Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 11/en

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Warm-up-exercises

Exercise

Determine explicitly the column rank and the row rank of the matrix

Describe linear dependencies (if they exist) between the rows and between the columns of the matrix.


Exercise

Show that the elementary operations on the rows do not change the column rank.


Exercise

Compute the determinant of the matrix


Exercise

Compute the determinant of the matrix


Exercise

Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.


Exercise

Check the multi-linearity and the property to be alternating, directly for the determinant of a -matrix.


Exercise

Let be the following square matrix

where and are square matrices. Prove that .


Exercise

Determine for which the matrix

is invertible.


Exercise

Linalg parallelogram area.png

Use the image to convince yourself that, given two vectors and , the determinant of the -matrix defined by these vectors is equal (up to sign) to the area of the plane parallelogram spanned by the vectors.


Exercise

Prove that you can develop the determinant according to each row and each column.


Exercise

Let be a field and . Prove that the transpose of a matrix satisfy the following properties (where , and ).

  1. .
  2. .
  3. .
  4. .


Exercise

Compute the determinant of the matrix

by developing the matrix along every column and along every row.


Exercise

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


Exercise

Let and let

be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map

.


The next exercises require the following definition.

Let be a field and let be a -vector space. For the linear map

is called the stretching

(or homothety) with extension factor .


Exercise

What is the determinant of a homothety?


Exercise

Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.


Exercise

Check the multiplication theorem for determinants of the following matrices




Hand-in-exercises

Exercise (4 points)

Let be a field and let and be vector spaces over of dimensions and . Let

be a linear map, described by the matrix with respect to two bases. Prove that


Exercise (3 points)

Compute the determinant of the matrix


Exercise (4 points)

Compute the determinant of the matrix


Exercise (2 points)

Compute the determinant of the elementary matrices.


Exercise (5 points)

Check the multiplication theorem for determinants of the following matrices



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