Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 3
- Exercise for the break
Binomial formula/R/Distributivity law/Exercise
- Exercises
Integers/Difference/Structural properties/Exercise
Binary operation/Table/4/2/Exercise
Show that taking powers of positive natural numbers, i.e., the assignment
is neither commutative nor associative. Does this composition have a neutral element?
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?
Real nonnegative numbers/Minimum/Structural properties/Exercise
Q mod Z/Directly/Operation/Exercise
Q mod Z/Direct/Operation/Group/Exercise
Binary operation/Associative/4 factors/Exercise
Binary operation/Associative/5 factors/1/Exercise
Intersection/Power set/Operation/Exercise
Operation table/Mappings on set with 2 elements/Exercise
Mappings/Bijective mappings/Group/Exercise
Group/Inverse/Self inverse/Exercise
Ring/(x^2-3yzy-2zy^2+4xy^2)(2xy^3x-z^2xyx)(1-3zyxz^2y)/Compute/Exercise
Commutative ring/Sum and product of polynomial terms/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.
Field/Binomi/Explained/Fact/Proof/Exercise
Complex numbers/General binomial formula/Exercise
Let be elements in a field and suppose that and are not zero. Prove the following fraction rules.
Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is
Show that the popular formula
does not hold.
Field/Reverse distributivity law/Exercise
Describe and prove rules for the addition and the multiplication of even and odd integer numbers. Define on the set with two elements
an "addition“ and a "multiplication“ which "represents“ these rules.
Field/Zero set/Nearly a field/Exercise
Let be a field. Show that for every natural number there exists a field element such that is the null element in and is the unit element in and such that
holds. Show that this assignment has the properties
Extend this assignment to all integer numbers and show that the stated structural properties hold again.
Let be a field with . Show that for the relation
holds.
Ring structure/Set of mappings to ring/Exercise
- Hand-in-exercises
Exercise (2 marks)
Discuss the operation
looking at associativity, commutativity, existence of a neutral element and existence of inverse element.
Power set/Ring structure with symmetric difference/Exercise
Field/Bijectivity of one-sided operations/Exercise
Exercise (3 marks)
Show that the "rule“
is for (and ) never true. Give an example with where this rule holds.
Exercise (5 marks)
Prove the general distributive property for a field.
<< | Linear algebra (Osnabrück 2024-2025)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF) |
---|