## Problem 446

Prove that there are infinitely many prime numbers in ONE-LINE.

## 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.

## 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
$\phi(g^{-1})=\phi(g)^{-1}.$

## 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 $\psi \circ \phi=\id_A$.

## Problem 442

Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}

(a) Find the coefficient matrix $A$ for this system.

(b) Find the inverse matrix of the coefficient matrix found in (a)

(c) Solve the system using the inverse matrix $A^{-1}$.

## Problem 441

Let $A$ and $B$ be $m\times n$ matrices.
Prove that
$\rk(A+B) \leq \rk(A)+\rk(B).$

## Problem 440

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.
Then prove that
$\dim(U+V) \leq \dim(U)+\dim(V).$

## Problem 439

Is every diagonalizable matrix invertible?

## Problem 438

Determine whether each of the following statements is True or False.

(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
$p(\lambda)=(\lambda-1)^n+2,$ then $A$ is invertible.

(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)

## Problem 437

Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.
Prove that $R$ is a field.

## Problem 436

Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.

1. The ring $R$ is a field.
2. The only ideals of $R$ are $(0)$ and $R$.
3. Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.

## Problem 435

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
$\left(\, f\left(\, \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.$ We put
$V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.$

(a) Prove that the map $f$ is a linear transformation.

(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)

(d) Define a map $g:V \to V$ by
$g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.$ Prove that the map $g$ is a linear transformation.

(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

## Problem 434

Let $R$ be a ring with $1$.
A nonzero $R$-module $M$ is called irreducible if $0$ and $M$ are the only submodules of $M$.
(It is also called a simple module.)

(a) Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element as its generator.

(b) Determine all the irreducible $\Z$-modules.

## Problem 433

Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.

(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
$T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)$ for any $f(x)\in P_3$ is a linear transformation.

(b) Let $B=\{1, x, x^2, x^3\}$ be a basis of $P_3$. With respect to the basis $B$, find the matrix representation of the linear transformation $T$ in part (a).

## Problem 432

(a) Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module.
Prove that the module $M$ has a nonzero annihilator.
In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$.
Here $r$ does not depend on $m$.

(b) Find an example of an integral domain $R$ and a torsion $R$-module $M$ whose annihilator is the zero ideal.

## Problem 431

Let $R$ be a commutative ring and let $I$ be a nilpotent ideal of $R$.
Let $M$ and $N$ be $R$-modules and let $\phi:M\to N$ be an $R$-module homomorphism.

Prove that if the induced homomorphism $\bar{\phi}: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.

## Problem 430

Let $V$ be a vector space over a field $K$.
If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
$W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}$ is a subspace of the vector space $V$.

## Problem 429

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

## Problem 428

Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix
$A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.$

(a) Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.

(b) Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.

(c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis
$\left\{\, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \,\right\}$ of the $x$-$z$ plane.

## Problem 427

Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.