Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 15

Exercise for the break

Exercise

A fruit seller sells apples, pears, and cherries. He cannot remember exactly his purchase prices, but he remembers that he has paid for ${}5$ kilograms of apples, the same as for ${}3$ kilograms of pears and one kilogram of cherries together. Moreover, naturally, the old fruit-seller-rule holds that the price of ${}3$ kilograms of apples equals the price of one kilogram of pears and one kilogram of cherries together. How does the orthogonal space for these price conditions look like? What is the price for one kilogram of apples if the price for one kilogram of cherries is ${}5$ Euro?

Exercises

Exercise

Determine a basis of the orthogonal space to ${}\mathbb {R} {\begin{pmatrix}5\\-4\\1\end{pmatrix}}\subseteq \mathbb {R} ^{3}$.

Exercise

Establish a system of linear equations, whose solution space is the line ${}U={\begin{pmatrix}2\\5\\-1\end{pmatrix}}\mathbb {R}$.

Exercise

Let ${}V$ be a ${}K$-vector space with its dual space ${}{V}^{*}$. Show that the orthogonal space ${}U^{\perp }\subseteq {V}^{*}$ to a linear subspace ${}U\subseteq V$, and the orthogonal space ${}F^{\perp }\subseteq V$ to a linear subspace ${}F\subseteq {V}^{*}$, are indeed linear subspaces.

Exercise

Let ${}U=\langle u_{1},\ldots ,u_{r}\rangle$ be a linear subspace of a ${}K$-vector space ${}V$. Show that

${}U^{\perp }={\left\{f\in {V}^{*}\mid f(u_{i})=0{\text{ for all }}i=1,\ldots ,r\right\}}\,$

holds.

Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces. Show that in the dual space ${}{V}^{*}$, the equality

${}{\left(U_{1}+U_{2}\right)}^{\perp }=U_{1}^{\perp }\cap U_{2}^{\perp }\,$

holds.

Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces. Show that in the dual space ${}{V}^{*}$, the equality

${}{\left(U_{1}\cap U_{2}\right)}^{\perp }=U_{1}^{\perp }+U_{2}^{\perp }\,$

holds.

Exercise

Prove Lemma 15.6   (4) using Lemma 13.13   (1).

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space and ${}U\subseteq V$ a linear subspace.

a) Show that there exist linear forms ${}L_{1},\ldots ,L_{r}$ on ${}V$ such that

${}U=\bigcap _{i=1}^{r}\operatorname {kern} L_{i}\,.$

b) Show that ${}U\subseteq V$ is the kernel of a linear mapping on ${}V$ (into some ${}K^{r}$).

c) Show that every linear subspace ${}U\subseteq K^{n}$ is the solution space of a system of linear equations.

Exercise

Describe the space of all symmetric ${}n\times n$-matrices using linear forms.

Exercise

Let ${}K$ be a field.

a) Let ${}L$ be an ${}\ell \times m$-matrix, and let ${}N$ be an ${}n\times p$-matrix. Show that

${\left\{M\mid M\in \operatorname {Mat} _{m\times n}(K),\,L\circ M\circ N=0\right\}}$

is a linear subspace of ${}\operatorname {Mat} _{m\times n}(K)$.

b) Let ${}V$ be an ${}n$-dimensional and ${}W$ be an ${}m$-dimensional ${}K$-vector space, and let ${}U\subseteq V$ and ${}T\subseteq W$ denote linear subspaces. Describe the linear subspace

${\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi (U)\subseteq T\right\}}$

using suitable bases and part a).

Exercise

Let

${}U=\langle {\begin{pmatrix}4\\-7\end{pmatrix}}\rangle \subseteq K^{2}\,$

and

${}T=\langle {\begin{pmatrix}-2\\5\end{pmatrix}}\rangle \subseteq K^{2}\,.$

a) Describe the linear subspace ${}W$ of the space of all ${}2\times 2$-matrices that map the linear subspace ${}U$ to the linear subspace ${}T$, as the solution space of a linear system.

b) Describe ${}W$ by an eliminated system.

c) Determine the dimension of ${}W$.

Exercise

Let

${}U=\langle {\begin{pmatrix}3\\4\\8\end{pmatrix}},\,{\begin{pmatrix}2\\1\\-1\end{pmatrix}}\rangle \subseteq K^{3}\,$

and

${}T=\langle {\begin{pmatrix}7\\1\\3\end{pmatrix}},\,{\begin{pmatrix}5\\4\\2\end{pmatrix}}\rangle \subseteq K^{3}\,.$

a) Describe the linear subspace ${}W$ of the space of all ${}3\times 3$-matrices that map the linear subspace ${}U$ into the linear subspace ${}T$, as a solution space of a linear system.

b) Describe ${}W$ by an eliminated linear system.

c) Determine the dimension of ${}W$.

Exercise

Let ${}V$ be a ${}K$-vector space, together with a direct sum decomposition

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

Show that

${}{V}^{*}=V_{1}^{\perp }\oplus V_{2}^{\perp }\,$

holds, and that the restriction of the dual mapping

${V}^{*}\longrightarrow {V_{1}}^{*}$

onto ${}V_{2}^{\perp }$ is an isomorphism.

Exercise

Describe the linear mapping

$\varphi \colon K^{3}\longrightarrow K^{2},$

given by the matrix

${\begin{pmatrix}6&5&2\\4&8&2\end{pmatrix}},$

in the sense of Lemma 15.10 , using the standard basis and the standard dual basis.

Exercise

Let

$\varphi \colon V\longrightarrow W$

be a linear mapping between the finite-dimensional ${}K$-vector spaces ${}V$ and ${}W$, and let

$\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$

be the dual mapping. Show

${}\operatorname {rk} \,\varphi =\operatorname {rk} \,\varphi ^{*}\,.$

Exercise

Let ${}\varphi \colon V\rightarrow W$ be an isomorphism between the ${}K$-vector spaces ${}V$ and ${}W$, and let ${}\varphi ^{*}\colon {W}^{*}\rightarrow {V}^{*}$ denote its dual mapping. Let ${}U\subseteq V$ be a linear subspace and ${}T:=\varphi (U)\subseteq W$ the corresponding linear subspace in ${}W$. Show that, in the dual spaces, the orthogonal spaces ${}U^{\perp }$ and ${}T^{\perp }$ correspond to each other.

Exercise

Let

$\varphi \colon V\longrightarrow W$

be a linear mapping between the finite-dimensional ${}K$-vector spaces ${}V$ and ${}W$, and let

$\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$

denote its dual mapping.

a) Show that, for a linear subspace ${}S\subseteq V$, the relation

${}\varphi ^{*-1}(S^{\perp })=(\varphi (S))^{\perp }\,$

holds.

b) Show that, for a linear subspace ${}T\subseteq W$, the relation

${}\varphi ^{*}(T^{\perp })=(\varphi ^{-1}(T))^{\perp }\,$

holds.

Exercise

Let

${}V=\mathbb {R} ^{(\mathbb {N} )}\,$

be the countably direct sum of ${}\mathbb {R}$ with itself, with the basis ${}e_{n}$, ${}n\in \mathbb {N}$. Let ${}p_{k}$, ${}k\in \mathbb {N}$, be the projections

$\mathbb {R} ^{(\mathbb {N} )}\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto a_{k}.$

a) Show that

$\varphi \colon V\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto \sum _{n\in \mathbb {N} }a_{n},$

is a linear form on ${}V$, which is not a linear combination of the projections.

b) Show that the natural mapping from ${}V$ into its bidual ${}{{V}^{*}}^{*}$ is not surjective.

Exercise

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

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

which assigns to a linear mapping its dual mapping, is linear.

Hand-in-exercises

Exercise (3 marks)

Let ${}V$ be a finite-dimensional ${}K$-vector space of dimension ${}n$, let ${}f_{1},\ldots ,f_{r}\in {V}^{*}$ be linear forms, and set

${}F=\langle f_{1},\ldots ,f_{r}\rangle \subseteq {V}^{*}\,.$

Show that these linear forms are linearly independent if and only if

${}\dim _{K}{\left(F^{\perp }\right)}=n-r\,$

holds.

Exercise (6 (4+1+1) marks)

Let

${}U=\langle {\begin{pmatrix}-2\\5\\-4\end{pmatrix}},\,{\begin{pmatrix}3\\4\\6\end{pmatrix}}\rangle \subseteq K^{3}\,$

and

${}T=\langle {\begin{pmatrix}6\\3\\4\end{pmatrix}},\,{\begin{pmatrix}1\\5\\-8\end{pmatrix}}\rangle \subseteq K^{3}\,.$

a) Describe the linear subspace ${}W$ of the space of all ${}3\times 3$-matrices that map the linear subspace ${}U$ into the linear subspace ${}T$, as a solution space of a linear system.

b) Describe ${}W$ by an eliminated linear system.

c) Determine the dimension of ${}W$.

Exercise (3 marks)

Describe the linear mapping

$\varphi \colon K^{4}\longrightarrow K^{3},$

given by the matrix

${\begin{pmatrix}3&6&4&5\\5&2&1&3\\8&1&2&7\end{pmatrix}},$

in the sense of Lemma 15.10 , using the standard basis and the standard dual basis.

Exercise (3 marks)

Let ${}V$ be a finite-dimensional ${}K$-vector space, with its dual space ${}{V}^{*}$ and the bidual ${}{{V}^{*}}^{*}$. Let ${}F\subseteq {V}^{*}$ be a linear subspace. Show that the two orthogonal spaces ${}F^{\perp }\subseteq V$ (in the sense of definition) and ${}F^{\perp }\subseteq {{V}^{*}}^{*}$ (in the sense of definition) are equal via the natural identification of the space and its bidual.

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