Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 6/en
- 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
(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
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
Exercise
Let be a field and an index set. Show that
Exercise
Let be a field and let
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 ).
- We have .
- We have .
- We have.
- 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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