Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 9/en

Aus Wikiversity
Zur Navigation springen Zur Suche springen



Warm-up-exercises

Exercise

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

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


Exercise

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

is linear.[1]


Exercise

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.


Exercise

Around the Earth along the equator is placed a ribbon. However, the ribbon is one meter longer than the equator, so that it is lifted up 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.


Exercise

Consider the linear map

such that

Compute


Exercise

Complete the proof of Theorem 9.3 to the compatibility with the scalar multiplication.


Exercise

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

be linear maps.

Prove that also the composite function

is a linear map.


Exercise

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.


Exercise

Prove that the functions

and

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

-linear?


Exercise

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

be a linear map.

Prove the following facts.

  1. Consider the subspace then also the image is a subspace of .
  2. In particular the image of the map is a subspace of .
  3. Consider the subspace then the preimage is a subspace of .
  4. In particular is a subspace of .


Exercise

Determine the kernel of the linear map


Exercise

Determine the kernel of the linear map

given by the matrix


Exercise

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


Exercise

Consider the function

which sends a rational number

into and all the irrational numbers into . Is this a linear map? Is it compatible with multiplication with a scalar?




Hand-in-exercises

Exercise (3 points)

Consider the linear map

such that

Compute


Exercise (3 points)

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


Exercise (3 points)

Determine the image and the kernel of the linear map


Exercise (3 points)

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

such that the image of is equal to .


Exercise (3 points)

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

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




Fußnoten
  1. Such a map is called a homothety with stretching or extension factor .
  2. Do not mind that there may exist negative numbers. In a mulled wine of course the ingredients do not come in 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.



<< | Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I | >>

PDF-Version dieses Arbeitsblattes (PDF englisch)

Zur Vorlesung (PDF)