Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 6/en

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Warm-up-exercises

Exercise

Compute the following product of matrices


Exercise

Compute over the complex numbers the following product of matrices


Exercise

Determine the product of matrices

where the -th standard vector

(of length ) is considered as a row vector and the -th standard vector (also of length ) is considered as a column vector.


Exercise

Let be a - matrix. Show that the product of matrices , with the -th standard vector (regarded as column vector) is the -th column of . What is , where is the -th standard vector (regarded as a row vector)?


Exercise

Compute the product of matrices

according to the two possible parantheses.

For the following statements we will soon give a simple proof using the relationship between matrices and linear maps.

Exercise

Show that the multiplication of matrices is associative. More precisely: Let be a field and let be an -matrix, an -matrix and a -matrix over . Show that .


For a matrix we denote by the -th product of with iself. This is also called the -th power of the matrix.

Exercise

Compute for the matrix

the powers


Exercise

Let be a field and let and be two vector spaces over . Show that the product

is also a -vector space.


Exercise

Let be a field and an index set. Show that

with pointwise addition and scalar multiplication is a -vector space.


Exercise

Let be a field and let

be a system of linear equations over . Show that the set of all solutions of the system is a subspace of . How is this solution space related to the solution spaces of the individual equations?


Exercise

Show that the addition and the scalar multiplication of a vector space can be restricted to a subspace and that this subspace with the inherited structures of is a vector space itself.


Exercise

Let be a field and let be a -vector space. Let be two subspaces of . Prove that the union is a subspace of if and only if or .




Hand-in-exercises

Exercise (3 points)

Compute over the complex numbers the following product of matrices


Exercise (4 points)

We consider the matrix

over a field . Show that the fourth power of is , that is


Exercise (3 points)

Let be a field and let be a -vector space. Show that the following properties hold (where and ).

  1. We have .
  2. We have .
  3. We have.
  4. If and then .


Exercise (3 points)

Give an example of a vector space and of three subsets of which satisfy two of the subspace axioms, but not the third.




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