Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 9
- 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.
- Mass is volume times density.
- Energy is mass times the calorific value.
- The distance is speed multiplied by time.
- Force is mass times acceleration.
- Energy is force times distance.
- Energy is power times time.
- Voltage is resistance times electric current.
- Charge is current multiplied by time.
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?
- An amoeba.
- An ant.
- A tit.
- A flounder.
- A boa constrictor.
- A guinea pig.
- A boa constrictor that has swallowed a guinea pig.
- A very good limbo dancer.
Consider the linear map
such that
Compute
Complete the proof of the theorem on determination on basis to 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.
- is injective if and only if are linearly independent.
- is surjective if and only if is a system of generators for .
- 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?
Let be a field, and let and be -vector spaces. Let
be a linear map. Prove the following facts.
- For a linear subspace , also the image is a linear subspace of .
- In particular, the image
of the map is a subspace of .
- For a linear subspace , also the preimage is a linear subspace of .
- 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
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?
- 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 comsider as the price function, and as the caloric function. Determine a basis for , one for and one for .