Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 9/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
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{

Let $K$ be a field and let \mathkor {} {V} {and} {W} {} be two $K$-vector spaces. Let \maabbdisp {\varphi} {V} {W } {} be a linear map. Prove that for all vectors
\mathl{v_1 , \ldots , v_n \in V}{} and coefficients
\mathl{s_1 , \ldots , s_n \in K}{} the relationship
\mathdisp {\varphi( \sum_{i=1}^n s_i v_i ) = \sum_{i=1}^n s_i \varphi (v_i)} { }
holds.

}
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\inputexercise
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{

Let $K$ be a field and let $V$ be a $K$-vector space. Prove that for $a \in K$ the map \maabbeledisp {} {V} {V } {v} { a v } {,} is linear\zusatzfussnote {Such a map is called a homothety with stretching or extension factor $a$} {.} {.}

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\inputexercise
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{

Interpret the following physical laws as linear functions from $\R$ to $\R$. Establish in each situation what is the measurable variable and what is the proportionality factor. \aufzaehlungacht{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. }

}
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\inputexercise
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{

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? \aufzaehlungacht{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. }

}
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\inputexercise
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{

Consider the linear map \maabbdisp {\varphi} {\R^2} {\R } {} such that
\mathdisp {\varphi \begin{pmatrix} 1 \\3 \end{pmatrix} = 5 \text{ and } \varphi \begin{pmatrix} 2 \\-3 \end{pmatrix} = 4} { . }
Compute
\mathdisp {\varphi \begin{pmatrix} 7 \\6 \end{pmatrix}} { . }

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\inputexercise
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{

Complete the proof of Theorem 9.3 to the compatibility with the scalar multiplication.

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\inputexercise
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{

Let $K$ be a field and let $U,V,W$ be vector spaces over $K$. Let
\mathdisp {\varphi \colon U\rightarrow V \text{ and } \psi \colon V\rightarrow W} { }
be linear maps. Prove that also the composite function \maabbdisp {\psi \circ \varphi} { U} {W } {} is a linear map.

}
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\inputexercise
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{

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 \maabbeledisp {\varphi} {K^n} {V } {(s_1 , \ldots , s_n) } { \sum_{i = 1}^n s_i v_i } {,} and prove the following statements. \aufzaehlungdrei{$\varphi$ 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. }

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\inputexercise
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{

Prove that the functions \maabbeledisp {} {{\mathbb C}} {\R } {z} { \operatorname{Re} \, { \left( z \right) } } {,} and \maabbeledisp {} {{\mathbb C}} {\R } {z} { \operatorname{Im} \, { \left( z \right) } } {,} are $\R$-linear maps. Prove that also the complex conjugation is $\R$-linear, but not ${\mathbb C}$-linear. Is the absolute value \maabbeledisp {} {{\mathbb C}} {\R } {z} {\betrag { z } } {,} $\R$-linear?

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\inputexercise
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{

Let $K$ be a field and let \mathkor {} {V} {and} {W} {} be two $K$-vector spaces. Let \maabbdisp {\varphi} {V} {W } {} be a linear map. Prove the following facts. \aufzaehlungvier{Consider the subspace
\mathl{S \subseteq V}{} then also the image $\varphi(S)$ is a subspace of $W$. }{In particular the image $\operatorname{bild} \varphi= \varphi(V)$ of the map is a subspace of $W$. }{Consider the subspace
\mathl{T \subseteq W}{} then the preimage
\mathl{\varphi^{-1}(T)}{} is a subspace of $V$. }{In particular
\mathl{\varphi^{-1}(0)}{} is a subspace of $V$. }

}
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\inputexercise
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{

Determine the kernel of the linear map \maabbeledisp {} {\R^4} {\R^3 } { \begin{pmatrix} x \\y\\ z\\w \end{pmatrix} } {\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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\inputexercise
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{

Determine the kernel of the linear map \maabbdisp {\varphi} {\R^4} {\R^2 } {,} given by the matrix
\mathdisp {M= \begin{pmatrix} 2 & 3 & 0 & -1 \\ 4 & 2 & 2 & 5 \end{pmatrix}} { . }

}
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\inputexercise
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{

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

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\inputexercise
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Consider the function \maabbdisp {f} {\R} {\R } {,} which sends a rational number $q\in \Q$ into $q$ and all the irrational numbers into $0$. Is this a linear map? Is it compatible with multiplication with a scalar?

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\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Consider the linear map \maabbdisp {\varphi} {\R^3} {\R^2 } {} such that
\mathdisp {\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
\mathdisp {\varphi \begin{pmatrix} 4 \\5\\ 6 \end{pmatrix}} { . }

}
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\inputexercise
{3}
{

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

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\inputexercise
{3}
{

Determine the image and the kernel of the linear map \maabbeledisp {f} {\R^4} {\R^4 } {\begin{pmatrix} x_1 \\x_2\\ x_3\\x_4 \end{pmatrix}} {\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} } {.}

}
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\inputexercise
{3}
{

Let
\mathl{E \subset \R^3}{} be the plane identified by the linear equation
\mathl{5x+7y-4z=0}{.} Determine a linear map \maabbdisp {\varphi} {\R^2} {\R^3 } {} such that the image of $\varphi$ is equal to $E$.

}
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\inputexercise
{3}
{

On the real vector space $G=\R^4$ of mulled wines we consider the two linear maps \maabbeledisp {\pi} {G} {\R } { \begin{pmatrix} z \\n\\ r\\s \end{pmatrix}} {8z+9n+5r+s } {,} and \maabbeledisp {\kappa} {G} {\R } { \begin{pmatrix} z \\n\\ r\\s \end{pmatrix}} {2z+n+4r+8s } {.} We put $\pi$ as the price function and $\kappa$ as the caloric function. Determine a basis for
\mathl{\operatorname{kern} \pi}{,} one for
\mathl{\operatorname{kern} \kappa}{} and one for
\mathl{\operatorname{kern} (\pi \times \kappa)}{}\zusatzfussnote {Do not mind that there may exist negative numbers. In a mulled wine of course the ingredients do not come in with a negative coefficient. But if you would like to consider for example, in how many ways you can change a particular recipe, without changing the total price or the total amount of energy, then the negative entries make sense} {.} {.}

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