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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 9

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Warm-up-exercises

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.


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.


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.


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

-linear?


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 .


Determine the kernel of the linear map


Determine the kernel of the linear map

given by the matrix


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


Consider the function

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




Hand-in-exercises

Exercise (3 marks)

Consider the linear map

such that

Compute


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 .