Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 33

Exercises

Exercise

Compute the cross product

${\begin{pmatrix}3\\5\\2\end{pmatrix}}\times {\begin{pmatrix}-4\\6\\-5\end{pmatrix}}$

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

Exercise

Compute the cross product

${\begin{pmatrix}4-7{\mathrm {i} }\\3+5{\mathrm {i} }\\-1-2{\mathrm {i} }\end{pmatrix}}\times {\begin{pmatrix}-6+4{\mathrm {i} }\\-2-8{\mathrm {i} }\\1-9{\mathrm {i} }\end{pmatrix}}$

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

Exercise

Compute the cross product

${\begin{pmatrix}3\\1\\4\end{pmatrix}}\times {\begin{pmatrix}2\\4\\3\end{pmatrix}}$

in ${}K^{3}$, where ${}K$ denotes the field with five elements.

Exercise

Compute the cross product

${\begin{pmatrix}4\\6\\1\end{pmatrix}}\times {\begin{pmatrix}5\\2\\5\end{pmatrix}}$

in ${}K^{3}$, where ${}K$ is the field with seven elements.

Exercise

Let ${}K$ be a field. Show that the cross product on ${}K^{3}$ is bilinear and alternating.

Exercise

Show that the cross product for vectors ${}x,y,z\in K^{3}$ fulfills the relation

${}x\times (y\times z)+y\times (z\times x)+z\times (x\times y)=0\,.$

Exercise

Let ${}u_{1},u_{2},u_{3}$ be an orthonormal basis of ${}\mathbb {R} ^{3}$. Show that ${}u_{1}\times u_{2}=\pm u_{3}$ holds.

Exercise

Show that, over an arbitrary field ${}K$, for linearly independent vectors ${}u={\begin{pmatrix}a\\b\\c\end{pmatrix}}$ and ${}v={\begin{pmatrix}d\\e\\f\end{pmatrix}}$, the family consisting of ${}u,v$, and the cross product ${}{\begin{pmatrix}a\\b\\c\end{pmatrix}}\times {\begin{pmatrix}d\\e\\f\end{pmatrix}}$ does not necessarily form a basis of ${}K^{3}$.

Exercise

Determine the isometries on ${}\mathbb {R}$.

Exercise

What kind of isometries on ${}\mathbb {R} ^{2}$ do you know from school?

Exercise

Let ${}U,V$ denote ${}{\mathbb {K} }$-vector spaces, both endowed with an inner product, and let ${}\varphi \colon U\rightarrow V$ be an isometry. Show that ${}\varphi$ is injective.

Exercise

Let ${}U,V,W$ be ${}{\mathbb {K} }$-vector spaces, each endowed with an inner product. Show the following statements.

a) The identity ${}V\rightarrow V$ is an isometry.

b) If ${}\varphi \colon U\rightarrow V$ is a bijective isometry, then the inverse mapping ${}\varphi ^{-1}$ is also an isometry.

c) If ${}\varphi \colon U\rightarrow V$ and ${}\psi \colon V\rightarrow W$ are isometries, then the composition ${}\psi \circ \varphi$ is also an isometry.

Exercise

Determine the isometries on ${}\mathbb {C}$.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product. Let

$\varphi \colon V\longrightarrow V$

be an isometry, and let ${}U\subseteq V$ be a ${}\varphi$-invariant linear subspace. Show that

$\varphi {|}_{U}\colon U\longrightarrow U$

is also an isometry.

Exercise

Let ${}V$ be a Euclidean vector space of dimension ${}n$. Show that a family of vectors ${}u_{1},\ldots ,u_{n}\in V$ is an orthonormal basis of ${}V$ if and only if the corresponding linear mapping

$\mathbb {R} ^{n}\longrightarrow V,e_{i}\longmapsto u_{i},$

is an isometry between ${}\mathbb {R} ^{n}$ and ${}V$.

Exercise

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

$\varphi \colon V\longrightarrow W$

denote a linear mapping. Show that the following statements are equivalent.

1. ${}\varphi$ is an isometry.
2. For every orthonormal basis ${}u_{i}$, ${}i=1,\ldots ,n$, of ${}V$, ${}\varphi (u_{i})$, ${}i=1,\ldots ,n$, is part of an orthonormal basis of ${}W$.
3. There exists an orthonormal basis ${}u_{i}$, ${}i=1,\ldots ,n$, of ${}V$ such that ${}\varphi (u_{i})$, ${}i=1,\ldots ,n$, is part of an orthonormal basis of ${}W$.

Exercise

Give an example of a bijective linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

with the property that, on one hand, there exists an orthogonal basis of ${}\mathbb {R} ^{2}$ that is mapped under ${}\varphi$ to an orthogonal basis, and, on the other hand, there exists an orthogonal basis that is not mapped to an orthogonal basis.

Exercise

Let ${}V$ be a Euclidean vector space. Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping with the property that the determinant of ${}\varphi$ equals ${}1$ or ${}-1$. Moreover, ${}\varphi$ satisfies the property that orthogonal vectors are mapped to orthogonal vectors. Show that ${}\varphi$ is an isometry.

Exercise

Give an example of a bijective linear mapping

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

that is not an isometry but fulfills for all ${}u,v\in V$ the relation

$\left\langle u,v\right\rangle =0{\text{ if and only if }}\left\langle \varphi (u),\varphi (v)\right\rangle =0.$

Exercise

Give an example of a linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

such that ${}\varphi$ is area-preserving but not an isometry.

Exercise

Give an example of a linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

that is not an isometry and such that its order is ${}2$.

Exercise

Let ${}K$ be a field, and ${}n\in \mathbb {N}$. Show that the set ${}\operatorname {GL} _{n}\!{\left(K\right)}$ of invertible matrices is a group. Moreover, show that this group is, for ${}n\geq 2$, not commutative.

Exercise

Let ${}V$ be a Euclidean vector space, and let

$\varphi \colon V\longrightarrow V$

denote a linear mapping. Show that the following statements are equivalent.

1. ${}\varphi$ is an isometry.
2. For every vector ${}v$ with ${}\Vert {v}\Vert =1$, we also have ${}\Vert {\varphi (v)}\Vert =1$.
3. For every orthonormal basis ${}u_{i},i=1,\ldots ,n$, also ${}\varphi (u_{i}),i=1,\ldots ,n$, is an orthonormal basis.
4. There exists an orthonormal basis ${}u_{i},i=1,\ldots ,n$, such that also ${}\varphi (u_{i}),i=1,\ldots ,n$, is an orthonormal basis.

Hand-in-exercises

Exercise (2 marks)

Compute the cross product

${\begin{pmatrix}3\\2\\5\end{pmatrix}}\times {\begin{pmatrix}4\\6\\2\end{pmatrix}}$

in ${}K^{3}$, where ${}K$ denotes the field with seven elements.

Exercise (3 marks)

Let ${}V$ be a Euclidean vector space. Show that the set of all isometries on ${}V$ forms a group under the composition of mappings.

Exercise (2 marks)

Let

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

be a proper isometry. Suppose that ${}f$ is trigonalizable. Show that ${}f$ is already diagonalizable.

Exercise (3 marks)

Let ${}V,W$ denote complex vector spaces, each endowed with an inner product, and let

$\varphi \colon V\longrightarrow W$

denote a linear mapping. Show that ${}\varphi$ is an isometry with respect to the given complex inner products if and only if ${}\varphi$ is an isometry with respect to the corresponding real inner products.

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