Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 24



Exercises

Determine the transformation matrices and , for the standard basis , and the basis in , which is given by


Determine the transformation matrices and , for the standard basis , and the basis of , which is given by the vectors


Let be a basis of a three-dimensional -vector space .

a) Show that is also a basis of .

b) Determine the transformation matrix .

c) Determine the transformation matrix .

d) Compute the coordinates with respect to the basis for the vector, which has the coordinates with respect to the basis .

e) Compute the coordinates with respect to the basis for the vector, which has the coordinates with respect to the basis .


Determine the transformation matrices and , for the standard basis , and the basis of , which is given by the vectors


We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point which has the coordinates with respect to the basis . What are the coordinates of this point with respect to the basis ?

c) Determine the transformation matrix which describes the change of bases from to .


We consider the linear map

Let be the subspace of , defined by the linear equation , and let be the restriction of on . On , there 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 ).


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

be a linear map. Prove that for all vectors and coefficients , the relationship

holds.


Let be a field, and let be a -vector space. Prove that for the map

is linear.[1]


Interpret the following physical laws as linear functions from to . Establish, in each situation, what is the measurable variable and what is the proportionality factor.

  1. Mass is volume times density.
  2. Energy is mass times the calorific value.
  3. The distance is speed multiplied by time.
  4. Force is mass times acceleration.
  5. Energy is force times distance.
  6. Energy is power times time.
  7. Voltage is resistance times electric current.
  8. Charge is current multiplied by time.


Around the Earth along the equator, a ribbon is placed. However, the ribbon is one meter longer than the equator, so that it is lifted uniformly all around to be tense. Which of the following creatures can run/fly/swim/dance under it?

  1. An amoeba.
  2. An ant.
  3. A tit.
  4. A flounder.
  5. A boa constrictor.
  6. A guinea pig.
  7. A boa constrictor that has swallowed a guinea pig.
  8. A very good limbo dancer.


Suppose that a linear function

has for the value . What is its vale for ?


Which of the following functions are linear?

  1. The real exponential function.
  2. The zero function.
  3. The constant function with value .
  4. The squaring function .
  5. The function which halves every real number.
  6. The function which subtracts from every real number.


Which of the following geometric shapes can be the image of a square under a linear mapping from to ?


Consider the linear map

such that

Compute


Complete the proof of the theorem on determination on basis, by proving the compatibility with the scalar multiplication.


Lucy Sonnenschein works as a bicycle messenger, she earns € per hour. At the fruit market, the price (per gram) for raspberries is €, for strawberries the price is €, and for apples the price is €. Describe the mapping, which assigns to any purchase of fruits, the time, how long Lucy has to work for it, as a composition of linear mappings.


Let be a field, and let be vector spaces over . Let and be linear maps. Prove that also the composite mapping

is a linear map.


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

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

is linear.


Let be a field, and let be a -vector space. Let be a family of vectors in . Consider the map

and prove the following statements.

  1. is injective if and only if are linearly independent.
  2. is surjective if and only if is a system of generators for .
  3. is bijective if and only if form a basis.


Prove that the functions

and

are -linear maps. Prove that also the complex conjugation is -linear, but not -linear. Is the absolute value

-linear?


Consider the function

which sends a rational number to , and all the irrational numbers to . Is this a linear map? Is it compatible with multiplication by a scalar?


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

be a linear map. Prove the following facts.

  1. For a linear subspace , also the image is a linear subspace of .
  2. In particular, the image

    of the map is a subspace of .

  3. For a linear subspace , also the preimage is a linear subspace of .
  4. In particular, is a subspace of .


Find, by elementary geometric considerations, a matrix describing a rotation by 45 degrees counter-clockwise in the plane.


Prove the addition theorems for sine and cosine, using the rotation matrices.


Determine the kernel of the linear map


Determine the kernel of the linear map

given by the matrix


How does the graph of a linear mapping

look like? How can you see in a sketch of the graph the kernel of the map?


Let be an -matrix over the field , let be the corresponding linear mapping, and let denote (depending on a vector ) the corresponding system of linear equations. Show that the solution set of the system equals the preimage of under the linear mapping .


Let and be vector spaces over a field , and let be linear mappings. Show that the mapping, defined by

is also linear.


Give an example for a linear mapping

which is not injective, but such that its restriction

is injective.


We consider the mapping

which assigns to a four-tuple the four-tuple

Describe this mapping with a matrix, under the condition




Hand-in-exercises

Exercise (3 marks)

Consider the linear map

such that

Compute


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

We consider the families of vectors

in .

a) Show that and are both a basis of .

b) Let denote the point which has the coordinates with respect to the basis . What are the coordinates of this point with respect to the basis ?

c) Determine the transformation matrix which describes the change of basis from to .


Exercise (3 marks)

Sketch the image of the pictured circles under the linear mapping given by the matrix from to itself.


Exercise (3 marks)

Find, by elementary geometric considerations, a matrix describing a rotation by 30 degrees counter-clockwise in the plane.


Exercise (3 marks)

Determine the image and the kernel of the linear map


Exercise (3 marks)

Let be the plane defined by the linear equation . Determine a linear map

such that the image of is equal to .


Exercise (3 marks)

On the real vector space of mulled wines, we consider the two linear maps

and

We consider as the price function, and as the caloric function. Determine a basis for , one for and one for .[2]




Footnotes
  1. Such a mapping is called a homothety, or a dilation with scale factor .
  2. Do not mind that there may exist negative numbers. In a mulled wine, of course the ingredients do not enter with a negative coefficient. But if you would like to consider, for example, in how many ways you can change a particular recipe, without changing the total price or the total amount of energy, then the negative entries make sense.


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