Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 13

Exercise for the break

Exercise

Determine, for a field ${}K$, the idempotent elements, that is, elements ${}e\in K$ with ${}e^{2}=e$. Determine the linear projections ${}\varphi \colon K\rightarrow K$.

Exercises

Exercise

We consider the basis

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

of ${}\mathbb {R} ^{2}$. Let ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ denote the projection from ${}\mathbb {R} ^{2}$ to

${}\mathbb {R} {\begin{pmatrix}2\\-5\end{pmatrix}}\subseteq \mathbb {R} ^{2}\,$

with respect to this basis. Determine the matrix of ${}\varphi$ with respect to the standard basis.

Exercise

Let ${}L\subseteq \mathbb {R} ^{3}$ be the solution space for the linear equation

${}3x+4y+6z=0\,,$

and set ${}W=\mathbb {R} {\begin{pmatrix}2\\0\\5\end{pmatrix}}$. Show that

${}\mathbb {R} ^{3}=L\oplus W\,$

holds, and describe the projections onto ${}L$ and onto ${}W$ with respect to the standard basis.

Exercise

Let ${}\varphi \colon V\rightarrow V$ denote a linear projection on a finite-dimensional ${}K$-vector space ${}V$. Show that ${}\varphi$ is described, with respect to a suitable basis ${}v_{1},\ldots ,v_{n}$ of ${}V$, by a matrix of the form

${\begin{pmatrix}1&0&\cdots &\cdots &\cdots &0\\0&\ddots &\ddots &\ddots &\vdots &\vdots \\\vdots &\ddots &1&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &0&\ddots &\vdots \\\vdots &\vdots &\ddots &\ddots &\ddots &0\\0&\cdots &\cdots &\cdots &0&0\end{pmatrix}}.$

Exercise

Let ${}C\subseteq \mathbb {R} ^{3}$ denote the image of the mapping

$\mathbb {R} \longrightarrow \mathbb {R} ^{3},t\longmapsto (t,t^{2},t^{3})=(x,y,z).$

Sketch the images of ${}C$ under the projections onto the different coordinate plans.

Exercise

Show that the sum of two linear projections

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

is in general not a projection.

Exercise

Simplify the proof of Lemma 13.5 with the help of the dimension formula.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Show that the homomorphism space

$\operatorname {Hom} _{K}{\left(V,W\right)}$

is a vector space.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Show that the homomorphism space

$\operatorname {Hom} _{K}{\left(V,W\right)}$

is a linear subspace of the mapping space ${}\operatorname {Map} \,{\left(V,W\right)}$.

Exercise

Let ${}W$ be a ${}K$-vector space over the field ${}K$. Show that the mapping

$\operatorname {Hom} _{K}{\left(K,W\right)}\longrightarrow W,\varphi \longmapsto \varphi (1),$

is an isomorphism of vector spaces.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let ${}\operatorname {Hom} _{K}{\left(V,W\right)}$ be the ${}K$-vector space of all linear mappings from ${}V$ to ${}W$, and let ${}v\in V$ denote a fixed vector. Show that the mapping

$F\colon \operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow W,\varphi \longmapsto F(\varphi ):=\varphi (v),$

is ${}K$-linear.

Exercise

We consider the space of matrices ${}\operatorname {Mat} _{2}(K)$, and we fix the matrix ${}A={\begin{pmatrix}9&-8\\-7&6\end{pmatrix}}$.

1. Show that the mapping

   $\varphi \colon \operatorname {Mat} _{2}(K)\longrightarrow \operatorname {Mat} _{2}(K),M\longmapsto A\circ M,$

   is linear.

2. Determine the describing matrix of ${}\varphi$ with respect to a suitable basis.
3. Determine the kernel and the image of ${}\varphi$.

Exercise

We consider the space of endomorphisms ${}\operatorname {End} _{}{\left(K^{2}\right)}=\operatorname {Hom} _{K}{\left(K^{2},K^{2}\right)}$. Is the mapping

$\operatorname {End} _{}{\left(K^{2}\right)}\longrightarrow \operatorname {End} _{}{\left(K^{2}\right)},\varphi \longmapsto \varphi \circ \varphi ,$

a linear mapping?

Exercise

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$. Show that the first ${}n^{2}+1$ powers

$M^{i},i=0,\ldots ,n^{2},$

are linearly independent in ${}\operatorname {Mat} _{n}(K)$.

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Show the following statements.

1. A linear mapping

   $\varphi \colon U\longrightarrow V$

   with another vector space ${}U$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(U,W\right)},f\longmapsto f\circ \varphi .$

2. A linear mapping

   $\psi \colon W\longrightarrow T$

   with another vector space ${}T$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V,T\right)},f\longmapsto \psi \circ f.$

Exercise

Formulate Lemma 13.8 for matrices with respect to given bases.

Exercise

Let ${}K$ be a field, let ${}V$ and ${}W$ be finite-dimensional ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

be a linear mapping.

a) Show that ${}\varphi$ is surjective if and only if there exists a linear mapping

$\psi \colon W\longrightarrow V$

such that

${}\varphi \circ \psi =\operatorname {Id} _{W}\,.$

b) Let now ${}\varphi$ be surjective, and set

${}S={\left\{\psi :W\rightarrow V\mid \psi {\text{ linear}},\,\varphi \circ \psi =\operatorname {Id} _{W}\right\}}\,,$

and let ${}\psi _{0}\in S$ be fixed. Define a bijection between ${}\operatorname {Hom} _{K}{\left(W,\operatorname {kern} \varphi \right)}$ and ${}S$, such that ${}0$ is mapped to ${}\psi _{0}$.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Show that

${}\operatorname {End} _{}{\left(V\right)}={\left\{\varphi :V\rightarrow V\mid \varphi {\text{ linear}}\right\}}\,$

is, with the natural addition and the composition of mappings, a ring.

The ring of the preceding exercise is called endomorphism ring of ${}V$.

Exercise

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

$g\colon V\longrightarrow V$

denote an isomorphism. Show that the mapping

$\Psi _{g}\colon \operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {End} _{}{\left(V\right)},f\longmapsto gfg^{-1},$

is an isomorphism of vector spaces, and that, moreover, the identities

${}\Psi _{g}(f_{1}\circ f_{2})=\Psi _{g}(f_{1})\circ \Psi _{g}(f_{2})\,$

and

${}\Psi _{g}(\operatorname {Id} _{V})=\operatorname {Id} _{V}\,$

hold.

Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}v_{1},\ldots ,v_{n}$ denote a basis of ${}V$. Determine the dimension of the space of all endomorphisms

$\varphi \colon V\longrightarrow V$

satisfying

${}\varphi (v_{i})\in Kv_{i}\,$

for all ${}i$. How do the matrices of such a ${}\varphi$ with respect to this basis look like?

Exercise

We consider the vectors ${}u={\begin{pmatrix}8\\7\\4\end{pmatrix}}$ and ${}v={\begin{pmatrix}6\\3\\9\end{pmatrix}}$ in ${}\mathbb {R} ^{3}$. Determine the dimension of the vector space, consisting of all linear mappings

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}$

that send ${}u$ to the ${}x$-axis and ${}v$ to the ${}y$-axis. Describe the corresponding linear subspace of the space of matrices with respect to suitable chosen bases.

Exercise

We consider the line ${}G=\mathbb {R} {\begin{pmatrix}-9\\8\\3\end{pmatrix}}\subseteq \mathbb {R} ^{3}$. Let ${}M\subseteq \operatorname {Hom} _{\mathbb {R} }{\left(\mathbb {R} ^{3},\mathbb {R} ^{2}\right)}$ be the set of all linear mappings that send this line to the axes of coordinates. Show that ${}M$ is not a linear subspace of the space of homomorphisms.

Exercise

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

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

denote automorphisms, such that, for every linear subspace ${}U\subseteq V$, the identity ${}\varphi (U)=\psi (U)$ holds. Show that ${}\varphi =a\psi$ with some ${}a\in K$.

Hand-in-exercises

Exercise (4 marks)

We consider the basis

${\begin{pmatrix}3\\7\end{pmatrix}},{\begin{pmatrix}4\\-9\end{pmatrix}}$

in ${}\mathbb {R} ^{2}$. Let ${}\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the projection of ${}\mathbb {R} ^{2}$ onto ${}\mathbb {R} {\begin{pmatrix}3\\7\end{pmatrix}}\subseteq \mathbb {R} ^{2}$ with respect to this basis. Determine the matrix of ${}\varphi$ with respect to the standard basis.

Exercise (5 (1+4) marks)

a) Show that the ${}2\times 2$-matrices

${\begin{pmatrix}{\frac {1\pm {\sqrt {1-4bc}}}{2}}&b\\c&{\frac {1\mp {\sqrt {1-4bc}}}{2}}\end{pmatrix}}$

describe projections. Here, ${}b,c\in K$ are such that a square root ${}{\sqrt {1-4bc}}$ exists.

b) Determine all ${}2\times 2$-matrices

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

that describe a projection.

Exercise (2 marks)

Let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces, and let ${}\varphi _{1},\ldots ,\varphi _{k}\in \operatorname {Hom} _{K}{\left(V,W\right)}$. Show

${}\operatorname {rk} \,{\left(\varphi _{1}+\cdots +\varphi _{k}\right)}\leq \sum _{i=1}^{k}\operatorname {rk} \,\varphi _{i}\,.$

Exercise (3 marks)

Let ${}G_{1}\neq G_{2}$ and ${}H_{1}\neq H_{2}$ denote lines in ${}K^{3}$. What is the dimension of the space

${}W={\left\{\varphi \in \operatorname {Hom} _{K}{\left(K^{3},K^{3}\right)}\mid \varphi (G_{1})\subseteq H_{1}{\text{ and }}\varphi (G_{2})\subseteq H_{2}\right\}}\,?$

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

Let the vectors

${}u={\begin{pmatrix}6\\9\\-2\end{pmatrix}}$ and ${}v={\begin{pmatrix}4\\-4\\7\end{pmatrix}}$ in ${}\mathbb {R} ^{3}$ be given. We consider the linear subspace

${}U\subseteq \operatorname {Hom} _{K}{\left(\mathbb {R} ^{3},\mathbb {R} ^{3}\right)}\,$

consisting of all linear mappings ${}\varphi$ satisfying simultaneously the following conditions:

a) ${}\varphi (u)\in \langle {\begin{pmatrix}2\\8\\3\end{pmatrix}},{\begin{pmatrix}7\\0\\5\end{pmatrix}}\rangle$.

b) ${}\varphi (v)\in \langle {\begin{pmatrix}6\\-5\\4\end{pmatrix}}\rangle$.

1. Determine the dimension of ${}U$.
2. Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by linear equations.
3. Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by a basis.

Footnotes

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