# Author: Yu

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

## Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix

## 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 $A^{-1}$.

## The Rank of the Sum of Two Matrices

## Problem 441

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

## Dimension of the Sum of Two Subspaces

## Problem 440

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

## True of False Problems on Determinants and Invertible Matrices

## 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)

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## Every Integral Domain Artinian Ring is a Field

## 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.

## Three Equivalent Conditions for a Ring to be a Field

## Problem 436

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

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

## Subspace Spanned By Cosine and Sine Functions

## 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)

Add to solve later## A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator

## 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.

## Differentiation is a Linear Transformation

## 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).

## Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator

## 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.

## Nilpotent Ideal and Surjective Module Homomorphisms

## 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.

Add to solve later## The Sum of Subspaces is a Subspace of a Vector Space

## 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 $V$.

## Idempotent Matrices are Diagonalizable

## Problem 429

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

Add to solve later## Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation

## 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.

## Union of Subspaces is a Subspace if and only if One is Included in Another

## 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 $W_2 \subset W_1$.

Add to solve later