Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 10

Exercise for the break

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.

Exercises

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a linear map. Prove that for all vectors ${}v_{1},\ldots ,v_{n}\in V$ and coefficients ${}s_{1},\ldots ,s_{n}\in K$, the relationship

${}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}\lambda _{i}\varphi (v_{i})\,$

holds.

Exercise

Linear mapping/0/Exercise

Exercise

Linear mapping/Negative vector/Exercise

Exercise

Proportionality/Gold price/Exercise

Exercise

Proportionality/Bread price/Exercise

Exercise

Proportionality/Velocity/Exercise

Exercise

Proportionality/Pedestrian/Exercise

Exercise

Interpret the following physical laws as linear functions from ${}\mathbb {R}$ to ${}\mathbb {R}$. 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

Lucy Sonnenschein/Bicycle in train/1/Exercise

Exercise

Matrix/Linear mapping/Exercise

Exercise

Let ${}K$ be a field and let ${}V$ be a ${}K$-vector space. Prove that for ${}v\in V$ the map

$K\longrightarrow V,\lambda \longmapsto \lambda v,$

is linear.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Prove that for ${}a\in K$ the map

$V\longrightarrow V,v\longmapsto av,$

is linear.

Exercise

Let ${}K$ be a field, and let ${}U,V,W$ be vector spaces over ${}K$. Let ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon V\rightarrow W$ be linear maps. Prove that also the composite mapping

$\psi \circ \varphi \colon U\longrightarrow W$

is a linear map.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a bijective linear map. Prove that also the inverse map

$\varphi ^{-1}\colon W\longrightarrow V$

is linear.

Exercise

Consider the linear map

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R}$

such that

$\varphi {\begin{pmatrix}1\\3\end{pmatrix}}=5{\text{ and }}\varphi {\begin{pmatrix}2\\-3\end{pmatrix}}=4.$

Compute

$\varphi {\begin{pmatrix}7\\6\end{pmatrix}}.$

Exercise

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

Exercise

Cookies/Linear mapping/Exercise

Thee following exercise relates to the exercises on the four number problem of exercise sheet 2.

Exercise

We consider the mapping

$\Psi \colon \mathbb {R} _{\geq 0}^{4}\longrightarrow \mathbb {R} _{\geq 0}^{4},$

which assigns for a four-tuple ${}(a,b,c,d)$ the four-tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

Describe this mapping with a matrix, under the condition

${}a\leq b\leq c\leq d\,.$

Exercise

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

Exercise

Elementary-geometric mappings/Inverse mappings/Exercise

Exercise

Prove that the functions

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Re} \,{\left(z\right)},$

and

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Im} \,{\left(z\right)},$

are ${}\mathbb {R}$-linear maps. Prove that also the complex conjugation is ${}\mathbb {R}$-linear, but not ${}\mathbb {C}$-linear. Is the absolute value

$\mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \vert {z}\vert ,$

${}\mathbb {R}$-linear?

Exercise

Matrix multiplication/Composition/Fact/Proof/Exercise

Exercise

Complete the proof of the theorem on determination on basis to the compatibility with the scalar multiplication.

Exercise

Vector space/Finite family/Linearity of substitution/Exercise

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}$ be a family of vectors in ${}V$. Consider the map

$\varphi \colon K^{n}\longrightarrow V,(s_{1},\ldots ,s_{n})\longmapsto \sum _{i=1}^{n}s_{i}v_{i},$

and prove the following statements.

1. ${}\varphi$ is injective if and only if ${}v_{1},\ldots ,v_{n}$ are linearly independent.
2. ${}\varphi$ is surjective if and only if ${}v_{1},\ldots ,v_{n}$ is a system of generators for ${}V$.
3. ${}\varphi$ is bijective if and only if ${}v_{1},\ldots ,v_{n}$ form a basis.

Exercise

Linear mappings/To product/Exercise

Exercise

Linear mappings/Product mapping/Linear/Exercise

Exercise

Let ${}K$ be a field and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}v_{1},\ldots ,v_{n}$ be a system of generators for ${}V$ and let ${}w_{1},\ldots ,w_{n}$ be a family of vectors in ${}W$.

a) Prove that there is at most one linear map

$\varphi \colon V\longrightarrow W$

such that ${}\varphi (v_{i})=w_{i}$ for all ${}i$.

b) Give an example of such a situation, where there is no linear mapping with ${}\varphi (v_{i})=w_{i}$ for all ${}i$.

Exercise

Linear mapping/Matrix/Commutative diagram/Fact/Proof/Exercise

Exercise

Consider the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$

which sends a rational number ${}q\in \mathbb {Q}$ to ${}q$, and all the irrational numbers to ${}0$. Is this a linear map? Is it compatible with multiplication by a scalar?

Exercise

Vector space/I to K/Contravariance in index set/Exercise

Exercise

Vector space/I to K/Direct/Co- and contravariance in index set/Exercise

Exercise

Vector space/Mapping set to K/Disjoint union/Exercise

Exercise

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces. Show that ${}V$ and ${}W$ are isomorphic to each other if and only if their dimension coincides.

Exercise

Finite field/Number of linear mappings/Exercise

Hand-in-exercises

Exercise (3 marks)

Consider the linear map

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}$

such that

$\varphi {\begin{pmatrix}2\\1\\3\end{pmatrix}}={\begin{pmatrix}4\\7\end{pmatrix}},\,\varphi {\begin{pmatrix}0\\4\\2\end{pmatrix}}={\begin{pmatrix}1\\1\end{pmatrix}}{\text{ and }}\varphi {\begin{pmatrix}3\\1\\1\end{pmatrix}}={\begin{pmatrix}5\\0\end{pmatrix}}.$

Compute

$\varphi {\begin{pmatrix}4\\5\\6\end{pmatrix}}.$

Exercise Q/Addition/Linear mapping/Matrix/Exercise

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 ${}K$ be a field and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a linear map. Prove that the graph of the map is a subspace of the Cartesian product ${}V\times W$.

The next exercise uses the following definition.

Let ${}(G,\circ ,e_{G})$ and ${}(H,\circ ,e_{H})$ denote groups. A mapping

$\psi \colon G\longrightarrow H$

is called group homomorphism, if the equality

${}\psi (g\circ g')=\psi (g)\circ \psi (g')\,$

holds for all

${}g,g'\in G$.

Exercise Vector spaces/Group homomorphism/Q-linear/Exercise

Vector spaces/Group homomorphism/Q-linear/Exercise

Exercise Linear subspaces/Same dimension/Automorphism/Exercise

Linear subspaces/Same dimension/Automorphism/Exercise

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