Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 47

Exercises

Exercise

Let ${}r,s\in \mathbb {R} \setminus \{0\}$ be two real numbers ${}\neq 0$. Show that these are equivalent to each other with respect to the equivalence relation given by the subgroup ${}(\mathbb {Q} ^{\times },1,\cdot )\subseteq (\mathbb {R} ^{\times },1,\cdot )$ if and only if there exist a real number ${}t\neq 0$ and integer numbers ${}a,b\in \mathbb {Z}$ satisfying ${}r=bt$ and ${}s=at$.

Exercise

We consider ${}\mathbb {R}$ as a ${}\mathbb {Q}$-vector space. Verify that in ${}\mathbb {R} /\mathbb {Q}$, the equality ${}[r]=[s]$ for two real numbers ${}r,s$ holds if and only if the difference ${}r-s$ is a rational number.

Exercise

For real numbers ${}r,s$, we set ${}r\sim s$ if there exist rational numbers ${}u,v\in \mathbb {Q}$ with ${}u\neq 0$ such that

${}r=us+v\,.$

a) Show that this is an equivalence relation on ${}\mathbb {R}$.

b) Determine the equivalence class of ${}{\frac {3}{7}}$.

c) Give an example of two real numbers that are not commensurable but are equivalent with respect to ${}\sim$.

Exercise

Show that in the residue class group ${}\mathbb {Q} /\mathbb {Z}$, for every ${}n\in \mathbb {N} _{+}$, there exists an element of order ${}n$.

Exercise

Show that there does not exist a subgroup ${}F\subseteq (\mathbb {Q} ,0,+)$ such that the composed mapping

$F\longrightarrow \mathbb {Q} /\mathbb {Z}$

is an isomorphism.

Exercise

Determine the residue class group of ${}\{1,-1\}\subset \mathbb {R} ^{\times }$.

Exercise

Find inside the permutation group ${}S_{3}$ a normal subgroup ${}N\neq 0,S_{3}$, and determine the corresponding residue class group.

Exercise

Let ${}G$ be a group, and ${}g\in G$ be an element, together with the (according to Lemma 44.12 ) corresponding group homomorphism

$\varphi \colon \mathbb {Z} \longrightarrow G,n\longmapsto g^{n}.$

Describe the canonical factorization of ${}\varphi$ in the sense of Theorem 47.11 .

Exercise

Let ${}G$ be a group, and let ${}g\in G$ an element of finite order. Show that the order of ${}g$ coincides with the minimal ${}d\in \mathbb {N} _{+}$ such that there exists a group homomorphism

$\mathbb {Z} /(d)\longrightarrow G$

with the property that ${}g$ belongs to the image.

Exercise

Show, using the homomorphism theorems, that cyclic groups with the same order are isomorphic.

Exercise

Let ${}G,H,$ and ${}F$ be groups, and let ${}\varphi \colon G\rightarrow H$ and ${}\psi \colon G\rightarrow F$ denote group homomorphisms. Suppose that ${}\psi$ is surjective and that ${}\operatorname {kern} \psi \subseteq \operatorname {kern} \varphi$ holds. Determine the kernel of the induced homomorphism

${\tilde {\varphi }}\colon F\longrightarrow H.$

Exercise

Show that for every real number ${}a\neq 0$, the residue class groups ${}\mathbb {R} /\mathbb {Z} a$ are isomorphic to each other.

For the following exercise, we use that every positive natural number has a unique factorization into prime numbers.

Exercise

Let ${}p$ be a prime number. Define a group homomorphism

$(\mathbb {Q} \setminus \{0\},\cdot ,1)\longrightarrow (\mathbb {Z} ,+,0),$

that maps ${}p\mapsto 1$ and all other prime numbers to ${}0$.

Determine also the kernel of this group homomorphism.

Exercise

Let ${}G_{1}$ and ${}G_{2}$ be groups, and let ${}N_{1}\subseteq G_{1}$ and ${}N_{2}\subseteq G_{2}$ be normal subgroup. Show that ${}N_{1}\times N_{2}$ is a normal subgroup in ${}G_{1}\times G_{2}$, and that we have an isomorphism

${}(G_{1}\times G_{2})/(N_{1}\times N_{2})\cong (G_{1}/N_{1})\times (G_{2}/N_{2})\,.$

Die following exercise uses the topological concept of denseness.

A subset ${}T\subseteq \mathbb {R}$ is called dense if for every real number ${}x\in \mathbb {R}$ and every ${}\epsilon >0$, there exists an element ${}t\in T$ such that

${}\vert {t-x}\vert <\epsilon \,$

holds.

Exercise

Let ${}H$ be an (additive) subgroup of the real numbers ${}\mathbb {R}$. Show that either ${}H={\mathbb {Z} }a$ with an uniquely determined nonnegative real number ${}a$, or ${}H$ is dense in ${}\mathbb {R}$.

Exercise

Let ${}K$ be a field, let ${}V$ be a ${}K$-vector space, and let ${}U\subseteq V$ denote a linear subspace. Let ${}u_{i}$, ${}i\in I$, denote a basis of ${}U$, and ${}v_{j}$, ${}j\in J$, a family of vectors in ${}V$. Show that the family ${}u_{i},i\in I,v_{j},j\in J$, is a basis of ${}V$ if and only if ${}[v_{j}]$, ${}j\in J$, is a basis of the residue class space ${}V/U$.

Exercise

Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a flag in a finite-dimensional ${}K$-vector space ${}V$. Show that

${}V_{i+1}/V_{i}\cong K\,$

holds for ${}i=0,\ldots ,n-1$.

Exercise

Suppose that the ${}K$-vector space ${}V$ is the direct sum of the linear subspaces ${}V_{1}$ and ${}V_{2}$, and let ${}U_{1}\subseteq V_{1}$ and ${}U_{2}\subseteq V_{2}$ be linear subspaces. Show that

${}V_{1}\oplus V_{2}/{\left(U_{1}\oplus U_{2}\right)}\cong V_{1}/U_{1}\oplus V_{2}/U_{2}\,.$

Interpret the statement of the following exercise in the context of the factorization theorem.

Exercise

Let ${}M$ be an ${}m\times n$-matrix over the field ${}K$ of rank ${}r$. Show that there exists an ${}r\times n$-matrix ${}A$, and an ${}m\times r$-matrix ${}B$, both of rank ${}r$, such that ${}M=B\circ A$ holds.

Exercise

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

denote a linear mapping, and ${}U\subseteq V$ a ${}\varphi$-invariant linear subspace. Show that this induces a uniquely determined linear mapping

$\varphi _{V/U}\colon V/U\longrightarrow V/U$

on the residue class space ${}V/U$ with the property that the diagram

${\begin{matrix}V&{\stackrel {\varphi }{\longrightarrow }}&V&\\\!\!\!\!\!\pi \downarrow &&\downarrow \pi \!\!\!\!\!&\\V/U&{\stackrel {\varphi _{V/U}}{\longrightarrow }}&V/U&\!\!\!\!\!\\\end{matrix}}$

commutes.

Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Let ${}u_{1},\ldots ,u_{s}$ be a basis of ${}U$, and ${}u_{1},\ldots ,u_{r},v_{1},\ldots ,v_{s}$ a basis of ${}V$, and suppose that the matrix ${}M$ describes ${}\varphi$ with respect to the given basis. Which matrix describes the linear mapping

$\varphi _{V/U}\colon V/U\longrightarrow V/U$

defined in Exercise 47.20 with respect to the basis ${}[v_{1}],\ldots ,[v_{s}]$ of ${}V/U$?

For the following exercise, compare Exercise 16.23 .

Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Let ${}\varphi _{U}$ be the restriction of ${}\varphi$ to ${}U$, and let

$\varphi _{V/U}\colon V/U\longrightarrow V/U$

be the linear mapping defined in Exercise 47.20 . Show

${}\det \varphi =\det \varphi _{U}\cdot \det \varphi _{V/U}\,.$

Exercise

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Let ${}\varphi _{U}$ be the restriction of ${}\varphi$ to ${}U$, and let

$\varphi _{V/U}\colon V/U\longrightarrow V/U$

be the linear mapping defined in Exercise 47.20 . Show that the characteristic polynomial satisfies the relation

${}\chi _{\varphi }=\chi _{\varphi _{U}}\cdot \chi _{\varphi _{V/U}}\,.$

Exercise

Let ${}\mathbb {R} ^{\mathbb {N} _{+}}$ be the real vector space of all sequences. Show that the following subsets are linear subspaces.

a) The set of the constant sequences.

b) The set ${}\mathbb {R} ^{(\mathbb {N} _{+})}$ of the sequences where only finitely many members are different from ${}0$.

c) The set ${}F$ of the sequences that are constant with the exception of finitely many members.

d) The set ${}E$ of the sequences that have only finitely many different values.

e) The set of all convergent sequences.

f) The set ${}N$ of all null sequences. What inclusions do hold between these linear subspaces?

Exercise

We consider the real sequences

${}x_{n}={\begin{cases}1\,,{\text{ if }}n{\text{ even}}\,,\\0\,,{\text{ else}}\,,\end{cases}}\,$

and

${}y_{n}={\begin{cases}1\,,{\text{ if }}n{\text{ odd}}\,,\\0\,,{\text{ else}}\,,\end{cases}}\,,$

and we use the notations from Exercise 47.24 .

a) Show that the two sequences ${}x_{n}$ and ${}y_{n}$ are linearly independent (considered) in ${}\mathbb {R} ^{\mathbb {N} _{+}}/\mathbb {R} ^{(\mathbb {N} _{+})}$.

b) Show that the two sequences ${}x_{n}$ and ${}y_{n}$ are linearly dependent in ${}\mathbb {R} ^{\mathbb {N} _{+}}/F$.

c) How is it in ${}\mathbb {R} ^{\mathbb {N} _{+}}/N$?

Exercise

Let ${}W\subseteq \mathbb {R} ^{\mathbb {N} _{+}}$ be the real vector space of all convergent sequences, and let ${}U\subseteq W$ denote the linear subspace of all null sequences. Show

${}W/U\cong \mathbb {R} \,.$

Exercise

Show that the mapping

$S^{1}\times \mathbb {R} \longrightarrow \mathbb {C} \setminus \{0\},(u,t)\longmapsto ue^{t},$

is a group isomorphism. How are the group structures given? Which distinctive subsets of the cylinder and of the punctured plane correspond to each other under this isomorphism.

Hand-in-exercises

Exercise (3 marks)

Let ${}G$ and ${}H$ be groups, and let ${}G\times H$ denote their product group. Show that the group ${}G\times \{e_{H}\}$ is a normal subgroup in ${}G\times H$, and that the residue class group ${}(G\times H)/(G\times \{e_{H}\})$ is canonically isomorphic to ${}H$.

Exercise (3 marks)

Determine the group homomorphisms between two cyclic groups. Which are injective and which are surjective?

Exercise (2 marks)

Show that there exists a group ${}G$ and a group homomorphism

$\varphi \colon (\mathbb {R} ,0,+)\longrightarrow G$

fulfilling the property that ${}r\in \mathbb {R}$ is rational if and only if ${}\varphi (r)=0$ holds.

Exercise (2 marks)

We consider ${}\mathbb {C}$ as a ${}\mathbb {R}$-vector space, and the linear subspace

${}\mathbb {R} =\mathbb {R} \cdot 1\subset \mathbb {C} \,.$

Show that in the residue class space ${}\mathbb {C} /\mathbb {R}$, two complex numbers become equal if and only if their imaginary parts coincide.

Exercise (3 marks)

Let ${}V$ be a ${}K$-vector space, and let ${}U_{1},U_{2},U$ denote linear subspaces. Let

$\varphi \colon V\longrightarrow V/U$

be the canonical projection. Show that the following statements are equivalent.

a) For the image spaces, we have

${}\varphi (U_{1})\cap \varphi (U_{2})=0\,.$

b) We have

${}U_{1}\cap (U_{2}+U)\subseteq U\,.$

c) We have

${}(U_{1}+U)\cap (U_{2}+U)\subseteq U\,.$

Exercise (4 marks)

Let ${}V$ be a ${}K$-vector space, together with a symmetric bilinear form ${}\left\langle -,-\right\rangle$, and let ${}T\subseteq V$ be the degeneracy space. Show that on the residue class space ${}V/T$, there exists a nondegenerate symmetric bilinear form ${}\left\langle -,-\right\rangle '$ such that

${}\left\langle [v],[w]\right\rangle '=\left\langle v,w\right\rangle \,$

holds for all ${}v,w\in V$.

Exercise (6 (1+3+2) marks)

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}U\subseteq V$ be a linear subspace. We define on ${}E$ a relation ${}\sim$ via

$P\sim Q{\text{ if and only if }}\exists v\in U{\text{ with }}P=Q+v.$

a) Show that ${}\sim$ is an equivalence relation.

b) Show that ${}F:=E/\sim$ is an affine space over the residue class space ${}V/U$.

c) Show that the canonical projection

$E\longrightarrow E/\sim$

is an affine mapping.

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