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

Exercise for the break

Exercise Create referencenumber

Let ${}E$ be an affine spaceMDLD/affine space of dimensionMDLD/dimension (affine) ${}d$, and let ${}F,G\subseteq E$ be affine subspacesMDLD/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 Create referencenumber

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.MDLD/affinely independent

===Exercise Exercise 30.3

change===

Let ${}E$ be an affine spaceMDLD/affine space over a ${}K$-vector spaceMDLD/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.MDLD/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.MDLD/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 basisMDLD/affine basis in the affine subspaceMDLD/affine subspace generatedMDLD/generated (affine) by them.

Exercise Create referencenumber

Let ${}E$ be an affine spaceMDLD/affine space over the ${}K$-vector spaceMDLD/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 basisMDLD/affine basis of ${}E$.
2. The points form a minimal affine generating systemMDLD/affine generating system of ${}E$.
3. The points are maximally affinely independent.MDLD/affinely independent

Exercise Create referencenumber

Let ${}E$ be an affine spaceMDLD/affine space over the ${}K$-vector spaceMDLD/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 basisMDLD/affine basis of ${}E$ if and only if they are affinely independent,MDLD/affinely independent and they are an affine generating systemMDLD/affine generating system for ${}E$.

Exercise Create referencenumber

Determine the polynomialsMDLD/polynomials ${}P\in \mathbb {R} [X]$, which define an affine-linear mappingMDLD/affine-linear mapping

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

===Exercise Exercise 30.7

change===

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

$\psi \colon E\longrightarrow F,$

a linear mappingMDLD/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.MDLD/affine-linear

Exercise Create referencenumber

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.MDLD/barycentric combinations

Exercise Create referencenumber

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

Exercise Create referencenumber

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

Exercise Create referencenumber

Describe the affine planeMDLD/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 preimageMDLD/preimage over ${}1$ of an affine mappingMDLD/affine mapping ${}\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.

Exercise Create referencenumber

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

Exercise Create referencenumber

Let ${}E$ and ${}F$ be affine spacesMDLD/affine spaces over the fieldMDLD/field ${}K$. Show that the projectionsMDLD/projections (product set)

$E\times F\longrightarrow E$

and

$E\times F\longrightarrow F$

are affine mappings.MDLD/affine mappings

Exercise Create referencenumber

Let ${}E$ and ${}F$ be affine spacesMDLD/affine spaces over the fieldMDLD/field ${}K$. Show that the spaces are isomorphicMDLD/isomorphic (affine) if and only if their dimensionsMDLD/dimensions (affine space) coincide.

Exercise Create referencenumber

Let ${}E$ be an affine space,MDLD/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 mappingMDLD/affine-linear mapping from ${}F$ to ${}E$.

Exercise Create referencenumber

Let

$\varphi \colon E\longrightarrow F$

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

Exercise Create referencenumber

Let

$\psi \colon E\longrightarrow E$

be an affine mappingMDLD/affine mapping on the affine spaceMDLD/affine space ${}E$. Show that the linear partMDLD/linear part (affine) ${}\psi _{0}$ is the identityMDLD/identity if and only if ${}\psi$ is a translation.MDLD/translation (affine)

Exercise Create referencenumber

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

$\psi \colon E\longrightarrow E$

its linear partMDLD/linear part (affine) ${}\psi _{0}$, 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 Create referencenumber

Let ${}E$ and ${}F$ be affine spacesMDLD/affine spaces over the fieldMDLD/field ${}K$, let ${}P_{1},\ldots ,P_{n}\in E$ be an affine basisMDLD/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 mappingMDLD/affine-linear mapping with

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

Show the following statements.

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

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

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

Exercise Create referencenumber

Let

$\varphi \colon E\longrightarrow F$

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

Exercise Create referencenumber

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

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Let

$\varphi \colon E\longrightarrow F$

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

Exercise (3 marks) Create referencenumber

Describe the affine planeMDLD/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 preimageMDLD/preimage over ${}1$ of an affine mappingMDLD/affine mapping ${}\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.

Exercise (2 marks) Create referencenumber

Let ${}E$ be an affine spaceMDLD/affine space of dimensionMDLD/dimension (affine space) ${}n$, and let

$\psi \colon E\longrightarrow E$

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

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

Let

$\varphi \colon E\longrightarrow F$

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

a) Show that the graphMDLD/graph (map) ${}G$ of ${}\varphi$ is an affine subspaceMDLD/affine subspace of the product spaceMDLD/product space (affine) ${}E\times F$.

b) Show that the mapping

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

is an isomorphismMDLD/isomorphism (affine) of affine spaces.

c) Show that

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

holds, where ${}p_{2}$ is the projectionMDLD/projection (product set) onto ${}F$.

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