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

Exercise for the break

Exercise

Symmetric 2x2-matrices/Vector space/Exercise

Exercises

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

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

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}\,.

Die followingn vier Aufgaben zeigen, dass keines the Axiome for the scalarmultiplikation eines vectorraumes überflüssig ist.

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 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

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

Vector space/Linear subspace/I to K/J in I/Exercise

Exercise

Vector space/I to K/Almost all zero/Exercise

The following four exercises use concepts from analysis.

Exercise

Vector space/Set of sequences in ordered K/Cauchy-sequences as linear subspace/Exercise

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

Function space/Differentiable/R/Subspace/Exercise/Exercise

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

Mapping set/R/Operations/Distributivity laws/Exercise

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

Linear combination/R/(1,0,0)/By (1,-2,5), (4,0,3) and (2,1,1)/Exercise

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.

Let ${}U_{j}$ , ${}j\in J$ , be a family of subspaces of ${}V$ . Prove that also the intersection
${}U=\bigcap _{j\in J}U_{j}\,$

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