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

Exercise for the break

Exercise

Binomial formula/R/Distributivity law/Exercise

Exercises

Exercise

Integers/Difference/Structural properties/Exercise

Exercise

Binary operation/Table/4/2/Exercise

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 composition have a neutral element?

Exercise

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

Exercise

Real nonnegative numbers/Minimum/Structural properties/Exercise

Exercise

Q mod Z/Directly/Operation/Exercise

Exercise

Q mod Z/Direct/Operation/Group/Exercise

Exercise

Binary operation/Associative/4 factors/Exercise

Exercise

Binary operation/Associative/5 factors/1/Exercise

Exercise

Intersection/Power set/Operation/Exercise

Exercise

Operation table/Mappings on set with 2 elements/Exercise

Exercise

Mappings/Bijective mappings/Group/Exercise

Exercise

Group/Inverse/Self inverse/Exercise

Exercise

Group/Inverse of xy/Exercise

Exercise

Group/3 elements/Exercise

Exercise

Ring/(x^2-3yzy-2zy^2+4xy^2)(2xy^3x-z^2xyx)(1-3zyxz^2y)/Compute/Exercise

Exercise

Commutative ring/Sum and product of polynomial terms/Exercise

Exercise

Graph (mapping)/R/Addition and multiplication/Exercise

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

Exercise

Field/Binomi/Explained/Fact/Proof/Exercise

Exercise

Complex numbers/General binomial formula/Exercise

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), which arises when one replaces addition by multiplication (and subtraction by division), 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

Field/Reverse distributivity law/Exercise

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“ which "represents“ these rules.

Exercise

Field/Zero set/Nearly a field/Exercise

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.

Exercise

Ring structure/Set of mappings to ring/Exercise

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 associativity, commutativity, existence of a neutral element and existence of inverse element.

Exercise Power set/Ring structure with symmetric difference/Exercise

Power set/Ring structure with symmetric difference/Exercise

Exercise Field/Bijectivity of one-sided operations/Exercise

Field/Bijectivity of one-sided operations/Exercise

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 property for a field.

Exercise Field/Qi/Exercise

Field/Qi/Exercise

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