Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 41

Exercises

Exercise

Determine the kernel of the matrix ${}{\begin{pmatrix}2&3\\4&6\end{pmatrix}}$, and the kernel of the transposed matrix ${}{\begin{pmatrix}2&4\\3&6\end{pmatrix}}$.

Exercise

Let

$\varphi \colon V\longrightarrow V$

be a bijective angle-preserving mapping on a Euclidean vector space ${}V$. Show that the adjoint mapping is also angle-preserving.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$, and let ${}v_{i}$, ${}i\in I$ denote a basis. Let

$\varphi ,\psi \colon V\longrightarrow V$

be linear mappings. Show that ${}\psi$ is the adjoint mapping of ${}\varphi$ if and only if

${}\left\langle \varphi (v_{i}),v_{j}\right\rangle =\left\langle v_{i},\psi (v_{j})\right\rangle \,$

holds for all ${}i,j\in I$.

Exercise

We consider the linear mapping ${}\varphi \colon \mathbb {C} ^{3}\rightarrow \mathbb {C} ^{3}$, given by the matrix

${\begin{pmatrix}5-2{\mathrm {i} }&3-7{\mathrm {i} }&4-7{\mathrm {i} }\\1-8{\mathrm {i} }&9-2{\mathrm {i} }&17{\mathrm {i} }\\6&8-9{\mathrm {i} }&2-2{\mathrm {i} }\end{pmatrix}}$

with respect to the standard basis. Determine the matrix of the adjoint endomorphism.

Exercise

We consider the linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$, given by the matrix ${}{\begin{pmatrix}-3&7\\4&5\end{pmatrix}}$ with respect to the standard basis. We define an inner product ${}\Psi$ on ${}\mathbb {R} ^{2}$ by ${}\Psi (e_{1},e_{1})=4$, ${}\Psi (e_{2},e_{2})=5$, and ${}\Psi (e_{1},e_{2})=3$. Determine the matrix of the adjoint endomorphism of ${}\varphi$ with respect to the given inner product, and with respect to the basis ${}{\begin{pmatrix}3\\-5\end{pmatrix}},{\begin{pmatrix}1\\4\end{pmatrix}}$.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the adjoint endomorphism fulfills the following properties (here, ${}\varphi ,\psi$ denote endomorphisms).

1. ${}{\left(\varphi +\psi \right)}^{\hat {}}={\hat {\varphi }}+{\hat {\psi }}\,.$

2. ${}{\left(s\varphi \right)}^{\hat {}}={\overline {s}}{\hat {\varphi }}\,.$

3. ${}{\left({\hat {\varphi }}\right)}^{\hat {}}=\varphi \,.$

4. ${}{\left(\varphi \circ \psi \right)}^{\hat {}}={\hat {\psi }}\circ {\hat {\varphi }}\,.$

Exercise

Let ${}V$ and ${}W$ be Euclidean vector spaces, and let ${}\varphi \colon V\rightarrow V$ be an endomorphism with the adjoint endomorphism ${}{\hat {\varphi }}$. Let ${}\psi \colon V\rightarrow W$ be an isometry. Show that the adjoint endomorphism of

$\psi \circ \varphi \circ \psi ^{-1}\colon W\longrightarrow W$

equals ${}\psi \circ {\hat {\varphi }}\circ \psi ^{-1}$.

Exercise

Let ${}V$ be a finite-dimensional complex vector space, endowed with an inner product, and let a ${}\mathbb {C}$-linear mapping ${}\varphi \colon V\rightarrow V$ be given. Let ${}{\hat {\varphi }}$ be the adjoint endomorphism of ${}\varphi$. Show that ${}{\hat {\varphi }}$ coincides with the adjoint endomorphism of ${}\varphi$, considered as a real-linear mapping, with respect to the corresponding real inner product.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with inner product ${}\left\langle -,-\right\rangle$. Show that the assignment

$\operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {End} _{}{\left(V\right)},\varphi \longmapsto {\hat {\varphi }},$

is antilinear.

Exercise

Show, using the matrix

${\begin{pmatrix}1&1\\0&1\end{pmatrix}},$

that a ${}\varphi$-invariant linear subspace ${}U\subseteq \mathbb {R} ^{2}$ is not necessarily invariant under the adjoint endomorphism ${}{\hat {\varphi }}$.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with inner product, and let

${}V=V_{1}\oplus V_{2}\,$

be a direct sum of the linear subspaces ${}V_{1}$ and ${}V_{2}$. Let

$\varphi _{1}\colon V_{1}\longrightarrow V_{1}$

and

$\varphi _{2}\colon V_{2}\longrightarrow V_{2}$

be linear mappings, and let

${}\varphi =\varphi _{1}\oplus \varphi _{2}\,$

denote their sum.

a) Suppose that the sum decomposition is orthogonal; that is, ${}V_{1}$ and ${}V_{2}$ are orthogonal to each other. Show

${}{\hat {\varphi }}={\hat {\varphi _{1}}}\oplus {\hat {\varphi _{2}}}\,.$

b) Show that the statement in part (a) does not hold when the sum decomposition is not orthogonal.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let ${}U\subseteq V$ denote a linear subspace with the orthogonal projection

$p_{U}\colon V\longrightarrow U.$

Let

$\varphi \colon V\longrightarrow V$

be a linear mapping, and let ${}{\hat {\varphi }}$ denote the corresponding adjoint endomorphism. Show the relation

${}p_{U}\circ {\hat {\varphi }}{|}_{U}={\left(p_{U}\circ \varphi {|}_{U}\right)}^{\hat {}}\,.$

Exercise

Let ${}V$ be a Euclidean vector space with the dual space ${}{V}^{*}$.

a) Show that via

${}\left\langle f,g\right\rangle :=\left\langle \operatorname {Grad} \,f,\operatorname {Grad} \,g\right\rangle \,,$

an inner product is defined on the dual space.

b) Show that the natural mapping

$V\longrightarrow {V}^{*},v\longmapsto \left\langle v,-\right\rangle ,$

is an isometry between ${}V$ and ${}{V}^{*}$.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show the following statements.

a) The identity is self-adjoint.

b) The composition of two commuting self-adjoint mappings is again self-adjoint.

c) For a bijective self-adjoint mapping, also the inverse mapping is self-adjoint.

Exercise

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

$\varphi \colon V\longrightarrow V$

be a self-adjoint endomorphism, and let ${}U\subseteq V$ denote a ${}\varphi$-invariant linear subspace. Show that also the restriction

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

is self-adjoint.

Exercise

Let ${}V$ be a finite-dimensional ${}\mathbb {C}$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that for every linear form ${}f\in {V}^{*}$, there exists a uniquely determined vector ${}y\in V$ with

${}f(v)=\left\langle y,v\right\rangle \,$

for all ${}v\in V$, and there exists a uniquely determined vector ${}z\in V$ with

${}f(v)=\left\langle v,z\right\rangle \,$

for all ${}v\in V$.

Exercise

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the set of all self-adjoint endomorphisms of ${}V$ is a linear subspace of ${}\operatorname {End} _{}{\left(V\right)}$.

Exercise

Let

$\varphi \colon V\longrightarrow V$

be an isometry on a Euclidean vector space ${}V$. Show that ${}\varphi$ is self-adjoint if and only if the order of ${}\varphi$ equals ${}1$ or ${}2$.

Exercise

Let ${}M$ be a real-symmetric ${}2\times 2$-matrix. Show that ${}M$ has an eigenvalue.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let ${}v_{1},\ldots ,v_{n}$ denote a basis of ${}V$. Let ${}\left\langle -,-\right\rangle$ be the inner product on ${}V$ defined by

${}\left\langle v_{i},v_{j}\right\rangle ={\begin{cases}1,{\text{ if }}i=j,\\0\,{\text{ else}}.\end{cases}}\,$

For a linear mapping ${}\varphi \colon V\rightarrow V$, let ${}\Psi _{\varphi }$ denote the corresponding (via $\left\langle -,-\right\rangle$) sesquilinear form. Show that the Gram matrix of ${}\Psi _{\varphi }$ with respect to the basis coincides with the describing matrix of ${}\varphi$ with respect to the basis.

Hand-in-exercises

Exercise (4 marks)

Let ${}V_{1},\ldots ,V_{n},V_{n+1}$ be vector spaces over ${}\mathbb {C}$, and let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i+1}$

denote linear or antilinear mappings, and let

${}\varphi =\varphi _{n}\circ \varphi _{n-1}\circ \cdots \circ \varphi _{2}\circ \varphi _{1}\,$

denote the composition of the mappings. Show by induction over ${}n$ the following two statements.

a) If the number of the antilinear mappings is even, then ${}\varphi$ is linear.

b) If the number of the antilinear mappings is odd, then ${}\varphi$ is antilinear.

Does the converse of this statement hold?

Exercise (6 marks)

We consider the linear mapping ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrix ${}{\begin{pmatrix}-6&-3\\4&5\end{pmatrix}}$ with respect to the standard basis. We equip ${}\mathbb {R} ^{2}$ with the inner product ${}\Psi$ given by ${}\Psi (e_{1},e_{1})=3$, ${}\Psi (e_{2},e_{2})=7$, and ${}\Psi (e_{1},e_{2})=2$. Determine the matrix of the adjoint endomorphism of ${}\varphi$ with respect to the given inner product, and with respect to the basis ${}{\begin{pmatrix}4\\-3\end{pmatrix}},{\begin{pmatrix}-2\\-7\end{pmatrix}}$.

Exercise (4 marks)

Let

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

be the linear mapping given by the matrix ${}{\begin{pmatrix}3&-2\\-4&7\end{pmatrix}}$ with respect to the basis ${}{\begin{pmatrix}2\\5\end{pmatrix}},{\begin{pmatrix}-1\\6\end{pmatrix}}$. Determine the matrix of the adjoint endomorphism of ${}\varphi$ with respect to the standard inner product, and with respect to this basis.

Exercise (4 marks)

Let

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

be the linear mapping that is given by the matrix ${}{\begin{pmatrix}4&-2+9{\mathrm {i} }\\-2-9{\mathrm {i} }&5\end{pmatrix}}$ with respect to the standard basis. Determine the eigenvalues, and the eigenvectors of ${}\varphi$.

Exercise (4 (1+1+1+1) marks)

Show that the matrix

${\begin{pmatrix}1&1\\0&2\end{pmatrix}},$

considered as a linear mapping from ${}\mathbb {R} ^{2}$ to ${}\mathbb {R} ^{2}$, is not self-adjoint, using the following methods.

a) Determine the adjoint mapping of ${}M$.

b) Lemma 41.10   (1) is not fulfilled.

c) Lemma 41.10   (3) is not fulfilled.

d) There does not exist an orthonormal basis of ${}\mathbb {R} ^{2}$ consisting of eigenvectors oh ${}M$ (that is, the conclusion of Theorem 41.11 is not fulfilled.)

<< | Linear algebra (Osnabrück 2024-2025)/Part II | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)