Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 25/refcontrol
- Exercises
Exercise Create referencenumber
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.
- The customers of remain for with while of them goes to and the same percentage goes to .
- The customers of remain for with while of them goes to and goes to .
- 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 Create referencenumber
The newspapers and sell subscriptions, and they compete in a local market with customers. Within a year, one can observe the following movements.
- The subscribers of stick with a percentage of to , switch to , switch to and become nonreaders.
- The subscribers of stick with a percentage of to , switch to , switch to and become nonreaders.
- The subscribers of stick with a percentage of to , nobody switches to , switch to and become nonreaders.
- Among the nonreaders, subscribe to or , the rest remains nonreaders.
a) Establish the matrix, which describes the movement of customers within a year.
b) In a certain year, each of the three newspapers has subscribers and there are nonreaders. How does the distribution look like after a year?
c) The three newspapers expand to another city, where there are no newspapers at all so far, but also potential customers. How many subscribers does each newspaper have (and how many nonreaders) after three years, if the same movements hold in the new city?
===Exercise * Exercise 25.3
change===
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 Create referencenumber
Let be an -matrixMDLD/matrix and the corresponding linear mapping. Show that is surjectiveMDLD/surjective if and only if there exists an -matrix such that .
Exercise Create referencenumber
Let
a) Show
b) Determine the inverse matrixMDLD/inverse matrix of .
c) Solve the equation
Exercise Create referencenumber
Determine the inverse matrixMDLD/inverse matrix of
Exercise Create referencenumber
Determine the inverse matrix of
Exercise Create referencenumber
Determine the inverse matrix of
Exercise Create referencenumber
Determine the inverse matrix of the complex matrix
Exercise * Create referencenumber
a) Determine if the complex matrix
is invertible.
b) Find a solution to the inhomogeneous linear system of equations
Exercise Create referencenumber
Determine the inverse matrixMDLD/inverse matrix of
Exercise Create referencenumber
Prove that the matrix
for all is the inverse of itself.
Let be a field. We denote by the -matrix,MDLD/matrix with entry at the position and entry everywhere else. Then the following matrices are called elementary matrices.
- .
Exercise Create referencenumber
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.
- exchange of the -th and the -th row of .
- multiplication of the -th row of by .
- addition of -times the -th row of to the -th row ().
Exercise Create referencenumber
Describe what happens when a matrix is multiplied from the right by an elementary matrix.
Exercise Create referencenumber
Prove that the elementary matrices are invertible. What are the inverse matrices of the elementary matrices?
Exercise Create referencenumber
Show that the shear matrix
may be written as a matrix productMDLD/matrix product ,where and are diagonal matricesMDLD/diagonal matrices and is a shear matrix of the form .
Exercise Create referencenumber
Let
Find elementary matricesMDLD/elementary matrices such that is the identity matrix.
- Hand-in-exercises
===Exercise (6 (3+1+2) marks) Create referencenumber=== 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 marks) Create referencenumber
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 complex numbers with corresponding real matrices and . Prove that the matrix product is the real matrix corresponding to .
Exercise (3 marks) Create referencenumber
Compute the inverse matrixMDLD/inverse matrix of
Exercise (3 marks) Create referencenumber
Perform the procedure to find the inverse matrix of the matrix
under the assumption that .
Exercise (3 marks) Create referencenumber
Let
Find elementary matricesMDLD/elementary matrices such that is the identity matrix.
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