Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 10/en

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Warm-up-exercises

Exercise

The telephone companies and compete for a market, where the market customers in a year are given by the customers-tuple (where is the number of customers of in the year etc.). There are customers passing from one provider to another one during a year.

  1. The customers of remain for with while of them goes to and the same percentage goes to .
  2. The customers of remain for with while of them goes to and goes to .
  3. The customers of remain for with while of them goes to and goes to .

a) Determine the linear map (i.e. the matrix), which expresses the customers-tuple with respect to .

b) Which customers-tuple arises from the customers-tuple within one year?

c) Which customers-tuple arises from the customers-tuple in four years?


Exercise

Let be a field and let and be vector spaces over of dimensions and . Let

be a linear map, described by the matrix with respect to two bases. Prove that is surjective if and only if the columns of the matrix form a system of generators for .


Exercise

Let be a field and let and be two -vector spaces. Let

be a bijective linear map. Prove that also the inverse map

is linear.


Exercise

Determine the inverse matrix of


Exercise

Determine the inverse matrix of


Exercise

Determine the inverse matrix of the complex matrix


Exercise

a) Determine if the complex matrix

is invertible.

b) Find a solution to the inhomogeneous linear system of equations


Exercise

Prove that the matrix

for all is the inverse of itself.


Exercise

We consider the linear map

Let be the subspace of defined by the linear equation , and let be the restriction of on . On are given vectors of the form

Compute the "change of basis" matrix between the bases

of and the transformation matrix of with respect to these three bases

(and the standard basis of ).


Exercise

Prove that the elementary matrices are invertible. What are the inverse matrices of the elementary matrices?


Exercise

Let be a field and a -matrix with entries in . Prove that the multiplication by the elementary matrices from the left with M has the following effects.

  1. exchange of the -th and the -th row of .
  2. multiplication of the -th row of by .
  3. addition of -times the -th row of to the -th row ().


Exercise

Describe what happens when a matrix is multiplied from the right by an elementary matrix.




Hand-in-exercises

Exercise (3 points)

Compute the inverse matrix of


Exercise (3 points)

Perform the procedure to find the inverse matrix of the matrix

under the assumption that .


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

An animal population consists of babies (first year), freshers (second year), Halbstarke (third year), mature ones (fourth year) and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year is given by a -tuple .

During a year of the babies become freshers, of the freshers become Halbstarke, of the Halbstarken become mature ones and of the mature ones reach the fifth year.

Babies and freshes can not reproduce yet, then they reach the sexual maturity and Halbstarke generate new pets and of the mature ones generate new babies, and the babies are born one year later.

a) Determine the linear map (i.e. the matrix), which expresses the total stock with respect to the stock .

b) What will happen to the stock in the next year?

c) What will happen to the stock in five years?


Exercise (3 points)

Let be a complex number and let

be the multiplication map, which is a -linear map. How does the matrix related to this map with respect to the real basis

and look like? Let and be two complex numbers with corresponding real matrices and . Prove that the matrix product is the real matrix corresponding to .




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