Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 24

Exercises

Exercise

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$, for the standard basis $𝔲$, and the basis $𝔳$ in ${ℝ}^4$, which is given by

$v_1=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),v_2=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),v_3=\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right),v_4=\left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right).$

Exercise

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$ for the standard basis $𝔲$ and the basis $𝔳$ of ${ℝ}^3$ that is given by the vectors

$v_1=\left(\begin{array}{c}4\\ 5\\ 1\end{array}\right),v_2=\left(\begin{array}{c}2\\ 3\\ -8\end{array}\right),\text{ and }{v}_3=\left(\begin{array}{c}5\\ 7\\ -3\end{array}\right)$

Exercise

Let $𝔳=v_1,v_2,v_3$ be a basis of a three-dimensional $K$-vector space $V$.

a) Show that $𝔴=v_1,v_1+v_2,v_2+v_3$ is also a basis of $V$.

b) Determine the transformation matrix $M_{𝔳}^{𝔴}$.

c) Determine the transformation matrix $M_{𝔴}^{𝔳}$.

d) Compute the coordinates with respect to the basis $𝔳$ for the vector, which has the coordinates $\left(\begin{array}{c}4\\ 8\\ -9\end{array}\right)$ with respect to the basis $𝔴$.

e) Compute the coordinates with respect to the basis $𝔴$ for the vector, which has the coordinates $\left(\begin{array}{c}3\\ -7\\ 5\end{array}\right)$ with respect to the basis $𝔳$.

Exercise

Determine the transformation matrices $M_{𝔳}^{𝔲}$ and $M_{𝔲}^{𝔳}$ for the standard basis $𝔲$ and the basis $𝔳$ of ${ℂ}^2$ that is given by the vectors

$v_1=\left(\begin{array}{c}3+5\mathrm{i}\\ 1-\mathrm{i}\end{array}\right)\text{ and }{v}_2=\left(\begin{array}{c}2+3\mathrm{i}\\ 4+\mathrm{i}\end{array}\right).$

Exercise

We consider the families of vectors

$𝔳=\left(\begin{array}{c}7\\ -4\end{array}\right),\left(\begin{array}{c}8\\ 1\end{array}\right)\text{ and }𝔲=\left(\begin{array}{c}4\\ 6\end{array}\right),\left(\begin{array}{c}7\\ 3\end{array}\right)$

in ${ℝ}^2$.

a) Show that $𝔳$ and $𝔲$ are both a basis of ${ℝ}^2$.

b) Let $P\in {ℝ}^2$ denote the point that has the coordinates $(-2,5)$ with respect to the basis $𝔳$. What are the coordinates of this point with respect to the basis $𝔲$?

c) Determine the transformation matrix that describes the change of bases from $𝔳$ to $𝔲$.

Exercise

We consider the linear map

$\phi :K^3⟶K^2,\left(\begin{array}{c}x\\ y\\ z\end{array}\right)⟼\left(\begin{array}{ccc}1& 2& 5\\ 4& 1& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right).$

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 $\phi$ 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

${𝔟}_1=v,w,{𝔟}_2=u,w\text{ and }{𝔟}_3=u,v$

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

Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Let

$\phi :V⟶W$

be a linear map. Prove that for all vectors $v_1,\dots ,v_n\in V$ and coefficients $s_1,\dots ,s_n\in K$, the relationship

$\phi \left(\sum _{i=1}^n{s}_i{v}_i\right)=\sum _{i=1}^n{\lambda }_i\phi (v_i)$

holds.

Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Prove that, for $a\in K$, the map

$V⟶V,v⟼av,$

is linear.

Exercise

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.

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

Suppose that a linear function

$\phi :\mathbb{Q}⟶\mathbb{Q}$

has for $\frac{11}{13}$ the value $\frac{7}{17}$. What is its vale for $\frac{3}{19}$?

Exercise

Which of the following functions $f:ℝ\to ℝ$ are linear?

1. The real exponential function.
2. The zero function.
3. The constant function with value $7$.
4. The squaring function $x\mapsto x^2$.
5. The function which halves every real number.
6. The function which subtracts $1$ from every real number.

Exercise

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

Exercise

Consider the linear map

$\phi :{ℝ}^2⟶ℝ$

such that

$\phi \left(\begin{array}{c}1\\ 3\end{array}\right)=5\text{ and }\phi \left(\begin{array}{c}2\\ -3\end{array}\right)=4.$

Compute

$\phi \left(\begin{array}{c}7\\ 6\end{array}\right).$

Exercise

Complete the proof of the theorem on determination on basis, by proving the compatibility with the scalar multiplication.

Exercise

Lucy Sonnenschein works as a bicycle messenger, she earns $12$ € per hour. At the fruit market, the price (per $100$ gram) for raspberries is $3$ €, for strawberries the price is $2$ €, and for apples the price is $0,4$ €. Describe the mapping, which assigns to any purchase of fruits, the time, how long Lucy has to work for it, as a composition of linear mappings.

Exercise

Let $K$ be a field, and let $U,V,W$ be vector spaces over $K$. Let $\phi :U\to V$ and $\psi :V\to W$ be linear maps. Prove that also the composite mapping

$\psi \circ \phi :U⟶W$

is a linear map.

Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Let

$\phi :V⟶W$

be a bijective linear map. Prove that also the inverse map

${\phi }^{-1}:W⟶V$

is linear.

Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_1,\dots ,v_n$ be a family of vectors in $V$. Consider the map

$\phi :K^n⟶V,(s_1,\dots ,s_n)⟼\sum _{i=1}^n{s}_i{v}_i,$

and prove the following statements.

1. $\phi$ is injective if and only if $v_1,\dots ,v_n$ are linearly independent.
2. $\phi$ is surjective if and only if $v_1,\dots ,v_n$ is a system of generators for $V$.
3. $\phi$ is bijective if and only if $v_1,\dots ,v_n$ form a basis.

Exercise

Prove that the functions

$ℂ⟶ℝ,z⟼\mathrm{Re}\left(z\right),$

and

$ℂ⟶ℝ,z⟼\mathrm{Im}\left(z\right),$

are $ℝ$-linear maps. Prove also that the complex conjugation is $ℝ$-linear, but not $ℂ$-linear. Is the absolute value

$ℂ⟶ℝ,z⟼|z|,$

$ℝ$-linear?

Exercise

Consider the function

$f:ℝ⟶ℝ$

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?

Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Let

$\phi :V⟶W$

be a linear map. Prove the following facts.

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

   $\mathrm{Im}\phi =\phi (V)$

   of the map is a subspace of $W$.

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

Exercise

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

Exercise

Prove the addition theorems for sine and cosine, using the rotation matrices.

Exercise *

Determine the kernel of the linear map

${ℝ}^4⟶{ℝ}^3,\left(\begin{array}{c}x\\ y\\ z\\ w\end{array}\right)⟼\left(\begin{array}{cccc}2& 1& 5& 2\\ 3& -2& 7& -1\\ 2& -1& -4& 3\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\\ w\end{array}\right).$

Exercise *

Determine the kernel of the linear mapping

$\phi :{ℝ}^4⟶{ℝ}^2,$

given by the matrix

$M=\left(\begin{array}{cccc}2& 3& 0& -1\\ 4& 2& 2& 5\end{array}\right).$

Exercise

How does the graph of a linear mapping

$f:ℝ⟶ℝ,$

$g:ℝ⟶{ℝ}^2,$

$h:{ℝ}^2⟶ℝ$

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

Exercise

Let $M$ be an $m\times n$-matrix over the field $K$, let $\phi :K^n\to K^m$ be the corresponding linear mapping, and let $Mx=c$ denote (depending on a vector $c\in K^m$) the corresponding system of linear equations. Show that the solution set of the system equals the preimage of $c$ under the linear mapping $\phi$.

Exercise

Let $V$ and $W$ be vector spaces over a field $K$, and let $\phi ,\psi :V\to W$ be linear mappings. Show that the mapping, defined by

$(\phi +\psi )(v):=\phi (v)+\psi (v),$

is also linear.

Exercise

Give an example for a linear mapping

$\phi :{ℝ}^2⟶{ℝ}^2$

that is not injective but such that its restriction

${\mathbb{Q}}^2⟶{ℝ}^2$

is injective.

Exercise

We consider the mapping

$\Psi :{ℝ}_{\ge 0}^4⟶{ℝ}_{\ge 0}^4$

that assigns to a four-tuple $(a,b,c,d)$ the four-tuple

$(|b-a|,|c-b|,|d-c|,|a-d|).$

Describe this mapping by a matrix, under the condition

$a\le b\le c\le d.$

Hand-in-exercises

Exercise (3 marks)

Consider the linear map

$\phi :{ℝ}^3⟶{ℝ}^2$

such that

$\phi \left(\begin{array}{c}2\\ 1\\ 3\end{array}\right)=\left(\begin{array}{c}4\\ 7\end{array}\right),\phi \left(\begin{array}{c}0\\ 4\\ 2\end{array}\right)=\left(\begin{array}{c}1\\ 1\end{array}\right)\text{ and }\phi \left(\begin{array}{c}3\\ 1\\ 1\end{array}\right)=\left(\begin{array}{c}5\\ 0\end{array}\right).$

Compute

$\phi \left(\begin{array}{c}4\\ 5\\ 6\end{array}\right).$

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

We consider the families of vectors

$𝔳=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right),\left(\begin{array}{c}4\\ 7\\ 1\end{array}\right),\left(\begin{array}{c}0\\ 2\\ 5\end{array}\right)\text{ and }𝔲=\left(\begin{array}{c}0\\ 2\\ 4\end{array}\right),\left(\begin{array}{c}6\\ 6\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 5\\ -2\end{array}\right)$

in ${ℝ}^3$.

a) Show that $𝔳$ and $𝔲$ are both a basis of ${ℝ}^3$.

b) Let $P\in {ℝ}^3$ denote the point that has the coordinates $(2,5,4)$ with respect to the basis $𝔳$. What are the coordinates of this point with respect to the basis $𝔲$?

c) Determine the transformation matrix that describes the change of basis from $𝔳$ to $𝔲$.

Exercise (3 marks)

Sketch the image of the pictured circles under the linear mapping given by the matrix $\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$ from ${ℝ}^2$ to itself.

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

$f:{ℝ}^4⟶{ℝ}^4,\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right)⟼\left(\begin{array}{cccc}1& 3& 4& -1\\ 2& 5& 7& -1\\ -1& 2& 3& -2\\ -2& 0& 0& -2\end{array}\right)\cdot \left(\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right).$

Exercise (3 marks)

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

$\phi :{ℝ}^2⟶{ℝ}^3,$

such that the image of $\phi$ is equal to $E$.

Exercise (3 marks)

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

$\pi :G⟶ℝ,\left(\begin{array}{c}z\\ n\\ r\\ s\end{array}\right)⟼8z+9n+5r+s,$

and

$\kappa :G⟶ℝ,\left(\begin{array}{c}z\\ n\\ r\\ s\end{array}\right)⟼2z+n+4r+8s.$

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

Footnotes

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