Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 34

Exercises

Exercise

Show that

${}\angle ({\begin{pmatrix}1\\0\end{pmatrix}},{\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}})={\begin{cases}\alpha \,,{\text{ if }}\alpha \leq \pi \,,\\2\pi -\alpha \,,{\text{ if }}\alpha >\pi \end{cases}}\,$

holds.

Exercise

Let ${}u,v\in V$ be vectors different from ${}0$ in a real vector space ${}V$, endowed with an inner product. Show that the angle between ${}u$ and ${}v$ equals the angle between ${}su$ and ${}tv$, where ${}s,t$ are positive real numbers.

The preceding statement says in particular that the angle only depends on the rays defined by the two vectors.

Exercise

Let ${}V$ be a real vector space, endowed with an inner product. Show that the angle

$\angle (u,v)$

does only depend on the restriction of the inner product to the linear subspace generated by ${}u$ and ${}v$.

Exercise

Let ${}u,v,w\in V$ vectors different from ${}0$ in a real vector space ${}V$, endowed with an inner product. Show that

${}\angle (u,w)\leq \angle (u,v)+\angle (v,w)\,$

holds.

Exercise

What angles exist on a line?

Exercise

Let

${}K={\left\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\right\}}\,$

be the unit circle. Show that we can obtain a metric on ${}K$ by defining ${}d(P,Q)$ (${}P,Q\in K$) to be the positive angle at the origin ${}(0,0)$ between the corresponding rays.

Exercise

Let ${}\alpha _{n}$ be the angle between the first standard vector ${}e_{1}$ and the vector ${}v_{n}=\sum _{i=1}^{n}e_{i}$ in ${}\mathbb {R} ^{n}$. Determine the limit

$\lim _{n\rightarrow \infty }\alpha _{n}.$

The following two exercises were already on exercise sheet 10.

Exercise

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

Exercise

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

Exercise

Determine, in an elementary-geometric way, the image vectors of the standard vectors ${}e_{1}$ and ${}e_{2}$ under a rotation about the origin counter-clockwise with an angle ${}\alpha$.

Exercise

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

Exercise

Let

${}M={\begin{pmatrix}{\frac {3}{5}}&-{\frac {4}{5}}\\{\frac {4}{5}}&{\frac {3}{5}}\end{pmatrix}}\,.$

a) Show that ${}M$ defines an isometry on ${}\mathbb {R} ^{2}$ and also on ${}\mathbb {C} ^{2}$.

b) Determine the complex eigenvalues of ${}M$.

c) Determine an orthonormal basis of ${}\mathbb {C} ^{2}$ consisting of eigenvectors of ${}M$.

An ellipse in ${}\mathbb {R} ^{2}$ that is symmetric to the axes is described by an equation of the form

${}ax^{2}+by^{2}=c\,,$

with ${}a,b,c\in \mathbb {R} _{+}$.

Exercise

Give an example of a (symmetric to the axes) ellipse ${}E$ in ${}\mathbb {R} ^{2}$ and a bijective linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ with ${}\varphi (E)=E$ that is not an isometry.

Exercise

Give an example of a bijective, continuous mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$, with ${}\varphi (S^{1})=S^{1}$ and with ${}\varphi (sv)=s\varphi (v)$ (for all ${}s\in \mathbb {R}$ and ${}v\in \mathbb {R} ^{2}$) that is not an isometry.

Exercise

Let

${}M={\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}\,$

be a plane reflection at an axis, ${}\alpha \neq 0$. Show that ${}{\begin{pmatrix}-\sin \alpha \\\cos \alpha -1\end{pmatrix}}$ is an eigenvector for the eigenvalue ${}1$, and that ${}{\begin{pmatrix}\cos \alpha -1\\\sin \alpha \end{pmatrix}}$ is an eigenvector for the eigenvalue ${}-1$ of ${}M$.

Exercise

Let

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be the rotation of the ${}3$-space around the ${}z$-axis by ${}45$ degree counter-clockwise. How does the describing matrix look with respect to the basis

${\begin{pmatrix}1\\2\\4\end{pmatrix}},\,{\begin{pmatrix}3\\3\\-1\end{pmatrix}},\,{\begin{pmatrix}5\\0\\7\end{pmatrix}}?$

Exercise

Let ${}\pi \in S_{n}$ be a permutation, and let

$M_{\pi }\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n}$

denote the corresponding permutation matrix and the corresponding linear mapping. Show that ${}M_{\pi }$ is an isometry. When is it a proper isometry?

Exercise

Determine, for every permutation ${}\pi \in S_{3}$, the eigenline of the corresponding permutation matrix.

Exercise

Show that the group of the spatial rotations is not commutative.

Exercise

Give an example of a spatial rotation where all matrix entries are ${}\neq 0,1$.

Exercise

Let ${}U_{1}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}}\rangle$ and ${}U_{2}=\langle {\begin{pmatrix}4\\-8\\7\end{pmatrix}},\,{\begin{pmatrix}3\\-5\\2\end{pmatrix}}\rangle$ be linear subspaces in ${}\mathbb {R} ^{3}$. Find an isometry ${}\varphi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}$ fulfilling ${}U_{1}=\langle \varphi (e_{1})\rangle$ and ${}U_{2}=\langle \varphi (e_{1}),\,\varphi (e_{2})\rangle$.

Exercise

The matrix

${\begin{pmatrix}\cos \alpha &\sin \alpha &0&0\\\sin \alpha &-\cos \alpha &0&0\\0&0&\cos \beta &\sin \beta \\0&0&\sin \beta &-\cos \beta \end{pmatrix}}$

defines a linear mapping ${}\varphi \colon \mathbb {R} ^{4}\rightarrow \mathbb {R} ^{4}$ (${}\alpha ,\beta \in [0,\pi [$). Determine the eigenvalues of ${}\varphi$ and their algebraic and geometric multiplicities.

Exercise

We consider ${}\mathbb {R} ^{n}$ with the maximum norm

${}\Vert {v}\Vert =\Vert {v}\Vert _{\rm {max}}\,.$

We want to understand real matrices

${}M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$

with the property

${}\Vert {Mv}\Vert =\Vert {v}\Vert \,$

for all ${}v\in \mathbb {R} ^{n}$. We call such a matrix ${}\Vert {-}\Vert _{\rm {max}}$-isometric.

a) Show that a ${}\Vert {-}\Vert _{\rm {max}}$-isometric matrix is invertible.

b) Show that the set of ${}\Vert {-}\Vert _{\rm {max}}$-isometric matrices is a subgroup of the general linear group.

c) Show that a permutation matrix is ${}\Vert {-}\Vert _{\rm {max}}$-isometric.

d) Under a sign-permutation matrix, we mean a matrix that arises from a permutation matrix by adding in front of some of the entries a ${}-$-sign. Give an example of a ${}3\times 3$-sign-permutation matrix that is not a permutation matrix, that is not an upper triangular matrix, and such that its determinant is ${}1$.

e) Show that a sign-permutation matrix is ${}\Vert {-}\Vert _{\rm {max}}$-isometric.

f) Show that every ${}\Vert {-}\Vert _{\rm {max}}$-isometric matrix is a sign-permutation matrix.

Hand-in-exercises

Exercise (2 marks)

Let ${}u,v\in V$ be normed vectors in a real vector space ${}V$, endowed with an inner product. Show that the vector ${}u+v$ bisects the two vectors at equal angles.

Exercise (4 marks)

We look at a clock with minute and second hands, both moving continuously. Determine a formula that calculates the angular position of the second hand from the angular position of the minute hand (each starting from the 12-clock position measured in the clockwise direction).

Exercise (4 marks)

Show that every proper linear isometry of ${}\mathbb {R} ^{3}$ can be realized as a composition of rotations around the three coordinate axes.

Exercise (4 marks)

Show that for ${}a,b,c,d\in \mathbb {R}$ satisfying ${}a^{2}+b^{2}+c^{2}+d^{2}=1$, the matrix

${\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(-ad+bc)&2(ac+bd)\\2(ad+bc)&a^{2}-b^{2}+c^{2}-d^{2}&2(-ab+cd)\\2(-ac+bd)&2(ab+cd)&a^{2}-b^{2}-c^{2}+d^{2}\end{pmatrix}}$

defines an isometry on ${}\mathbb {R} ^{3}$.

Exercise (4 marks)

Let ${}M$ be a complex ${}2\times 2$-matrix such that the columns form an orthonormal basis of ${}\mathbb {C} ^{2}$ and such that the determinant equals ${}1$. Show that ${}M$ has the form

${}M={\begin{pmatrix}u&-{\overline {v}}\\v&{\overline {u}}\end{pmatrix}}\,,$

with ${}u,v\in \mathbb {C}$ and ${}\Vert {\begin{pmatrix}u\\v\end{pmatrix}}\Vert =1$.

Exercise (5 (1+2+2) marks)

Let

${}M={\begin{pmatrix}{\frac {5}{13}}&-{\frac {12}{13}}\\{\frac {12}{13}}&{\frac {5}{13}}\end{pmatrix}}\,.$

a) Show that ${}M$ defines an isometry on ${}\mathbb {R} ^{2}$ and also on ${}\mathbb {C} ^{2}$.

b) Determine the complex eigenvalues of ${}M$.

c) Determine an orthonormal basis of ${}\mathbb {C} ^{2}$ consisting of eigenvectors of ${}M$.

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