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

Exercises

Exercise

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

Exercise

We consider $ℝ$ as a $\mathbb{Q}$-vector space. Verify that in $ℝ/\mathbb{Q}$, 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 $u\ne 0$ such that

$r=us+v.$

a) Show that this is an equivalence relation on $ℝ$.

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

$F⟶\mathbb{Q}/\mathbb{Z}$

is an isomorphism.

Exercise

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

Exercise

Find inside the permutation group $S_3$ a normal subgroup $N\ne 0,S_3$, 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

$\phi :\mathbb{Z}⟶G,n⟼g^n.$

Describe the canonical factorization of $\phi$ 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

$\mathbb{Z}/(d)⟶G$

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 $\phi :G\to H$ and $\psi :G\to F$ denote group homomorphisms. Suppose that $\psi$ is surjective and that $\mathrm{kern}\psi \subseteq \mathrm{kern}\phi$ holds. Determine the kernel of the induced homomorphism

$\tilde{\phi }:F⟶H.$

Exercise

Show that for every real number $a\ne 0$, the residue class groups $ℝ/\mathbb{Z}a$ 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

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

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 $T\subseteq ℝ$ is called dense if for every real number $x\in ℝ$ and every $ϵ>0$, there exists an element $t\in T$ such that

$|t-x|<ϵ$

holds.

Exercise

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

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

$0=V_0\subset V_1\subset \dots \subset V_{n-1}\subset V_n=V$

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

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

holds for $i=0,\dots ,n-1$.

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

$V_1\oplus V_2/(U_1\oplus U_2)\cong V_1/U_1\oplus V_2/U_2.$

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

$\phi :V⟶V$

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

${\phi }_{V/U}:V/U⟶V/U$

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

$\begin{array}{cccc}V& \stackrel{\phi }{⟶}& V& \\ \pi \downarrow & & \downarrow \pi & \\ V/U& \stackrel{{\phi }_{V/U}}{⟶}& V/U& \end{array}$

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 $\phi$-invariant linear subspace. Let $u_1,\dots ,u_s$ be a basis of $U$, and $u_1,\dots ,u_r,v_1,\dots ,v_s$ a basis of $V$, and suppose that the matrix $M$ describes $\phi$ with respect to the given basis. Which matrix describes the linear mapping

${\phi }_{V/U}:V/U⟶V/U$

defined in Exercise 47.20 with respect to the basis $[v_1],\dots ,[v_s]$ 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 $\phi$-invariant linear subspace. Let ${\phi }_U$ be the restriction of $\phi$ to $U$, and let

${\phi }_{V/U}:V/U⟶V/U$

be the linear mapping defined in Exercise 47.20 . Show

$det\phi =det{\phi }_U\cdot det{\phi }_{V/U}.$

Exercise

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

${\phi }_{V/U}:V/U⟶V/U$

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

${\chi }_{\phi }={\chi }_{{\phi }_U}\cdot {\chi }_{{\phi }_{V/U}}.$

Exercise

Let ${ℝ}^{{\mathbb{N}}_{+}}$ 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 ${ℝ}^{({\mathbb{N}}_{+})}$ 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

$x_n=\{\begin{array}{c}1,\text{ if }n\text{ even},\\ 0,\text{ else},\end{array}$

and

$y_n=\{\begin{array}{c}1,\text{ if }n\text{ odd},\\ 0,\text{ else},\end{array},$

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 ${ℝ}^{{\mathbb{N}}_{+}}/{ℝ}^{({\mathbb{N}}_{+})}$.

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

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

Exercise

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

$W/U\cong ℝ.$

Exercise

Show that the mapping

$S^1\times ℝ⟶ℂ\setminus \{0\},(u,t)⟼u{e}^t,$

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

$\phi :(ℝ,0,+)⟶G$

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

Exercise (2 marks)

We consider $ℂ$ as a $ℝ$-vector space, and the linear subspace

$ℝ=ℝ\cdot 1\subset ℂ.$

Show that in the residue class space $ℂ/ℝ$, 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

$\phi :V⟶V/U$

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

a) For the image spaces, we have

$\phi (U_1)\cap \phi (U_2)=0.$

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 $⟨-,-⟩$, 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 ${⟨-,-⟩}^{\prime }$ such that

${⟨[v],[w]⟩}^{\prime }=⟨v,w⟩$

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

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

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

$E⟶E/\sim$

is an affine mapping.

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