- 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
-
![{\displaystyle {}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}\lambda _{i}\varphi (v_{i})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ff2db7d80c6e142750a0df29850256e14f0139)
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
-
![{\displaystyle {}\operatorname {Im} \varphi =\varphi (V)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0120cfbb7c2866625316e78e6a8b6d3d14df64d5)
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
-
![{\displaystyle {}M={\begin{pmatrix}2&3&0&-1\\4&2&2&5\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d362d40da5b4ba48822f7a42f28399ee3fe164d0)
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
Consider the linear map
-
such that
-
Compute
-
Find, by elementary geometric considerations, a matrix describing a rotation by 30 degrees counter-clockwise in the plane.
Determine the image and the kernel of the linear map
-
Let
be the plane defined by the linear equation
.
Determine a linear map
-
such that the image of
is equal to
.
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
.