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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 6

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Warm-up-exercises

Compute the following product of matrices


Compute, over the complex numbers, the following product of matrices


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.


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


Compute the product of matrices

according to the two possible parentheses.


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

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 itself. This is also called the -th power of the matrix.

Compute, for the matrix

the powers


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

is also a -vector space.


Let be a field, and an index set. Show that

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


Let be a field, and let

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


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


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




Hand-in-exercises

Exercise (3 marks)

Compute, over the complex numbers, the following product of matrices


Exercise (3 marks)

We consider the matrix

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


Exercise (3 marks)

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

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


Exercise (3 marks)

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