Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 6

Exercise for the break

Exercise

Show that the set of all "symmetric“ ${}2\times 2$-matrices over a field ${}K$, that is, matrices of the form

${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$

satisfying the condition

${}a_{12}=a_{21}\,,$

is, with componentwise addition and componentwise scalar multiplication, a ${}K$-vector space.

Exercises

Exercise

Let ${}K$ be a field, and let ${}V$ and ${}W$ be vector spaces over ${}K$. Show that the product set

$V\times W$

is also a ${}K$-vector space.

Exercise

Let ${}V$ be a vector space over a field ${}K$. Let ${}s_{1},\ldots ,s_{k}\in K$ and ${}v_{1},\ldots ,v_{n}\in V$. Show

${}{\left(\sum _{i=1}^{k}s_{i}\right)}\cdot {\left(\sum _{j=1}^{n}v_{j}\right)}=\sum _{1\leq i\leq k,\,1\leq j\leq n}s_{i}\cdot v_{j}\,.$

The following four exercises show that, in the definition of a vector space, no axiom for the scalar multiplication is redundant.

Exercise

Give an example of a field ${}K$, a commutative group ${}(V,+,0)$, and a mapping

$K\times V\longrightarrow V,(s,v)\longmapsto sv,$

such that this structure fulfills all vector space axioms, with the exception of

$(5)\,\,\,1u=u.$

Exercise

Give an example of a field ${}K$, a commutative group ${}(V,+,0)$, and a mapping

$K\times V\longrightarrow V,(s,v)\longmapsto sv,$

such that this structure fulfills all vector space axioms, with the exception of

$(6)\,\,\,r(su)=(rs)u.$

Exercise

Give an example of a field ${}K$, a commutative group ${}(V,+,0)$, and a mapping

$K\times V\longrightarrow V,(s,v)\longmapsto sv,$

such that this structure fulfills all vector space axioms, with the exception of

$(7)\,\,\,r(u+v)=ru+rv.$

Exercise

Give an example of a field ${}K$, a commutative group ${}(V,+,0)$, and a mapping

$K\times V\longrightarrow V,(s,v)\longmapsto sv,$

such that this structure fulfills all vector space axioms, with the exception of

$(8)\,\,\,(r+s)u=ru+su.$

Exercise

Check whether the following subsets of ${}\mathbb {R} ^{2}$ are linear subspaces:

1. ${}V_{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x+2y=0\right\}}$,
2. ${}V_{2}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x\geq y\right\}}$,
3. ${}V_{3}={\left\{(x,y)\in \mathbb {R} ^{2}\mid y=x+1\right\}}$,
4. ${}V_{4}={\left\{(x,y)\in \mathbb {R} ^{2}\mid xy=0\right\}}$.

Exercise

Let ${}K$ be a field, and let

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}$

be a system of linear equations over ${}K$. Show that the set of all solutions of this system is a linear subspace of ${}K^{n}$. How is this solution space related to the solution spaces of the individual equations?

Exercise

Show that the addition and the scalar multiplication of a vector space ${}V$ can be restricted to a linear subspace, and that this subspace with the inherited structures of ${}V$ is a vector space itself.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}U,W\subseteq V$ be linear subspaces of ${}V$. Prove that the union ${}U\cup W$ is a linear subspace of ${}V$ if and only if ${}U\subseteq W$ or ${}W\subseteq U$.

Exercise

Let ${}D$ be the set of all real ${}2\times 2$-matrices

${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}},$

which fulfill the condition

${}a_{11}a_{22}-a_{21}a_{12}=0\,.$

Show that ${}D$ is not a linear subspace in the space of all ${}2\times 2$-matrices.

Exercise

We consider in ${}\mathbb {Q} ^{3}$ the linear subspaces

${}U=\langle {\begin{pmatrix}2\\1\\4\end{pmatrix}},{\begin{pmatrix}3\\-2\\7\end{pmatrix}}\rangle \,$

and

${}W=\langle {\begin{pmatrix}5\\-1\\11\end{pmatrix}},{\begin{pmatrix}1\\-3\\3\end{pmatrix}}\rangle \,.$

Show that ${}U=W$.

Exercise

Let ${}K$ be a field, and ${}I$ an index set. Show that

${}K^{I}:=\operatorname {Maps} \,(I,K)\,,$

with pointwise addition and scalar multiplication, is a ${}K$-vector space.

Exercise

let ${}K$ be a field and let ${}J\subseteq I$ denote two index sets. Show that ${}K^{J}=\operatorname {Map} \,{\left(J,K\right)}$ is, in a natural way, a linear subspace of ${}K^{I}$.

Exercise

Let ${}K$ be a field, let ${}I$ denote an index set, and let ${}K^{I}=\operatorname {Map} \,{\left(I,K\right)}$ be the corresponding vector space.

1. Show that

   ${}E={\left\{f\in K^{I}\mid f(i)=0{\text{ for all }}i\in I{\text{ up to finite exceptions}}\right\}}\,$

   is a linear subspace of ${}K^{I}$.

2. For every ${}i\in I$, let ${}e_{i}\in K^{I}$ be defined by

   ${}e_{i}(j)={\begin{cases}1,{\text{ if }}j=i\,,\\0{\text{ else}}\,.\end{cases}}\,$

   Show that every element ${}f\in E$ can be expressed uniquely as a linear combination of the family ${}e_{i}$, ${}i\in I$.

The following four exercises use concepts from analysis.

Exercise

Let ${}K$ denote an ordered field, and set

${}C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}K\right\}}\,.$

Show that ${}C$ is a linear subspace of the space of sequences

${}F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}K\right\}}\,.$

Exercise

Show that the subset

${}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ continuous}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is a linear subspace.

Exercise

Show that the subset

${}S={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ differentiable}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is a linear subspace.

Exercise

Show that the subset

${}M={\left\{f:\mathbb {R} \rightarrow \mathbb {R} \mid f{\text{ monotonic}}\right\}}\subseteq \operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,$

is not a linear subspace.

Exercise

We consider the set

${}M=\operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,,$

which is, with the pointwise addition ${}+$ of functions, a commutative group. On this set, the composition of mappings ${}\circ$ gives an associative operation, with the identity as neutral element.

1. Show that the distributive law in the form

   ${}(f+g)\circ h=f\circ h+g\circ h\,$

   holds.

2. Show that the distributive law in the form

   ${}h\circ (f+g)=h\circ f+h\circ g\,$

   does not hold.

Exercise

Write in ${}\mathbb {Q} ^{2}$ the vector

$(2,-7)$

as a linear combination of the vectors

$(5,-3){\text{ and }}(-11,4).$

Exercise

Write in ${}\mathbb {C} ^{2}$ the vector

$(1,0)$

as a linear combination of the vectors

$(3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).$

Exercise

Express, in ${}\mathbb {R} ^{3}$, the vector

$(1,0,0)$

as a linear combination of the vectors

$(1,-2,5),(4,0,3){\text{ and }}(2,1,1).$

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{i}$, ${}i\in I$, be a family of vectors in ${}V$, and let ${}w\in V$ be another vector. Assume that the family

$w,v_{i},i\in I,$

is a system of generators of ${}V$, and that ${}w$ is a linear combination of the ${}v_{i}$, ${}i\in I$. Prove that also ${}v_{i}$, ${}i\in I$, is a system of generators of ${}V$.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Prove the following facts.

1. Let ${}U_{j}$, ${}j\in J$, be a family of linear subspaces of ${}V$. Prove that also the intersection

   ${}U=\bigcap _{j\in J}U_{j}\,$

   is a subspace.

2. Let ${}v_{i}$, ${}i\in I$, be a family of elements of ${}V$, and consider the subset ${}W$ of ${}V$ that is given by all linear combinations of these elements. Show that ${}W$ is a subspace of ${}V$.
3. The family ${}v_{i}$, ${}i\in I$, is a system of generators of ${}V$ if and only if

   ${}\langle v_{i},\,i\in I\rangle =V\,.$

Hand-in-exercises

Exercise (3 marks)

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Show that the following properties hold (for ${}v\in V$ and ${}s\in K$).

1. We have ${}0v=0$.
2. We have ${}s0=0$.
3. We have ${}(-1)v=-v$.
4. If ${}s\neq 0$ and ${}v\neq 0$, then ${}sv\neq 0$.

Exercise (4 marks)

We consider in ${}\mathbb {Q} ^{4}$ the linear subspaces

${}U=\langle {\begin{pmatrix}3\\1\\-5\\2\end{pmatrix}},{\begin{pmatrix}2\\-2\\4\\-3\end{pmatrix}},{\begin{pmatrix}1\\0\\3\\2\end{pmatrix}}\rangle \,$

and

${}W=\langle {\begin{pmatrix}6\\-1\\2\\1\end{pmatrix}},{\begin{pmatrix}0\\-2\\-2\\-7\end{pmatrix}},{\begin{pmatrix}9\\2\\-1\\10\end{pmatrix}}\rangle \,.$

Show that ${}U=W$.

Exercise (3 marks)

Give an example of a vector space ${}V$ and of three subsets of ${}V$ that satisfy two of the subspace axioms, but not the third.

Exercise (3 marks)

Write in ${}\mathbb {Q} ^{3}$ the vector

$(2,5,-3)$

as a linear combination of the vectors

$(1,2,3),(0,1,1),{\text{ and }}(-1,2,4).$

Prove that it cannot be expressed as a linear combination of two of the three vectors.

Exercise (3 marks)

Let ${}K$ be a field, ${}V$ a ${}K$-vector space, and ${}M$ a set with a binary operation

$+\colon M\times M\longrightarrow M,$

and a mapping

$\cdot \colon K\times M\longrightarrow M.$

Let

$\varphi \colon V\longrightarrow M$

be a surjective mapping satisfying

$\varphi (x+y)=\varphi (x)+\varphi (y)\,\,{\text{ and }}\,\,\varphi (sx)=s\varphi (x)$

for all ${}x,y\in V$ and ${}s\in K$. Show that ${}M$ is a ${}K$-vector space.

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