[\pi=\frac{C}{d}.\]

$\pi$ in decimal starts with 3.14… and never end.

I will show you several beautiful formulas for $\pi$.

Add to solve later 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 representation of $T$ with respect to $B$. Find the determinant of $P$.

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.

Prove the followings.

**(a)** The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

**(b)** Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.

Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then

\[B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}\]
is a basis of $\R^n$.

**(c)** Each vector $\mathbf{u}\in \R^n$ can be expressed as

\[\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}\]
for some vector $\mathbf{v}\in \calN(T)$.

**(a)** Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$.

Consider the subset $F$ of $C[-1, 1]$ defined by

\[F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.\]
Prove or disprove that $F$ is a subspace of $C[-1, 1]$.

**(b)** Let $n$ be a positive integer.

An $n\times n$ matrix $A$ is called **skew-symmetric** if $A^{\trans}=-A$.

Let $M_{n\times n}$ be the vector space over $\R$ of all $n\times n$ real matrices.

Consider the subset $W$ of $M_{n\times n}$ defined by

\[W=\{A\in M_{n\times n} \mid A \text{ is skew-symmetric}\}.\]
Prove or disprove that $W$ is a subspace of $M_{n\times n}$.

Let $A$ be the matrix for a linear transformation $T:\R^n \to \R^n$ with respect to the standard basis of $\R^n$.

We assume that $A$ is idempotent, that is, $A^2=A$.

Then prove that

\[\R^n=\im(T) \oplus \ker(T).\]

Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd.

Add to solve laterLet $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$.

Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism.

Let $T$ be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations.

\begin{align*}

T\left(\, \begin{bmatrix}

1 \\

1 \\

1

\end{bmatrix} \,\right)

=\begin{bmatrix}

1 \\

0 \\

1

\end{bmatrix}, \qquad T\left(\, \begin{bmatrix}

2 \\

3 \\

5

\end{bmatrix} \, \right) =

\begin{bmatrix}

0 \\

2 \\

-1

\end{bmatrix}, \qquad

T \left( \, \begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix} \, \right)=

\begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}.

\end{align*}

Then for any vector

\[\mathbf{x}=\begin{bmatrix}

x \\

y \\

z

\end{bmatrix}\in \R^3,\]
find the formula for $T(\mathbf{x})$.

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.

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

**(a)** Prove that the map $\exp:\R \to \R^{\times}$ defined by

\[\exp(x)=e^x\]
is an injective group homomorphism.

**(b)** Prove that the additive group $\R$ is isomorphic to the multiplicative group

\[\R^{+}=\{x \in \R \mid x > 0\}.\]

Let $V$ be a real vector space of all real sequences

\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\]
Let $U$ be a subspace of $V$ defined by

\[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\]
Let $T$ be the linear transformation from $U$ to $U$ defined by

\[T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). \]

**(a)** Find the eigenvalues and eigenvectors of the linear transformation $T$.

**(b)** Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.