Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 48

Exercises

Exercise

Let ${}R$ and ${}S$ be commutative rings, and let

$\varphi \colon R\longrightarrow S$

be a ring homomorphism. Show that the kernel

${}\operatorname {kern} \varphi ={\left\{f\in R\mid \varphi (f)=0\right\}}\,$

is an ideal in ${}R$.

Exercise

Show directly, and using Theorem 44.3 , that every subgroup of ${}\mathbb {Z}$ is an ideal.

Exercise

Show that ${}\mathbb {Z} \subseteq \mathbb {Q}$ is a subgroup but not an ideal.

Exercise

Show that a commutative ring is a field if and only if it contains exactly two ideals.

Exercise

Let ${}k$ and ${}n$ be integers. Show that the following statements are equivalent.

1. ${}k$ divides ${}n$.
2. We have ${}\mathbb {Z} n\subseteq \mathbb {Z} k$.
3. There exists a ring homomorphism

   $\mathbb {Z} /(n)\longrightarrow \mathbb {Z} /(k).$

4. There exists a surjective group homomorphism

   $\mathbb {Z} /(n)\longrightarrow \mathbb {Z} /(k).$

Exercise

Let ${}n\in \mathbb {N}$, and let ${}\mathbb {Z} /(n)$ denote the corresponding residue class ring. Show that ${}a\in \mathbb {Z}$ is a unit modulo ${}n$ if and only if ${}a$ and ${}n$ are coprime.

Exercise

Let ${}n\geq 1$ be a natural number, and let ${}\mathbb {Z} /(n)$ be the corresponding residue class ring. Show that the following statements are equivalent.

1. ${}\mathbb {Z} /(n)$ is a field.
2. ${}\mathbb {Z} /(n)$ is a domain.
3. ${}n$ is a prime number.

Exercise

Let ${}C$ be the set all Cauchy sequences in ${}\mathbb {Q}$.

a) Show that ${}C$ is a commutative ring with the componentwise addition and multiplication.

b) Show that the subset ${}N\subseteq C$, consisting of all null sequences is an ideal.

c) Show that ${}C/N$ is a field.

Exercise

Look into a mirror. Does the mirror interchange left and right, up and down, front and back? Which linear mapping describes the reflection at a mirror?

Exercise

Is there any reason to produce different screws for left-handers?

Exercise

Does the right-hand rule also for left-handers?

Exercise

Let ${}V$ be a finite-dimensional real vector space. Show that on the set of (ordered) bases, the property representing the same orientation is an equivalence relation, which has (for ${}V\neq 0$) exactly two equivalence classes.

Exercise

Let ${}V\neq 0$ be a finite-dimensional real vector space, together with a basis ${}v_{1},\ldots ,v_{n}$. Show that, after replacing one vector ${}v_{i}$ by its negative ${}-v_{i}$, the new basis represents the opposite orientation.

Exercise

Determine whether the following two bases of ${}\mathbb {R} ^{2}$,

${\begin{pmatrix}2\\4\end{pmatrix}},\,{\begin{pmatrix}-5\\7\end{pmatrix}}\,\,{\text{ and }}\,\,{\begin{pmatrix}-3\\6\end{pmatrix}},\,{\begin{pmatrix}2\\-5\end{pmatrix}},$

represent the same orientation, or not.

Exercise

Determine whether the following two bases of the ${}\mathbb {R} ^{3}$,

${\begin{pmatrix}1\\0\\4\end{pmatrix}},\,{\begin{pmatrix}2\\4\\-3\end{pmatrix}},\,{\begin{pmatrix}0\\3\\-5\end{pmatrix}}\,\,{\text{ and }}\,\,{\begin{pmatrix}-3\\7\\2\end{pmatrix}},\,{\begin{pmatrix}-4\\5\\-1\end{pmatrix}},\,{\begin{pmatrix}-6\\0\\11\end{pmatrix}},$

represent the same orientation, or not.

Exercise

We consider in ${}\mathbb {R} ^{3}$ the three vectors

${\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}0\\2\\-2\end{pmatrix}},\,{\begin{pmatrix}x\\5\\7\end{pmatrix}}.$

a) How do we have to choose ${}x$ such that these three vectors represent the standard orientation of the ${}\mathbb {R} ^{3}$?

b) How do we have to choose ${}x$ such that these three vectors represent the orientation of the ${}\mathbb {R} ^{3}$ opposite to the standard orientation?

Exercise

Right now, Lucy Sonnenschein is in position ${}(-2,3)\in \mathbb {Z} ^{2}$ (the coordinates are denoted by ${}x$ and ${}y$), and looks in the direction of the positive ${}x$-axis. The following instructions refer always to her present position and her looking direction, clockwise refers to plan view. Lucy does the following movements one after another. She makes a step to the right, then two steps backwards, she turns around by ${}180$ degree, she makes three steps to the left, she make a clockwise quarter rotation, she makes four steps to the right, and two steps backwards, turns around by ${}360$ degree, and makes a step to the left.

Where is she in the end and in what direction does she look?

Exercise

Discuss whether it is useful to denote the vertices of a triangle in the plane always counter-clockwise by ${}A,B,C$. Take also the image on the right under consideration.

Exercise

Let ${}V$ and ${}W$ be finite-dimensional oriented real vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a bijective linear mapping. Show that ${}\varphi$ is orientation-preserving if and only if there exists a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ representing the orientation of ${}V$ such that the image vectors ${}\varphi (v_{1}),\ldots ,\varphi (v_{n})$ represent the orientation of ${}W$.

Exercise

Let ${}M=\{1,\ldots ,n\}$, let ${}\pi$ be a permutation on ${}M$, and let ${}M_{\pi }$ denote the corresponding permutation matrix. Show that ${}M$ is orientation-preserving if and only if

${}\operatorname {sgn} (\pi )=1\,$

holds.

Exercise

Let ${}V$ be a finite-dimensional real vector space, and let ${}G=\operatorname {GL} _{}{\left(V\right)}$ denote the group of all bijective linear mappings on ${}V$. Show that the set of the orientation-preserving mappings is a normal subgroup in ${}G$. What relation exists to the modulus of the determinant?

Hand-in-exercises

Exercise (3 marks)

Determine the multiplicative order of all units in the residue class field ${}\mathbb {Z} /(11)$.

Exercise (4 marks)

Determine whether the two bases of ${}\mathbb {R} ^{3}$,

${\begin{pmatrix}2\\4\\-5\end{pmatrix}},\,{\begin{pmatrix}7\\6\\-1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\-3\end{pmatrix}}\,\,{\text{ and }}\,\,{\begin{pmatrix}-3\\6\\2\end{pmatrix}},\,{\begin{pmatrix}-4\\4\\-2\end{pmatrix}},\,{\begin{pmatrix}-5\\0\\13\end{pmatrix}},$

represent the same orientation, or not.

Exercise (6 marks)

Let ${}V$ be a finite-dimensional complex vector space. Show that there exists a natural orientation on ${}V$, considered as a real vector space.

Exercise (4 marks)

Let ${}V$ be a Euclidean vector space of dimension ${}n$, and consider the product ${}V^{n}=V\times \cdots \times V$, equipped with the product topology. Let ${}I$ be a real interval, and let

$\varphi \colon I\longrightarrow V^{n}$

denote a continuous mapping with the property that

${}\varphi (t)=(\varphi _{1}(t),\varphi _{2}(t),\ldots ,\varphi _{n}(t))\,$

is a basis of ${}V$ for every ${}t\in I$. Show that all bases ${}\varphi (t)$, ${}t\in I$, represent the same orientation of ${}V$.

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