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

Exercise for the break

Exercise

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.

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 *

Let

$\varphi \colon V\longrightarrow W$

be a linear mapping between the ${}K$-vector spaces ${}V$ and ${}W$. Show that ${}\varphi (0)=0$ holds.

Exercise

Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping between the ${}K$-vector spaces ${}V$ and ${}W$. Let ${}v\in V$. Show that ${}\varphi (-v)=-\varphi (v)$ holds.

Exercise

The price of an ounce of gold is ${}1100$ €.

a) What is the price for seven ounces of gold?

b) How much gold do we get for ${}10000$ €?

Exercise

Lucy Sonnenschein rides her bicycle with 10 meter per second.

a) How many kilometers does she ride per hour?

b) How long does it take for her to ride 100 kilometers?

Exercise

Five pedestrians walk a certain distance in 35 minutes. The next day, seven pedestrians walk the same distance with the same speed. How long does it take for them?

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

Let ${}M={\left(a_{ij}\right)}_{1\leq i\leq m,\,1\leq j\leq n}$ be an ${}m\times n$-matrix over a field ${}K$. Show that the corresponding mapping

$K^{n}\longrightarrow K^{m},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto M{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}s_{j}\\\sum _{j=1}^{n}a_{2j}s_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}s_{j}\end{pmatrix}},$

is linear.

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

Let a linear mapping

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

satisfying

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

be given. Compute

$\varphi {\begin{pmatrix}3\\-5\\4\end{pmatrix}}.$

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

Exercise

We consider the mapping

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

that assigns to 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 by 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

Describe the inverse mappings of the elementary-geometric mappings such as reflection at an axis, point reflection, rotation, homothety, translation.

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 also that 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

Let ${}B$ be an ${}n\times p$-Matrix, and ${}A$ be an ${}m\times n$-matrix. Let

$K^{p}{\stackrel {B}{\longrightarrow }}K^{n}{\stackrel {A}{\longrightarrow }}K^{m}$

denote the corresponding linear mappings. Show that the matrix product ${}A\circ B$ describes the composition of the linear mappings.

Exercise

Complete the proof of the theorem on determination on basis, by proving the compatibility with the scalar multiplication.

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 ${}V$. Show that the mapping

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

is linear.

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

Let ${}K$ be a field, and let ${}U,V,W$ denote vector spaces over ${}K$. Let ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon U\rightarrow W$ be linear mappings. Show that the mapping

$U\longrightarrow V\times W,u\longmapsto \varphi (u)\times \psi (u)$

into the product space ${}V\times W$ is also a linear mapping.

Exercise

Let ${}K$ be a field. For ${}i\in \{1,\ldots ,n\}$, let ${}K$-vector spaces ${}V_{i}$ and ${}W_{i}$, and linear mappings

$\varphi _{i}\colon V_{i}\longrightarrow W_{i}$

be given. Show that the product mapping

$\varphi =\varphi _{1}\times \varphi _{2}\times \cdots \times \varphi _{n}\colon V_{1}\times V_{2}\times \cdots \times V_{n}\longrightarrow W_{1}\times W_{2}\times \cdots \times W_{n},\left(v_{1},\,v_{2},\,\ldots ,\,v_{n}\right)\longmapsto \left(\varphi _{1}(v_{1}),\,\varphi _{2}(v_{2}),\,\ldots ,\,\varphi _{n}(v_{n})\right),$

is also a linear mapping between the product spaces.

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

Proof Lemma 10.14 .

Exercise

Consider the function

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

that 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

Let ${}K$ be a field, and let ${}S$ and ${}T$ be sets. Show that a mapping

$\psi \colon S\longrightarrow T$

defines a linear mapping

$\operatorname {Map} \,{\left(T,K\right)}\longrightarrow \operatorname {Map} \,{\left(S,K\right)},\varphi \longmapsto \varphi \circ \psi .$

Exercise

Let ${}K$ be a field, and let ${}S$ and ${}T$ denote sets. Let

$\psi \colon S\longrightarrow T$

be a mapping.

a) Show that, by ${}e_{s}\mapsto e_{\psi (s)}$, a linear mapping

$K^{(S)}\longrightarrow K^{(T)}$

is determined.

b) Suppose now that ${}\psi$ has also the property that all its fibers are finite. Show that this defines a linear mapping

$K^{(T)}\longrightarrow K^{(S)},\varphi \longmapsto \varphi \circ \psi .$

Exercise

Let ${}K$ be a field, and let ${}I$ denote an index set, together with a partition

${}I=I_{1}\uplus I_{2}\,.$

Show that there exists a natural isomorphism

${}\operatorname {Map} \,{\left(I,K\right)}\cong \operatorname {Map} \,{\left(I_{1},K\right)}\oplus \operatorname {Map} \,{\left(I_{2},K\right)}\,.$

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

Let ${}K$ be a finite field with ${}q$ elements. Determine the number of linear mappings

$\varphi \colon K^{n}\longrightarrow K^{m}.$

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 (2 marks)

Show that the addition

$+\colon \mathbb {Q} ^{2}=\mathbb {Q} \times \mathbb {Q} \longrightarrow \mathbb {Q}$

is a linear mapping. How does its matrix with respect to the standard basis look like?

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 (3 marks)

Let ${}V$ and ${}W$ ${}\mathbb {Q}$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

be a group homomorphism. Show that ${}\varphi$ is already ${}\mathbb {Q}$-linear.

Exercise (3 marks)

Let ${}V$ be a finite-dimensional ${}K$-vector space, and let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces of the same dimension. Show that there exists an ${}K$-automorphism

$\varphi \colon V\longrightarrow V$

such that

${}\varphi (U_{1})=U_{2}\,.$

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