Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 35

Exercises

Exercise

Let ${}V$ be a Euclidean vector space, and let ${}\varphi \colon V\rightarrow V$ denote a homothety with factor ${}s\neq 0$. Show that ${}\varphi$ is angle-preserving.

Exercise

Let ${}V$ and ${}W$ be Euclidean vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote an injective linear mapping. Show that ${}\varphi$ is angle-preserving if and only if for all ${}u,v\in V$, ${}u,v\neq 0$, the equation

${}\left\langle \varphi (u),\varphi (v)\right\rangle ={\frac {\Vert {\varphi (u)}\Vert }{\Vert {u}\Vert }}\cdot {\frac {\Vert {\varphi (v)}\Vert }{\Vert {v}\Vert }}\cdot \left\langle u,v\right\rangle \,$

holds.

Exercise

Let ${}U,V$ and ${}W$ be Euclidean vector spaces. Show that the following statements hold.

a) The identity

$\operatorname {Id} _{V}\colon V\longrightarrow V$

is angle-preserving.

b) The composition of angle-preserving mappings

$\varphi \colon U\longrightarrow V$

and

$\psi \colon V\longrightarrow W$

is again angle-preserving.

c) For a bijective angle-preserving mapping

$\varphi \colon U\longrightarrow V,$

the inverse mapping is also angle-preserving.

Exercise

Let ${}V$ be a Euclidean vector space. Show that the set of all angle-preserving mappings

$\varphi \colon V\longrightarrow V$

is a subgroup of ${}\operatorname {GL} _{}{\left(V\right)}$.

Exercise

Let

${}M={\begin{pmatrix}1&a_{12}&\ldots &\ldots &a_{1n}\\0&1&a_{23}&\ldots &a_{2n}\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\ldots &0&1&a_{n-1n}\\0&\ldots &\ldots &0&1\end{pmatrix}}\,$

be an upper triangular matrix such that the corresponding linear mapping

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

is angle-preserving. Show

${}M=E_{n}\,.$

Exercise

Let

${}M={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$

be a diagonal matrix. Show that the corresponding linear mapping

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

is angle-preserving if and only if ${}\vert {d_{ii}}\vert$ is constant and different from ${}0$.

Exercise

Give, for every ${}r$, ${}0\leq r<n$, a linear mapping

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

of rank ${}r$ with the property that orthogonal vectors are mapped to orthogonal vectors but such that it is not angle-preserving.

Exercise

Let ${}V$ and ${}W$ be Euclidean vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote an injective linear mapping, fulfilling the property that orthogonal vectors are mapped to orthogonal vectors. Show that ${}\varphi$ is angle-preserving.

Exercise

Let

$\varphi \colon V\longrightarrow V$

be an angle-preserving linear mapping on the Euclidean vector space ${}V$. Show that there exists an isometry

$\psi \colon V\longrightarrow V$

and a homothety

$\sigma \colon V\longrightarrow V$

such that

${}\varphi =\psi \circ \sigma \,$

holds.

Exercise

Determine the distance between the point ${}{\begin{pmatrix}7\\-3\\4\end{pmatrix}}$ and each linear subspace ${}U_{I}=\langle e_{i},\,i\in I\rangle$ for ${}I\subseteq \{1,2,3\}$.

Exercise

Determine the distance between the point ${}{\begin{pmatrix}5\\-6\end{pmatrix}}$ and the line given by

${}3x-7y=8\,.$

Determine also the foot of the perpendicular line of the point to this line.

Exercise

Determine the distance between the point ${}{\begin{pmatrix}3\\-5\\7\end{pmatrix}}$ and the plane given by

${}4x-7y+6z=3\,$

in ${}\mathbb {R} ^{3}$.

Exercise

Determine the minimal distance between ${}(4,1,-5)$ and a point of the plane ${}E$ given by the equation ${}2x-7y+3z=0$.

Exercise

Establish for the skew lines

$G={\left\{{\begin{pmatrix}2\\-5\\6\end{pmatrix}}+t{\begin{pmatrix}7\\-3\\3\end{pmatrix}}\mid t\in \mathbb {R} \right\}}\,\,{\text{ and }}\,\,H={\left\{{\begin{pmatrix}-4\\-4\\-1\end{pmatrix}}+t{\begin{pmatrix}8\\6\\5\end{pmatrix}}\mid t\in \mathbb {R} \right\}}$

a linear system of equations, and determine with it the perpendicular line, its feet, and the distance between the two lines.

Exercise

Compute the distance

between the two skew lines

$G={\left\{{\begin{pmatrix}1\\3\\0\end{pmatrix}}+t{\begin{pmatrix}4\\4\\-3\end{pmatrix}}\mid t\in \mathbb {R} \right\}}\,\,{\text{ and }}\,\,H={\left\{{\begin{pmatrix}2\\-6\\3\end{pmatrix}}+t{\begin{pmatrix}5\\7\\2\end{pmatrix}}\mid t\in \mathbb {R} \right\}}.$

The following exercises discuss distances between nonlinear objects.

Exercise

Let ${}A$ be the circle in ${}\mathbb {R} ^{2}$ with center ${}(3,-4)$ and radius ${}2$, and let ${}B$ be the circle in ${}\mathbb {R} ^{2}$ with center ${}(7,2)$ and radius ${}3$. Determine the distance between these circles and where this distance is obtained.

Exercise

Let ${}A$ be the circle in ${}\mathbb {R} ^{2}$ with center ${}(1,-3)$ and radius ${}2$, and let ${}B$ denote the line given by

${}6x+5y=30\,.$

Determine the distance between the circle and the line, and where the distance is obtained.

Exercise

Determine the distance between the hyperbola

${\left\{(x,y)\mid xy=1\right\}}$

and the union of axes

${\left\{(x,y)\mid xy=0\right\}}.$

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a sequence in a metric space ${}M$, and suppose that all members are different. Let ${}A={\left\{x_{n}\mid n\in \mathbb {N} \right\}}$, and let ${}P$ be a point from ${}M$ different from all members. Show that ${}P$ is an accumulation point of the sequence if and only if

${}d{\left(A,P\right)}=0\,.$

The following exercise needs analysis 1 (finding extrema with derivation).

Exercise

For what point ${}(t,t^{2})$ of the standard parabola becomes the distance to the point ${}(0,1)$ minimal?

Hand-in-exercises

Exercise (3 marks)

Let ${}\varphi \colon V\rightarrow V$ be an angle-preserving linear mapping on the Euclidean vector space ${}V$. Show that there exists a real number ${}s$ such that only ${}s$ or ${}-s$ might be an eigenvalue of ${}\varphi$.

Exercise (2 marks)

Let ${}V$ and ${}W$ denote Euclidean vector spaces, and let

$\varphi \colon V\longrightarrow W$

be a linear mapping. Show that ${}\varphi$ is an isometry if and only if for arbitrary subsets ${}A,B\subseteq V$ the equation

${}d(\varphi (A),\varphi (B))=d(A,B)\,$

holds.

Exercise (4 marks)

Determine the distance between the point ${}{\begin{pmatrix}4\\1\end{pmatrix}}$ and the line given by

${}-6x+3y=11\,.$

Determine also the foot of the perpendicular line of the point to this line.

Exercise (5 marks)

Establish for the skew lines

$G={\left\{{\begin{pmatrix}5\\3\\-2\end{pmatrix}}+t{\begin{pmatrix}6\\-4\\-1\end{pmatrix}}\mid t\in \mathbb {R} \right\}}\,\,{\text{ and }}\,\,H={\left\{{\begin{pmatrix}-4\\9\\-5\end{pmatrix}}+t{\begin{pmatrix}8\\2\\-6\end{pmatrix}}\mid t\in \mathbb {R} \right\}}$

a linear system of equations, and determine with it the perpendicular line, its feet, and the distance between the two lines.

Exercise (3 marks)

Compute the distance between the two skew lines

$G={\left\{{\begin{pmatrix}2\\-1\\4\end{pmatrix}}+t{\begin{pmatrix}5\\3\\-2\end{pmatrix}}\mid t\in \mathbb {R} \right\}}\,\,{\text{ and }}\,\,H={\left\{{\begin{pmatrix}4\\-3\\5\end{pmatrix}}+t{\begin{pmatrix}6\\7\\-4\end{pmatrix}}\mid t\in \mathbb {R} \right\}}.$

Exercise (4 marks)

Let two disjoint circles ${}K_{1}$ and ${}K_{2}$ in the Euclidean plane be given, with centers ${}M_{1}\neq M_{2}$ and radii ${}r_{1}$ and ${}r_{2}$. Show that the distance between the two circles is obtained in points that lie on the connecting line segment of the two centers.

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