## The Group of Rational Numbers is Not Finitely Generated

## Problem 461

**(a)** Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.

**(b)** Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.

## Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic

## Problem 460

Let $\Q=(\Q, +)$ be the additive group of rational numbers.

**(a)** Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.

**(b)** Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.

## Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable

## Problem 459

Let

\[A=\begin{bmatrix}

1-a & a\\

-a& 1+a

\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is a complex number.

Determine the values of $a$ such that the matrix $A$ is diagonalizable.

(*Nagoya University, Linear Algebra Exam Problem*)

## Prove that a Group of Order 217 is Cyclic and Find the Number of Generators

## Problem 458

Let $G$ be a finite group of order $217$.

**(a)** Prove that $G$ is a cyclic group.

**(b)** Determine the number of generators of the group $G$.

## A Matrix Equation of a Symmetric Matrix and the Limit of its Solution

## Problem 457

Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.

**(a)** Prove that for sufficiently small positive real $\epsilon$, the equation

\[A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}\]
has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.

**(b)** Evaluate

\[\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)\]
in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.

(*University of California, Berkeley, Linear Algebra Qualifying Exam*)

## Diagonalize the 3 by 3 Matrix if it is Diagonalizable

## Problem 456

Determine whether the matrix

\[A=\begin{bmatrix}

0 & 1 & 0 \\

-1 &0 &0 \\

0 & 0 & 2

\end{bmatrix}\]
is diagonalizable.

If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

Add to solve later## The Order of a Conjugacy Class Divides the Order of the Group

## Problem 455

Let $G$ be a finite group.

The **centralizer** of an element $a$ of $G$ is defined to be

\[C_G(a)=\{g\in G \mid ga=ag\}.\]

A **conjugacy class** is a set of the form

\[\Cl(a)=\{bab^{-1} \mid b\in G\}\]
for some $a\in G$.

**(a)** Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

**(b)** Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

## All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$

## Problem 454

Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.

Add to solve later## Differentiating Linear Transformation is Nilpotent

## Problem 453

Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.

Consider the differentiation linear transformation $T: P_n\to P_n$ defined by

\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]

**(a)** Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.

**(b)** Compute $A^3$, where $A$ is the matrix obtained in part (a).

**(c)** If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?

**(d)** Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$.

Prove that without any calculation that the matrix $A$ is nilpotent.

## Eigenvalues of Similarity Transformations

## Problem 452

Let $A$ be an $n\times n$ complex matrix.

Let $S$ be an invertible matrix.

**(a)** If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.

**(b)** If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.

**(c)** Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.

## Inequality about Eigenvalue of a Real Symmetric Matrix

## Problem 451

Let $A$ be an $n\times n$ real symmetric matrix.

Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality

\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]

## Null Space, Nullity, Range, Rank of a Projection Linear Transformation

## Problem 450

Let $\mathbf{u}=\begin{bmatrix}

1 \\

1 \\

0

\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation

\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]

**(a)** Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.

**(b)** Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.

**(c)** Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.

**(d)** Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.

**(e)** Let

\[B=\left\{\, \begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}, \begin{bmatrix}

-1 \\

1 \\

0

\end{bmatrix}, \begin{bmatrix}

0 \\

-1 \\

1

\end{bmatrix} \,\right\}\]
be a basis for $\R^3$.

Calculate the coordinates of $\begin{bmatrix}

x \\

y \\

z

\end{bmatrix}$ with respect to $B$.

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

## A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.

## Problem 449

Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.

Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.

## The Product of a Subgroup and a Normal Subgroup is a Subgroup

## Problem 448

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

The **product** of $H$ and $N$ is defined to be the subset

\[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\]
Prove that the product $H\cdot N$ is a subgroup of $G$.

## If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix

## Problem 447

Let $A$ be a square matrix such that

\[A^{\trans}A=A,\]
where $A^{\trans}$ is the transpose matrix of $A$.

Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.

## A One-Line Proof that there are Infinitely Many Prime Numbers

## Inverse Map of a Bijective Homomorphism is a Group Homomorphism

## Problem 445

Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.

Suppose that $f:G\to H$ is bijective.

Then there exists a map $\psi:H\to G$ such that

\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\]
Then prove that $\psi:H \to G$ is also a group homomorphism.

## Group Homomorphism Sends the Inverse Element to the Inverse Element

## Problem 444

Let $G, G’$ be groups. Let $\phi:G\to G’$ be a group homomorphism.

Then prove that for any element $g\in G$, we have

\[\phi(g^{-1})=\phi(g)^{-1}.\]

## Injective Group Homomorphism that does not have Inverse Homomorphism

## Problem 443

Let $A=B=\Z$ be the additive group of integers.

Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.

**(a)** Prove that $\phi$ is a group homomorphism.

**(b)** Prove that $\phi$ is injective.

**(c)** Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.