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

Exercise for the break

Exercise

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

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 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 *

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 *

Determine the kernel of the linear map

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

Exercise

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

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

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

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

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

Exercise

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

$\varphi \colon V\longrightarrow W$

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

Exercise

Let a 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 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?

Exercise

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.

1. Determine a basis and the dimension of ${}\operatorname {kern} f$, and of ${}\operatorname {Im} f$.
2. Find a 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 ${}\mathbb {R} ^{2}=U\oplus \operatorname {Im} f$?

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and ${}f\colon V\rightarrow V$ be an 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

We consider the mapping

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

Show that for this mapping, neither the image of a 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, and let ${}V$ be a ${}K$-vector space. For a vector ${}v\in V$, the 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, we will come back later to this concept.

Exercise

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 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. Show that the image ${}\varphi (F')$ arises from the image ${}\varphi (F)$ by a translation.

Exercise

How does the graph of a 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?

Exercise

Give an example for a linear mapping

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

that is not injective but such that its restriction

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

is injective.

Exercise

Let ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the 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 ${}{\begin{pmatrix}4\\2\end{pmatrix}}$.

Exercise

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) that 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

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 that 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

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 ${}K^{2}$).

Exercise

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 bases by ${}{\mathfrak {e}}_{4}$ and ${}{\mathfrak {e}}_{3}$. We consider the linear mapping

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

given by the 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 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 ${}{\mathfrak {v}}$.

Exercise

Prove Theorem 9.7 using Theorem 11.5 .

Exercise

Let

$\varphi \colon V\longrightarrow V$

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

Exercise

Prove Corollary 8.10 using Corollary 11.9 and Exercise 10.23 .

Exercise

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.

Exercise

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

Hand-in-exercises

Exercise (3 marks)

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

Exercise

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)

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)

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)

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 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) that 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)

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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