Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 3

Exercise for the break

Exercise

Formulate the binomial formula for two real numbers, and prove it using the distributive law.

Exercises

Exercise

Consider the integers ${}\mathbb {Z}$ together with the difference as a binary operation, that is, the mapping

$\mathbb {Z} \times \mathbb {Z} \longrightarrow \mathbb {Z} ,(a,b)\longmapsto a-b.$

Does there exist a neutral element for this operation? Is this operation associative, commutative, does there exist for every element an inverse element?

Exercise

On the set

${}M=\{a,b,c,d\}\,,$

we consider the binary operation ${}\star$ given by the table

${}\star$	${}a$	${}b$	${}c$	${}d$
${}a$	${}b$	${}a$	${}c$	${}a$
${}b$	${}d$	${}a$	${}b$	${}b$
${}c$	${}a$	${}b$	${}c$	${}c$
${}d$	${}b$	${}d$	${}d$	${}d$

1. Compute

   $b\star (a\star (d\star a)).$

2. Does there exist a neutral element for ${}\star$?

Exercise

Show that taking powers of positive natural numbers, i.e., the assignment

$\mathbb {N} \times \mathbb {N} \longrightarrow \mathbb {N} ,(a,b)\longmapsto a^{b},$

is neither commutative nor associative. Does this operation have a neutral element?

Exercise

Show that the operation on a line that assigns to two points their midpoint, is commutative, but not associative. Does there exist a neutral element?

Exercise

Study the binary operation

$\mathbb {R} _{\geq 0}\times \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},(x,y)\longmapsto \operatorname {min} \,(x,y),$

with respect to associativity, commutativity, existence of a neutral element, existence of inverse elements.

Exercise

Let ${}M$ be a set, with an associative binary operation ${}\star$ defined on it. Show that

${}(a\star b)\star (c\star (d\star e))=a\star ((b\star (c\star d))\star e)\,$

holds for arbitrary ${}a,b,c,d,e\in M$.

Exercise

Let ${}G$ be a set, and let ${}M={\mathfrak {P}}\,(G)$ be the corresponding power set. Consider the intersection of subsets of ${}G$ as a binary operation on ${}M$. Is this operation commutative, associative, does there exist a neutral element?

Exercise

Let ${}M$ be the set of all mappings from the set ${}\{0,1\}$ to itself, that is

${}M={\left\{F:\{0,1\}\rightarrow \{0,1\}\mid F{\text{ mapping}}\right\}}\,.$

Denote the elements of ${}M$ with certain symbols, and establish a value table for the binary operation on ${}M$ given by the composition of mappings.

Exercise

Let ${}S$ be a set and define

${}G={\left\{F:S\rightarrow S\mid F{\text{ bijective}}\right\}}\,.$

Show that ${}G$ is, with the composition of mappings as operation, a group.

Exercise

Let ${}G$ be a group. Show that

${}{\left(x^{-1}\right)}^{-1}=x\,$

holds for all ${}x\in G$.

Exercise

Let ${}G$ be a group, and ${}x,y\in G$. Express the inverse of ${}xy$ with the inverses of ${}x$ and ${}y$.

Exercise

Construct a group with three elements.

Exercise

Let ${}R$ be a ring, and let ${}\spadesuit ,\heartsuit$ and ${}\clubsuit$ denote elements in ${}R$. Compute the product

${\left(\spadesuit ^{2}-3\heartsuit \clubsuit \heartsuit -2\clubsuit \heartsuit ^{2}+4\spadesuit \heartsuit ^{2}\right)}{\left(2\spadesuit \heartsuit ^{3}\spadesuit -\clubsuit ^{2}\spadesuit \heartsuit \spadesuit \right)}{\left(1-3\clubsuit \heartsuit \spadesuit \clubsuit ^{2}\heartsuit \right)}.$

What is the result, when the ring is commutative?

Exercise

Let ${}R$ be a commutative ring, and ${}f,a_{i},b_{j}\in R$. Show the following equations:

$\sum _{i=0}^{n}a_{i}f^{i}+\sum _{j=0}^{m}b_{j}f^{j}=\sum _{k=0}^{\max(n,m)}(a_{k}+b_{k})f^{k}$

and

${\left(\sum _{i=0}^{n}a_{i}f^{i}\right)}\cdot {\left(\sum _{j=0}^{m}b_{j}f^{j}\right)}=\sum _{k=0}^{n+m}c_{k}f^{k}{\text{ where }}c_{k}=\sum _{r=0}^{k}a_{r}b_{k-r}.$

Exercise

Sketch the graph of the real addition

$+\colon \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,(x,y)\longmapsto x+y,$

and the graph of the real multiplication

$\cdot \colon \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,(x,y)\longmapsto x\cdot y.$

The following exercise is proved by induction. This is a proof method, usually introduced in analysis. See also the appendix to this course.

Exercise

Prove the general binomial formula, that is, the formula

$(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}$

for ${}n\in \mathbb {N}$ and arbitrary elements ${}a,b\in K$ in a field ${}K$.

The following exercise refers to the complex numbers ${}\mathbb {C}$.

Exercise

Compute

$(x+{\mathrm {i} }y)^{n}.$

Exercise

Let ${}x,y,z,w$ be elements in a field, and suppose that ${}z$ and ${}w$ are not zero. Prove the following fraction rules.

1. ${}{\frac {x}{1}}=x\,,$

2. ${}{\frac {1}{z}}=z^{-1}\,,$

3. ${}{\frac {1}{-1}}=-1\,,$

4. ${}{\frac {0}{z}}=0\,,$

5. ${}{\frac {z}{z}}=1\,,$

6. ${}{\frac {x}{z}}={\frac {xw}{zw}}\,$

7. ${}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$

8. ${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$

Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is

${}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?$

Show that the popular formula

${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,$

does not hold.

Exercise

Show that, in a field, the "reversed distributive law“, that is,

${}a+(bc)=(a+b)\cdot (a+c)\,,$

does not hold.

Exercise

Describe and prove rules for the addition and the multiplication of even and odd integer numbers. Define on the set with two elements

$\{E,O\}$

an "addition“ and a "multiplication“ that "represent“ these rules.

Exercise

Show that the set ${}\{0\}$ satisfies all axioms for a field, except that ${}0=1$ holds.

Exercise

Let ${}K$ be a field. Show that, for every natural number ${}n\in \mathbb {N}$, there exists a field element ${}n_{K}$ such that ${}0_{K}$ is the null element in ${}K$, and ${}1_{K}$ is the unit element in ${}K$, and such that

${}(n+1)_{K}=n_{K}+1_{K}\,$

holds. Show that this assignment has the properties

$(n+m)_{K}=n_{K}+m_{K}{\text{ and }}(nm)_{K}=n_{K}\cdot m_{K}.$

Extend this assignment to all integer numbers ${}\mathbb {Z}$, and show that the stated structural properties hold again.

Exercise

Let ${}K$ be a field with ${}2\neq 0$. Show that for ${}f,g\in K$, the relation

${}fg={\frac {1}{4}}{\left({\left(f+g\right)}^{2}-{\left(f-g\right)}^{2}\right)}\,$

holds.

Hand-in-exercises

Exercise (2 marks)

Discuss the operation

$\mathbb {R} _{\geq 0}\times \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},(x,y)\longmapsto \operatorname {max} \,(x,y),$

looking at associative law, commutative law, existence of a neutral element and existence of inverse element.

Exercise (3 marks)

Let ${}M$ be a set. Show that the power set ${}{\mathfrak {P}}\,(M)$ is a commutative ring, if we consider the intersection ${}\cap$ as multiplication and the symmetric difference

${}A\triangle B={\left(A\setminus B\right)}\cup {\left(B\setminus A\right)}\,$

as addition (what are the neutral elements?).

Exercise (2 marks)

Show that in a field ${}K$, the following properties hold.

(1) For every ${}a\in K$, the mapping

$\alpha _{a}\colon K\longrightarrow K,x\longmapsto x+a,$

is bijective.

(2) For every ${}b\in K$, ${}b\neq 0$, the mapping

$\mu _{b}\colon K\longrightarrow K,x\longmapsto bx,$

is bijective.

Exercise (3 marks)

Show that the "rule“

${}{\frac {a}{b}}+{\frac {c}{d}}={\frac {a+c}{b+d}}\,$

is for ${}a,c\in \mathbb {N} _{+}$ (and ${}b,d,b+d\in \mathbb {Z} \setminus \{0\}$) never true. Give an example with ${}a,b,c,d,b+d\neq 0$ where this rule holds.

Exercise (5 marks)

Prove the general distributive law for a field.

Exercise (4 marks)

We consider the set

${}K=\mathbb {Q} \times \mathbb {Q} ={\left\{(a,b)\mid a,b\in \mathbb {Q} \right\}}\,$

with the special elements

$0=(0,0){\text{ and }}1=(1,0),$

the addition

${}(a,b)+(c,d):=(a+c,b+d)\,$

and the multiplication

${}(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,.$

Show that ${}K$ is, with these operations, a field.

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