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

    Exercise for the break

### Exercise Create referencenumber

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.MDLD/vector space
    Exercises

### Exercise Create referencenumber

Let $K$ be a field,MDLD/field and let $V$ and $W$ be vector spacesMDLD/vector spaces over $K$. Show that the product setMDLD/product set
    $V\times W$
is also a $K$-vector space.
===Exercise Exercise 6.3
Let $K$ be a fieldMDLD/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 subspaceMDLD/linear subspace of $K^{n}$. How is this solution space related to the solution spaces of the individual equations?

### Exercise Create referencenumber

Check whether the following subsets of $\mathbb {R} ^{2}$ are linear subspaces:MDLD/linear subspaces
* $V_{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x+2y=0\right\}}$,
* $V_{2}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x\geq y\right\}}$,
* $V_{3}={\left\{(x,y)\in \mathbb {R} ^{2}\mid y=x+1\right\}}$,
* $V_{4}={\left\{(x,y)\in \mathbb {R} ^{2}\mid xy=0\right\}}$.

### Exercise Create referencenumber

Let $V$ be a vector spaceMDLD/vector space over a fieldMDLD/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 exercise show that, in the definition of a vector space, no axiom for the scalar multiplication is redundant.

### Exercise Create referencenumber

Give an example of a fieldMDLD/field $K$, a commutative groupMDLD/commutative group $(V,+,0)$, and a mappingMDLD/mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms,MDLD/vector space axioms with the exception of
    $(5)\,\,\,1u=u.$

### Exercise Create referencenumber

Give an example of a fieldMDLD/field $K$, a commutative groupMDLD/commutative group $(V,+,0)$, and a mappingMDLD/mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms,MDLD/vector space axioms with the exception of
    $(6)\,\,\,r(su)=(rs)u.$

### Exercise Create referencenumber

Give an example of a fieldMDLD/field $K$, a commutative groupMDLD/commutative group $(V,+,0)$, and a mappingMDLD/mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms,MDLD/vector space axioms with the exception of
    $(7)\,\,\,r(u+v)=ru+rv.$

### Exercise Create referencenumber

Give an example of a fieldMDLD/field $K$, a commutative groupMDLD/commutative group $(V,+,0)$, and a mappingMDLD/mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms,MDLD/vector space axioms with the exception of
    $(8)\,\,\,(r+s)u=ru+su.$
===Exercise Exercise 6.10
Show that the addition and the scalar multiplication of a vector spaceMDLD/vector space $V$ can be restricted to a linear subspace,MDLD/linear subspace and that this subspace with the inherited structures of $V$ is a vector space itself.

### Exercise Create referencenumber

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

### Exercise Create referencenumber

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 subspaceMDLD/linear subspace in the space of all $2\times 2$-matrices.

### Exercise Create referencenumber

We consider in $\mathbb {Q} ^{3}$ the linear subspacesMDLD/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 Create referencenumber

Let $K$ be a field,MDLD/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.MDLD/vector space

### Exercise Create referencenumber

let $K$ be a fieldMDLD/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 subspaceMDLD/linear subspace of $K^{I}$.

### Exercise Create referencenumber

field,MDLD/field let $I$ denote an index set, and let $K^{I}=\operatorname {Map} \,{\left(I,K\right)}$ the corresponding vector space.MDLD/vector space
* Show that is a linear subspaceMDLD/linear subspace of $K^{I}$.
    $E={\left\{f\in K^{I}\mid f(i)=0{\text{ for all }}i\in I{\text{ up to finite exceptions}}\right\}}\,$
* For every $i\in I$, let $e_{i}\in K^{I}$ be defined by Show that every element $f\in E$ can be expressed uniquely as a linear combinationMDLD/linear combination of the family $e_{i}$, $i\in I$.
    $e_{i}(j)={\begin{cases}1,{\text{ if }}j=i\,,\\0{\text{ else}}\,.\end{cases}}\,$
The following four exercises use concepts from analysis.

### Exercise Create referencenumber

Let $K$ denote an ordered field,MDLD/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 subspaceMDLD/linear subspace of the space of sequences
    $F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}K\right\}}\,.$

### Exercise Create referencenumber

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.MDLD/linear subspace

### Exercise Create referencenumber

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.MDLD/linear subspace

### Exercise Create referencenumber

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.MDLD/linear subspace

### Exercise Create referencenumber

We consider the set
    $M=\operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}\,,$
which is, with the pointwise addition $+$ of functions, a commutative group.MDLD/commutative group On this set, the compositionMDLD/composition (map) of mappings $\circ$ gives an associative operation,MDLD/associative operation with the identityMDLD/identity as neutral element.MDLD/neutral element
* Show that the distributivity law in the form holds.
    $(f+g)\circ h=f\circ h+g\circ h\,$
* Show that the distributivity law in the form does not hold.
    $h\circ (f+g)=h\circ f+h\circ g\,$

### Exercise Create referencenumber

Write in $\mathbb {Q} ^{2}$ the vector
    $(2,-7)$
as a linear combination of the vectors
    $(5,-3){\text{ and }}(-11,4).$

### Exercise Create referencenumber

Write in $\mathbb {C} ^{2}$ the vector
    $(1,0)$
as a linear combinationMDLD/linear combination of the vectors
    $(3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).$

### Exercise Create referencenumber

Express, in $\mathbb {R} ^{3}$, the vector
    $(1,0,0)$
as a linear combinationMDLD/linear combination of the vectors
    $(1,-2,5),(4,0,3){\text{ and }}(2,1,1).$
===Exercise Exercise 6.25
Let $K$ be a field,MDLD/field and let $V$ be a $K$-vector space.MDLD/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 Exercise 6.26
Let $K$ be a field and let $V$ be a $K$-vector space.MDLD/vector space Prove the following facts.
* Let $U_{j}$, $j\in J$, be a family of subspaces of $V$. Prove that also the intersection is a subspace.
    $U=\bigcap _{j\in J}U_{j}\,$
* Let $v_{i}$, $i\in I$, be a family of elements of $V$ and consider the subset $W$ of $V$ which is given by all linear combinations of these elements. Show that $W$ is a subspace of $V$.
* 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) Exercise 6.27
Let $K$ be a field,MDLD/field and let $V$ be a $K$-vector space.MDLD/vector space Show that the following properties hold (for $v\in V$ and $s\in K$).
* We have $0v=0$.
* We have $s0=0$.
* We have $(-1)v=-v$.
* If $s\neq 0$ and $v\neq 0$ then $sv\neq 0$.

### Exercise (4 marks) Create referencenumber

We consider in $\mathbb {Q} ^{4}$ the linear subspacesMDLD/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) Create referencenumber

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

### Exercise (3 marks) Create referencenumber

Write in $\mathbb {Q} ^{3}$ the vector
    $(2,5,-3)$
as a linear combination of the vectors
    $(1,2,3),(0,1,1){\text{ und }}(-1,2,4).$
Prove that it cannot be expressed as a linear combination of two of the three vectors.

### Exercise (3 marks) Create referencenumber

Let $K$ be a field,MDLD/field $V$ a $K$-vector space,MDLD/vector space and $M$ a set with a binary operationMDLD/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 mappingMDLD/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.