Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 11/refcontrol

Exercise for the break

Exercise Create referencenumber

Which of the following geometric shapes can be the image of a square under a linear mappingMDLD/linear mapping from ${}\mathbb {R} ^{2}$ to ${}\mathbb {R} ^{2}$?

Exercises

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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 the following facts.

1. For a linear subspace ${}S\subseteq V$, also the image ${}\varphi (S)$ is a linear subspace of ${}W$.
2. In particular, the image

   ${}\operatorname {Im} \varphi =\varphi (V)\,$

   of the map is a subspace of ${}W$.

3. For a linear subspace ${}T\subseteq W$, also the preimage ${}\varphi ^{-1}(T)$ is a linear subspace of ${}V$.
4. In particular, ${}\varphi ^{-1}(0)$ is a subspace of ${}V$.

Exercise * Create referencenumber

Determine the kernel of the linear map

$\varphi \colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{2},$

given by the matrix

${}M={\begin{pmatrix}2&3&0&-1\\4&2&2&5\end{pmatrix}}\,.$

Exercise * Create referencenumber

Determine the kernel of the linear map

$\mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{3},{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}\longmapsto {\begin{pmatrix}2&1&5&2\\3&-2&7&-1\\2&-1&-4&3\end{pmatrix}}{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}.$

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We consider the linear mappingMDLD/linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrixMDLD/matrix ${}{\begin{pmatrix}8&4\\5&9\end{pmatrix}}$.

1. Determine the imageMDLD/image of the line defined by the equation

   ${}5x-7y=0\,.$

2. Determine the preimageMDLD/preimage of the line given by the equation

   ${}6x-11y=0\,.$

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Let ${}K$ be a field,MDLD/field and let ${}V$ denote a finite-dimensionalMDLD/finite-dimensional ${}K$-vector space.MDLD/vector space Let ${}U\subseteq V$ be a linear subspace. Show that there exists a ${}K$-vector space ${}W$ and a surjective ${}K$-linear mappingMDLD/linear mapping

$\varphi \colon V\longrightarrow W$

such that ${}U=\operatorname {kern} \varphi$ holds.

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Let a linear mappingMDLD/linear mapping ${}\varphi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{2}$ as in Example 10.11 be given, in order to represent three-dimensional shapes in the plane. Imagine the preimageMDLD/preimage for a point ${}P\in \mathbb {R} ^{2}$! How do the corresponding line equations look like? What points ${}Q\in \mathbb {R} ^{3}$ have the same image point as the corner ${}(1,1,1)$ of the unit cube?

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Let ${}A={\begin{pmatrix}3&1&5\\-1&0&2\end{pmatrix}}$, and let

$f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},x\longmapsto A\cdot x$

denote the corresponding linear mapping.MDLD/linear mapping

1. Determine a basisMDLD/basis (vs) and the dimensionMDLD/dimension (vs) of ${}\operatorname {kern} f$, and of ${}\operatorname {Im} f$.
2. Find a linear subspaceMDLD/linear subspace ${}V\subseteq \mathbb {R} ^{3}$ such that

   ${}\mathbb {R} ^{3}=V\oplus \operatorname {kern} f\,$

   holds.

3. Does there also exist a linear subspace ${}U\subset \mathbb {R} ^{2}$, ${}U\neq 0$, such that ${}\mathbb {R} ^{2}=U\oplus \operatorname {Im} f$?

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Let ${}V$ be a finite-dimensionalMDLD/finite-dimensional ${}K$-vector space,MDLD/vector space and ${}f\colon V\rightarrow V$ be an endomorphism.MDLD/endomorphism Show that the following statements are equivalent.

1. ${}\operatorname {kern} f=\operatorname {kern} {\left(f\circ f\right)}$,
2. ${}\operatorname {kern} f\cap \operatorname {Im} f=\{0\}$,
3. ${}V=\operatorname {kern} f\oplus \operatorname {Im} f$,
4. ${}\operatorname {Im} f=\operatorname {Im} {\left(f\circ f\right)}$.

Exercise Create referencenumber

We consider the mappingMDLD/mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}-1.$

Show that for this mapping, neither the image of a linear subspaceMDLD/linear subspace ${}U\subseteq \mathbb {R}$ must be a linear subspace, nor the preimage of a linear subspace must be a linear subspace.

Let ${}K$ be a field,MDLD/field and let ${}V$ be a ${}K$-vector space.MDLD/vector space For a vector ${}v\in V$, the mappingMDLD/mapping

$V\longrightarrow V,x\longmapsto x+v,$

is called the translation by the vector ${}v$.

A translation is in general not a linear mapping, as ${}0$ is not sent to ${}0$. It is, however, an affine-linear mapping,MDLD/affine-linear mapping we will come back later to this concept.

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Let ${}F\subseteq \mathbb {R} ^{n}$ be a "geometric shape“, for example, a circle or a rectangle in the plane. Let

$\theta _{w}\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n},x\longmapsto x+w,$

be the translationMDLD/translation with the vector ${}w$, and let ${}F'=\theta _{w}(F)$ denote the translated shape. Let

$\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$

be a linear mapping.MDLD/linear mapping Show that the imageMDLD/image ${}\varphi (F')$ arises from the image ${}\varphi (F)$ by a translation.

Exercise Create referencenumber

How does the graphMDLD/graph (map) of a linear mappingMDLD/linear mapping

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

$g\colon \mathbb {R} \longrightarrow \mathbb {R} ^{2},$

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

look like? How can you see in a sketch of the graph the kernel of the map?

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Give an example for a linear mappingMDLD/linear mapping

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

which is not injective,MDLD/injective but such that its restrictionMDLD/restriction

$\mathbb {Q} ^{2}\longrightarrow \mathbb {R} ^{2}$

is injective.

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Let ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the linear mappingMDLD/linear mapping determined by the matrix ${}M={\begin{pmatrix}5&1\\2&3\end{pmatrix}}$ (with respect to the standard basis). Determine the describing matrix of ${}\varphi$ with respect to the basis ${}{\begin{pmatrix}1\\4\end{pmatrix}}$ and ${}{\begin{pmatrix}4\\2\end{pmatrix}}$.

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The telephone companies ${}A,B$ and ${}C$ compete for a market, where the market customers in a year ${}j$ are given by the customers-tuple ${}K_{j}=(a_{j},b_{j},c_{j})$ (where ${}a_{j}$ is the number of customers of ${}A$ in the year ${}j$ etc.). There are customers passing from one provider to another one during a year.

1. The customers of ${}A$ remain for ${}80\%$ with ${}A$, while ${}10\%$ of them goes to ${}B$, and the same percentage goes to ${}C$.
2. The customers of ${}B$ remain for ${}70\%$ with ${}B$, while ${}10\%$ of them goes to ${}A$, and ${}20\%$ goes to ${}C$.
3. The customers of ${}C$ remain for ${}50\%$ with ${}C$, while ${}20\%$ of them goes to ${}A$, and ${}30\%$ goes to ${}B$.

a) Determine the linear map (i.e. the matrix), which expresses the customers-tuple ${}K_{j+1}$ with respect to ${}K_{j}$.

b) Which customers-tuple arises from the customers-tuple ${}(12000,10000,8000)$ within one year?

c) Which customers-tuple arises from the customers-tuple ${}(10000,0,0)$ in four years?

Exercise Create referencenumber

The newspapers ${}A,B$ and ${}C$ sell subscriptions, and they compete in a local market with ${}100000$ customers. Within a year, one can observe the following movements.

1. The subscribers of ${}A$ stick with a percentage of ${}80\%$ to ${}A$, ${}10\%$ switch to ${}B$, ${}5\%$ switch to ${}C$ and ${}5\%$ become nonreaders.
2. The subscribers of ${}B$ stick with a percentage of ${}60\%$ to ${}B$, ${}10\%$ switch to ${}A$, ${}20\%$ switch to ${}C$ and ${}10\%$ become nonreaders.
3. The subscribers of ${}C$ stick with a percentage of ${}70\%$ to ${}C$, nobody switches to ${}A$, ${}10\%$ switch to ${}B$ and ${}20\%$ become nonreaders.
4. Among the nonreaders, ${}10\%$ subscribe to ${}A,B$ or ${}C$, the rest remains nonreaders.

a) Establish the matrix, which describes the movement of customers within a year.

b) In a certain year, each of the three newspapers has ${}20000$ subscribers and there are ${}40000$ nonreaders. How does the distribution look like after a year?

c) The three newspapers expand to another city, where there are no newspapers at all so far, but also ${}100000$ potential customers. How many subscribers does each newspaper have (and how many nonreaders) after three years, if the same movements hold in the new city?

Exercise Create referencenumber

We consider the linear map

$\varphi \colon K^{3}\longrightarrow K^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}1&2&5\\4&1&1\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}.$

Let ${}U\subseteq K^{3}$ be the subspace of ${}K^{3}$, defined by the linear equation ${}2x+3y+4z=0$, and let ${}\psi$ be the restriction of ${}\varphi$ on ${}U$. On ${}U$, there are given vectors of the form

$u=(0,1,a),\,v=(1,0,b){\text{ and }}w=(1,c,0).$

Compute the "change of basis" matrix between the bases

${\mathfrak {b}}_{1}=v,w,\,{\mathfrak {b}}_{2}=u,w{\text{ and }}{\mathfrak {b}}_{3}=u,v$

of ${}U$, and the transformation matrix of ${}\psi$ with respect to these three bases (and the standard basis of ${}K^{2}$).

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We consider the families of vectors

${\mathfrak {u}}={\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {v}}={\begin{pmatrix}1\\0\\1\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\end{pmatrix}}$

in ${}\mathbb {R} ^{4}$ and in ${}\mathbb {R} ^{3}$. We denote the standard basesMDLD/standard bases by ${}{\mathfrak {e}}_{4}$ and ${}{\mathfrak {e}}_{3}$. We consider the linear mappingMDLD/linear mapping

$f\colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{3}$

given by the matrixMDLD/matrix

${}A={\begin{pmatrix}-2&3&2&3\\-3&5&0&1\\-1&2&-2&-2\end{pmatrix}}\,$

with respect to the standard bases. Determine the describing matricesMDLD/describing matrices of ${}f$ with respect to the bases

a) ${}{\mathfrak {e}}_{4}$ and ${}{\mathfrak {v}}$,

b) ${}{\mathfrak {u}}$ and ${}{\mathfrak {e}}_{3}$,

c) ${}{\mathfrak {u}}$ and ${}{\mathfrak {v}}$.

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Prove Theorem 9.7 using Theorem 11.5 .

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Let

$\varphi \colon V\longrightarrow V$

be an endomorphismMDLD/endomorphism on a finite-dimensionalMDLD/finite-dimensional ${}K$-vector spaceMDLD/vector space ${}V$. Show that ${}\varphi$ is a homothetyMDLD/homothety if and only if the describing matrixMDLD/describing matrix is independent of the chosen basis.MDLD/basis

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Prove Corollary 8.10 using Corollary 11.9 and Exercise 10.23 .

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Prove Lemma 9.5 with the help of Lemma 11.10 and Example 10.13 .

The following exercise uses the concept of an isomorphism for groups, it also needs the concept of cardinality. The result might be somehow confusing.

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Let ${}V$ and ${}W$ be finite-dimensionalMDLD/finite-dimensional real vector spaces,MDLD/real vector spaces both not the zero-space. Are these spaces isomorphicMDLD/isomorphic (group) as commutative groups?MDLD/commutative groups

Hand-in-exercises

Exercise (3 marks) Create referencenumber

Sketch the imageMDLD/image of the pictured circles under the linear mappingMDLD/linear mapping given by the matrixMDLD/matrix ${}{\begin{pmatrix}2&0\\0&3\end{pmatrix}}$ from ${}\mathbb {R} ^{2}$ to itself.

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Determine the image and the kernel of the linear map

$f\colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{4},{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}1&3&4&-1\\2&5&7&-1\\-1&2&3&-2\\-2&0&0&-2\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}.$

Exercise (3 marks) Create referencenumber

Let ${}E\subseteq \mathbb {R} ^{3}$ be the plane defined by the linear equation ${}5x+7y-4z=0$. Determine a linear map

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

such that the image of ${}\varphi$ is equal to ${}E$.

Exercise (3 marks) Create referencenumber

On the real vector space ${}G=\mathbb {R} ^{4}$ of mulled wines, we consider the two linear maps

$\pi \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 8z+9n+5r+s,$

and

$\kappa \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 2z+n+4r+8s.$

We consider ${}\pi$ as the price function, and ${}\kappa$ as the caloric function. Determine a basis for ${}\operatorname {ker} {\left(\pi \right)}$, one for ${}\operatorname {ker} {\left(\kappa \right)}$ and one for ${}\operatorname {ker} {\left(\pi \times \kappa \right)}$.

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

An animal population consists of babies (first year), freshers (second year), rockers (third year), mature ones (fourth year) and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year ${}j$ is given by a ${}5$-tuple ${}B_{j}=(b_{1,j},b_{2,j},b_{3,j},b_{4,j},b_{5,j})$.

During a year, ${}7/8$ of the babies become freshers, ${}9/10$ of the freshers become rockers, ${}5/6$ of the rockers become mature ones, and ${}2/3$ of the mature ones reach the fifth year.

Babies and freshers can not reproduce yet, then they reach the sexual maturity, and ${}10$ rockers generate ${}5$ new pets, and ${}10$ of the mature ones generate ${}8$ new babies, and the babies are born one year later.

a) Determine the linear map (i.e. the matrix), which expresses the total stock ${}B_{j+1}$ with respect to the stock ${}B_{j}$.

b) What will happen to the stock ${}(200,150,100,100,50)$ in the next year?

c) What will happen to the stock ${}(0,0,100,0,0)$ in five years?

Exercise (3 marks) Create referencenumber

Let ${}z\in \mathbb {C}$ be a complex number and let

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the multiplication map, which is a ${}\mathbb {C}$-linear map. How does the matrix related to this map with respect to the real basis ${}1$ and ${}{\mathrm {i} }$ look like? Let ${}z_{1}$ and ${}z_{2}$ be complex numbers with corresponding real matrices ${}M_{1}$ and ${}M_{2}$. Prove that the matrix product ${}M_{2}\circ M_{1}$ is the real matrix corresponding to ${}z_{1}z_{2}$.

Footnotes

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