Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 10
- Exercise for the break
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.
- 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.
Linear mapping/Negative vector/Exercise
Proportionality/Gold price/Exercise
Proportionality/Bread price/Exercise
Proportionality/Velocity/Exercise
Proportionality/Pedestrian/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.
- 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.
Lucy Sonnenschein/Bicycle in train/1/Exercise
Matrix/Linear mapping/Exercise
Let be a field and let be a -vector space. Prove that for the map
is linear.
Let be a field, and let be a -vector space. Prove that for the map
is linear.
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.
Consider the linear map
such that
Compute
Linear mapping/Basis/(0,1,2) to (3,-2) and (1,4,1) to (1,0) and (2,1,3) to (7,2)/(3,-5,4)/Exercise
Cookies/Linear mapping/Exercise
Thee following exercise relates to the exercises on the four number problem of exercise sheet 2.
We consider the mapping
which assigns for a four-tuple the four-tuple
Describe this mapping with a matrix, under the condition
Find, by elementary geometric considerations, a matrix describing a rotation by 45 degrees counter-clockwise in the plane.
Elementary-geometric mappings/Inverse mappings/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?
Matrix multiplication/Composition/Fact/Proof/Exercise
Complete the proof of the theorem on determination on basis to the compatibility with the scalar multiplication.
Vector space/Finite family/Linearity of substitution/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.
- 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.
Linear mappings/To product/Exercise
Linear mappings/Product mapping/Linear/Exercise
Let be a field and let and be -vector spaces. Let be a system of generators for and let be a family of vectors in .
a) Prove that there is at most one linear map
such that for all .
b) Give an example of such a situation, where there is no linear mapping with for all .
Linear mapping/Matrix/Commutative diagram/Fact/Proof/Exercise
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?
Vector space/I to K/Contravariance in index set/Exercise
Vector space/I to K/Direct/Co- and contravariance in index set/Exercise
Vector space/Mapping set to K/Disjoint union/Exercise
Let denote a field, and let and denote finite-dimensional -vector spaces. Show that and are isomorphic to each other if and only if their dimension coincides.
Finite field/Number of linear mappings/Exercise
- Hand-in-exercises
Exercise (3 marks)
Consider the linear map
such that
Compute
Q/Addition/Linear mapping/Matrix/Exercise
Exercise (3 marks)
Find, by elementary geometric considerations, a matrix describing a rotation by 30 degrees counter-clockwise in the plane.
Exercise ( marks)
Let be a field and let and be -vector spaces. Let
be a linear map. Prove that the graph of the map is a subspace of the Cartesian product .
The next exercise uses the following definition.
Vector spaces/Group homomorphism/Q-linear/Exercise
Linear subspaces/Same dimension/Automorphism/Exercise
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