Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 8/en

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Warm-up-exercises

Exercise

Let be a field and let be a -vector space of dimension . Suppose that vectors in are given. Prove that the following facts are equivalent.

  1. form a basis for .
  2. form a system of generators for .
  3. are linearly independent.


Exercise

Let be a field and let denote the polynomial ring over . Let . Show that the set of all polynomials of degree is a finite dimensional subspace of . What is its dimension?


Exercise

Show that the set of real polynomials of degree which have a zero at and a zero at is a finite dimensional subspace of . Determine the dimension of this vector space.


Exercise

Let be a field and let and be two finite-dimensional -vector spaces with and . What is the dimension of the Cartesian product ?


Exercise

Let be a finite-dimensional vector space over the complex numbers, and let be a basis of . Prove that the family of vectors

form a basis for , considered as a real vector space.


Exercise

Consider the standard basis in and the three vectors

Prove that these vectors are linearly independent and extend them to a basis by adding an appropriate standard vector as shown in Theorem 8.2. Can one take any standard vector?


Exercise

Determine the transformation matrices and for the standard basis and the basis in which is given by


Exercise

Determine the transformation matrices and for the standard basis and the basis of which is given by the vectors


Exercise

We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point which has the coordinates with respect to the basis . What are the coordinates of this point with respect to the basis ?

c) Determine the transformation matrix which describes the change of basis from to .




Hand-in-exercises

Exercise (4 points)

Show that the set of all real polynomials of degree which have a zero at , at and at is a finite dimensional subspace of . Determine the dimension of this vector space.


Exercise (3 points)

Let be a field and let be a -vector space. Let be a family of vectors in and let

be the subspace they span. Prove that the family is linearly independent if and only if the dimension of is exactly .


Exercise (4 points)

Determine the transformation matrices and for the standard basis and the basis of which is given by the vectors


Exercise (6 points)

We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point which has the coordinates with respect to the basis . What are the coordinates of this point with respect to the basis ?

c) Determine the transformation matrix which describes the change of basis from to .




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