Jump to content

Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 18

From Wikiversity



Exercise for the break

Show that one can represent every finite permutation by an arrow diagram without crossings.




Exercises

Compute, for the permutation

the number of inversions and the sign.


Compute, for the permutation given by

the powers and , and determine the cycle representation of these three permutations.


We consider the permutation , given by the value table

  1. Determine the cycle representation of , and the range of action.
  2. Compute and the order of .
  3. Determine the inversions of and the sign of .
  4. Express as a product of transpositions, and determine again the sign of


We consider the two permutations given by

and

Compute and . Determine the number of inversions and the sign of . Describe the cycle representation of and of . What is the order of ?


We consider the permutation on

given by

  1. Establish a value table for .
  2. Establish a value table for .
  3. Show that all iterated compositions are bijective.
  4. Determine, for every , the minimal such that

    holds.

  5. Determine the minimal such that

    holds for all .


Show that the assignment

given by

is well-defined and bijective.


Gabi Hochster, Heinz Ngolo, Lucy Sonnenschein and Mustafa Müller want to play Secret Santa. That is, every child gets a gift from exactly one of the other (!) children. How many possibilities are there?


Determine the fixed points of the mapping


Let denote a set, and let

be a mapping. Show that has an fixed point if and only if the intersection of the graph of with the diagonal is not empty.


Compute the determinant of all the -matrices, such that in each column and in each row, there are exactly one and two s.


Let and let be a permutation on . The corresponding permutation matrix is given by

all other entries being . Show that



a) Give an example of an -permutation matrix such that in every diagonal (diagonal and antidiagonal, including all parallel diagonals), there is at most one .


b) Show that there is no solution for a) where holds.


Let a field, and let

the set of all invertible -matrices.

a) Show that (without referring to the determinant), is, with matrix multiplication as operation, a group.


b) Show that (without referring to the determinant), the mapping

is a group homomorphism.


Determine with the Leibniz-formula the determinant of the matrix


Let be a group. A subset is called a subgroup of , when the following conditions hold.

  1. .
  2. If , then also .
  3. If , then also .



The Christmas exercise for the whole family

Which construction principle is behind the sequence


(Some people claim that this exercise is very easy for primary school children, but quite hard for mathematicians.)




Hand-in-exercises

Exercise (2 marks)

Determine the sign of the permutation given by the image (the left hand represents the domain, the right hand represents the codomain).


Exercise (2 marks)

Let be a set, and let be a partition of , that is, every is a subset of , and is the disjoint union of the . Show that the product group

is a subgroup of .


Exercise (2 marks)

Show that every even permutation , , can be written as a product of cycles of length .


Exercise (5 marks)

Let be a cycle of length . Show that can be written as a product of transpositions, but not with a smaller number of transpositions.


Exercise (4 marks)

Let . How many injective mappings do exist from to , and how many surjective mappings do exist from to ?


Exercise (3 marks)

Determine with the Leibniz-formula the determinant of the matrix


Exercise (4 marks)

We consider the mapping

that is described in Exercise 18.16 (the natural numbers are given as finite sequences in the decimal system).

  1. Is increasing?
  2. Is surjective?
  3. Is injective?
  4. Does have a fixed point?



<< | Linear algebra (Osnabrück 2024-2025)/Part I | >>
PDF-version of this exercise sheet
Lecture for this exercise sheet (PDF)