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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 8

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Warm-up-exercises

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.


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 linear subspace of . What is its dimension?


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.


Let be a field, and let and be two finite-dimensional -vector spaces with

and

What is the dimension of the Cartesian product ?


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

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


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 the base exchange theorem. Can one take any standard vector?


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


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


We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point that 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 that describes the change of bases from to .




Hand-in-exercises

Exercise (4 marks)

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


Exercise (2 marks)

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

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


Exercise (4 marks)

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


Exercise (6 (3+1+2) marks)

We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point that 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 that describes the change of basis from to .