# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 22

*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 a - matrix. Show that the matrix product of 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)?

Let

be a diagonal matrix, and an -matrix. Describe and .

Let

be a diagonal matrix, and an -tuple over a field , and let be a tuple of variables. What is specific about the system of linear equations

and how can you solve it?

Compute the product of matrices

according to the two possible parantheses.

For the following statement we will get soon a simpler proof via the relation between matrices and linear mappings.

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 .

Show that the matrix multiplication of square matrices is, in general, not commutative.

For a matrix , we denote by the -th fold composition
(matrix multiplication)
of with itself. is called the -th *power* of the matrix.

Compute, for the matrix

the powers

Out of the resources and , several commodities are produced. The following table shows, how much of the resources is needed to produce the commodities (always in suitable units).

a) Establish a matrix, which computes, applied to a four-tuple of commodities, the required resources.

b) The following table shows how much of each commodity shall be produced in a month.

What resources are necessary?

c) The following table shows how much of each resource is delivered on a certain day.

Which tuples of commodities can be produced from this without waste?

Determine (approximately) the coordinates of the sketched point (the side length of a box represents a unit).

Draw the following points in the Cartesian plane .

Let a point be given in the plane . Sketch the points

Let a point be given in the plane . Sketch the set of all points

Draw two points and in the Cartesian plane and add them.

Show that the product space , for a field , is, with componentwise addition and scalar multiplication, the properties

- ,
- ,
- ,

hold.

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

is also a -vector space.

Let be a vector space over a field . Let and . Show

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.

Check whether the following subsets of are linear subspaces:

- ,
- ,
- ,
- .

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?

Let be the set of all real -matrices

which fulfill the condition

Show that is not a linear subspace in the space of all -matrices.

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

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

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

Let

be the set of all real Cauchy sequences. Show that is a linear subspace of the space of all sequences

Show that the subset

is a linear subspace.

Show that the subset

is a linear subspace.

Show that the subset

is not a linear subspace.

*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 (4 marks)

Let . Find and prove a formula for the -th power of the matrix

### Exercise (2 marks)

Find, appart from the matrices and , four more matrices fulfilling the property .

### Exercise (3 marks)

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

- We have .
- We have .
- We have .
- If and then .

### Exercise (3 marks)

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

<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF) |
---|