Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 9

Warm-up-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 ${}K$ 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

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

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.

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

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 ${}U,V,W$ 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 ${}V$ 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

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 ${}K$ 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

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

Exercise *

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

Find, by elementary geometric considerations, a matrix describing a rotation by 45 degrees counter-clockwise in the plane.

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?

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 (3 marks)

Find, by elementary geometric considerations, a matrix describing a rotation by 30 degrees counter-clockwise in the plane.

Exercise (3 marks)

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 ${}\operatorname {ker} {\left(\pi \times \kappa \right)}$.