# 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.
* Mass is volume times density.
* Energy is mass times the calorific value.
* The distance is speed multiplied by time.
* Force is mass times acceleration.
* Energy is force times distance.
* Energy is power times time.
* Voltage is resistance times electric current.
* Charge is current multiplied by time.

### Exercise

Around the Earth along the equator is placed a ribbon. However, the ribbon is one meter longer than the equator, so that it is lifted up uniformly all around to be tense. Which of the following creatures can run/fly/swim/dance under it?
* An amoeba.
* An ant.
* A tit.
* A flounder.
* A boa constrictor.
* A guinea pig.
* A boa constrictor that has swallowed a guinea pig.
* 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 to 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.
* $\varphi$ is injective if and only if $v_{1},\ldots ,v_{n}$ are linearly independent.
* $\varphi$ is surjective if and only if $v_{1},\ldots ,v_{n}$ is a system of generators for $V$.
* $\varphi$ 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 that also 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.
* For a linear subspace $S\subseteq V$, also the image $\varphi (S)$ is a linear subspace of $W$.
* In particular, the image of the map is a subspace of $W$.
    $\operatorname {Im} \varphi =\varphi (V)\,$
* For a linear subspace $T\subseteq W$, also the preimage $\varphi ^{-1}(T)$ is a linear subspace of $V$.
* 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
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$
which 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 comsider $\pi$ 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 $\operatorname {ker} {\left(\pi \times \kappa \right)}$.