Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 30

Exercise for the break

Exercise

Let ${}E$ be an affine space of dimension ${}d$, and let ${}F,G\subseteq E$ be affine subspaces of dimension ${}r$ and ${}s$, respectively. Show that ${}F\cap G=\emptyset$ is either empty, or that its dimension is at least ${}r+s-n$.

Exercises

Exercise

Check whether the points

${\begin{pmatrix}5\\4\\7\end{pmatrix}},\,{\begin{pmatrix}-2\\1\\6\end{pmatrix}},\,{\begin{pmatrix}3\\-9\\4\end{pmatrix}},\,{\begin{pmatrix}-8\\8\\3\end{pmatrix}}$

in ${}\mathbb {R} ^{3}$ are affinely independent.

Exercise

Let ${}E$ be an affine space over a ${}K$-vector space ${}V$, and let

$P_{1},\ldots ,P_{n}$

denote a finite family of points in ${}E$. Show that the following statements are equivalent.

1. The points ${}P_{1},\ldots ,P_{n}$ are affinely independent.
2. For every ${}i\in \{1,\ldots ,n\}$, the family of vectors

   ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$

   is linearly independent.

3. There exists some ${}i\in \{1,\ldots ,n\}$ such that the family of vectors

   ${\overrightarrow {P_{i}P_{1}}},\ldots ,{\overrightarrow {P_{i}P_{i-1}}},\,{\overrightarrow {P_{i}P_{i+1}}},\ldots ,{\overrightarrow {P_{i}P_{n}}}$

   is linearly independent.

4. The points ${}P_{1},\ldots ,P_{n}$ form an affine basis in the affine subspace generated by them.

Exercise

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}P_{1},\ldots ,P_{n}$ be a finite family of points in ${}E$. Show that the following statements are equivalent.

1. The points form an affine basis of ${}E$.
2. The points form a minimal affine generating system of ${}E$.
3. The points are maximally affinely independent.

Exercise

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$, and let ${}P_{1},\ldots ,P_{n}$ be a finite family of points from ${}E$. Show that these points form an affine basis of ${}E$ if and only if they are affinely independent, and they are an affine generating system for ${}E$.

Exercise

Determine the polynomials ${}P\in \mathbb {R} [X]$ that define an affine-linear mapping

$P\colon \mathbb {R} \longrightarrow \mathbb {R} .$

Exercise

Let ${}K$ be a field, and let ${}E$ and ${}F$ denote affine spaces over the vector spaces ${}V$ and ${}W$, respectively. Let a mapping

$\psi \colon E\longrightarrow F,$

a linear mapping

$\varphi \colon V\longrightarrow W,$

and a point ${}P\in E$ be given such that

${}\psi (P+v)=\psi (P)+\varphi (v)\,$

holds for all ${}v\in V$. Show that ${}\psi$ is affine-linear.

Exercise

Let ${}\psi \colon \mathbb {R} \rightarrow \mathbb {R}$ be a mapping of the form

${}\psi (x)=ax+b\,$

with certain ${}a,b\in \mathbb {R}$. Show directly that ${}\psi$ is compatible with barycentric combinations.

Exercise

Determine, by a drawing, the image point of ${}P$ under the affine mapping ${}\varphi$ given by ${}\varphi (P_{i})=Q_{i}$.

Exercise

Determine, by a drawing, the image point of ${}P$ under the affine mapping ${}\varphi$ that is determined by ${}\varphi (P_{i})=Q_{i}$.

Exercise

Describe the affine plane

${}E={\left\{{\begin{pmatrix}3\\1\\4\end{pmatrix}}+s{\begin{pmatrix}2\\7\\-6\end{pmatrix}}+t{\begin{pmatrix}-1\\5\\1\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$

as the preimage over ${}1$ of an affine mapping ${}\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.

Exercise

Describe the affine line

${}G={\left\{{\begin{pmatrix}6\\2\\3\end{pmatrix}}+s{\begin{pmatrix}-2\\5\\4\end{pmatrix}}\mid s\in \mathbb {R} \right\}}\,$

as the preimage over ${}(1,0)$ of an affine mapping ${}\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{2}$.

Exercise

Let ${}E$ and ${}F$ be affine spaces over the field ${}K$. Show that the projections

$E\times F\longrightarrow E$

and

$E\times F\longrightarrow F$

are affine mappings.

Exercise

Let ${}E$ and ${}F$ be affine spaces over the field ${}K$. Show that the spaces are isomorphic if and only if their dimensions coincide.

Exercise

Let ${}E$ be an affine space, and let ${}P_{1},\ldots ,P_{n}$ be a finite family of points in ${}E$. Let

${}F={\left\{\left(a_{1},\,\ldots ,\,a_{n}\right)\in K^{n}\mid \sum _{i=1}^{n}a_{i}=1\right\}}\subset K^{n}\,.$

Show that the assignment

$\left(a_{1},\,\ldots ,\,a_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}P_{i}$

defines a well-defined affine-linear mapping from ${}F$ to ${}E$.

Exercise

Let

$\varphi \colon E\longrightarrow F$

be an affine-linear mapping between the affine spaces ${}E$ and ${}F$ over ${}K$. Show that, for every affine subspace ${}H\subseteq E$, the image ${}\varphi (H)$ is an affine subspace of ${}F$.

Exercise

Let

$\psi \colon E\longrightarrow E$

be an affine mapping on the affine space ${}E$. Show that the linear part ${}\psi _{0}$ is the identity if and only if ${}\psi$ is a translation.

Exercise

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$. Show that the mapping that assigns to an affine mapping

$\psi \colon E\longrightarrow E$

its linear part ${}\psi _{0}$, satisfies the following properties.

1. ${}{\left(\operatorname {Id} _{E}\right)}_{0}=\operatorname {Id} _{V}\,,$

2. ${}(\psi \circ \varphi )_{0}=\psi _{0}\circ \varphi _{0}\,.$

Exercise

Let ${}E$ and ${}F$ be affine spaces over the field ${}K$, let ${}P_{1},\ldots ,P_{n}\in E$ be an affine basis of ${}E$, and let ${}Q_{1},\ldots ,Q_{n}\in F$ denote points. Let

$\psi \colon E\longrightarrow F$

be the corresponding affine-linear mapping with

${}\psi (P_{i})=Q_{i}\,.$

Show the following statements.

a) ${}\psi$ is bijective if and only if ${}Q_{1},\ldots ,Q_{n}$ is an affine basis of ${}F$.

b) ${}\psi$ is injective if and only if ${}Q_{1},\ldots ,Q_{n}$ are affinely independent.

c) ${}\psi$ is surjective if and only if ${}Q_{1},\ldots ,Q_{n}$ is an affine generating system of ${}F$.

Exercise

Let

$\varphi \colon E\longrightarrow F$

be an affine-linear mapping between the affine spaces ${}E$ and ${}F$ over ${}K$. Show that the preimages ${}\varphi ^{-1}(Q)$ for all ${}Q\in F$ are parallel to each other.

Exercise

Compare several concepts for vector spaces and affine spaces, including their mappings.

Hand-in-exercises

Exercise (3 marks)

Let

$\varphi \colon E\longrightarrow F$

be an affine-linear mapping between the affine spaces ${}E$ and ${}F$ over ${}K$. Show that, for every affine subspace ${}G\subseteq F$, the preimage ${}\varphi ^{-1}(G)$ is an affine subspace of ${}E$.

Exercise (3 marks)

Describe the affine plane

${}E={\left\{{\begin{pmatrix}5\\6\\-2\end{pmatrix}}+s{\begin{pmatrix}3\\-4\\8\end{pmatrix}}+t{\begin{pmatrix}5\\4\\7\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$

as the preimage over ${}1$ of an affine mapping ${}\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.

Exercise (2 marks)

Let ${}E$ be an affine space of dimension ${}n$, and let

$\psi \colon E\longrightarrow E$

denote an affine mapping. Let ${}P_{1},\ldots ,P_{n+1}\in E$ be affinely independent points, and suppose that they are also fixed points of ${}\psi$. Show that ${}\psi$ is the identity.

Exercise (6 (3+2+1) marks)

Let

$\varphi \colon E\longrightarrow F$

be an affine-linear mapping between the affine spaces ${}E$ and ${}F$ over ${}K$.

a) Show that the graph ${}G$ of ${}\varphi$ is an affine subspace of the product space ${}E\times F$.

b) Show that the mapping

$\psi \colon E\longrightarrow G,P\longmapsto (P,\varphi (P)),$

is an isomorphism of affine spaces.

c) Show that

${}\varphi =p_{2}\circ \psi \,$

holds, where ${}p_{2}$ is the projection onto ${}F$.

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