Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 6/en/latex

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\setcounter{section}{6}

\zwischenueberschrift{Warm-up-exercises}

\inputexercise
{}
{

Compute the following product of matrices
\mathdisp {\begin{pmatrix} Z & E & I & L & E \\ R & E & I & H & E \\ H & O & R & I & Z \\ O & N & T & A & L \end{pmatrix} \cdot \begin{pmatrix} S & E & I \\ P & V & K \\ A & E & A \\ L & R & A \\ T & T & L \end{pmatrix}} { . }

}
{} {}

\inputexercise
{}
{

Compute over the complex numbers the following product of matrices
\mathdisp {\begin{pmatrix} 2- { \mathrm i} & -1-3 { \mathrm i} & -1 \\ { \mathrm i} & 0 & 4-2 { \mathrm i} \end{pmatrix} \begin{pmatrix} 1+ { \mathrm i} \\1- { \mathrm i} \\ 2+5 { \mathrm i} \end{pmatrix}} { . }

}
{} {}

\inputexercise
{}
{

Determine the product of matrices
\mathdisp {e_i \circ e_j} { , }
where the $i$-th standard vector \zusatzklammer {of length $n$} {} {} is considered as a row vector and the $j$-th standard vector \zusatzklammer {also of length $n$} {} {} is considered as a column vector.

}
{} {}

\inputexercise
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{

Let $M$ be a $m\times n$- matrix. Show that the product of matrices $M e_j$, with the $j$-th standard vector \zusatzklammer {regarded as column vector} {} {} is the $j$-th column of $M$. What is $e_i M$, where $e_i$ is the $i$-th standard vector \zusatzklammer {regarded as a row vector} {} {}?

}
{} {}

\inputexercise
{}
{

Compute the product of matrices
\mathdisp {\begin{pmatrix} 2+ { \mathrm i} & 1- \frac{1}{2} { \mathrm i} & 4 { \mathrm i} \\ -5+7 { \mathrm i} & \sqrt{2} + { \mathrm i} & 0 \end{pmatrix} \begin{pmatrix} -5+4 { \mathrm i} & 3-2 { \mathrm i} \\ \sqrt{2} - { \mathrm i} & e + \pi { \mathrm i} \\ 1 & - { \mathrm i} \end{pmatrix} \begin{pmatrix} 1+ { \mathrm i} \\2-3 { \mathrm i} \end{pmatrix}} { }
according to the two possible parantheses.

}
{} {} For the following statements we will soon give a simple proof using the relationship between matrices and linear maps.

\inputexercise
{}
{

Show that the multiplication of matrices is associative. More precisely: Let $K$ be a field and let $A$ be an
\mathl{m \times n}{-}matrix, $B$ an
\mathl{n \times p}{-}matrix and $C$ a
\mathl{p\times r}{-}matrix over $K$. Show that
\mathl{(A B)C=A(BC)}{.}

}
{} {}

For a matrix $M$ we denote by $M^n$ the $n$-th product of $M$ with iself. This is also called the $n$-th \stichwort {power} {} of the matrix.

\inputexercise
{}
{

Compute for the matrix
\mathdisp {M= \begin{pmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\0 & 1 & 2 \end{pmatrix}} { }
the powers
\mathbeddisp {M^{i}} {}
{\, i = 1 , \ldots , 4} {}
{} {} {} {.}

}
{} {}

\inputexercise
{}
{

Let ${\displaystyle K}$ be a field and let ${\displaystyle V}$ and ${\displaystyle W}$ be two vector spaces over ${\displaystyle K}$. Show that the product
\mathdisp {V\times W} { }
is also a ${\displaystyle K}$-vector space.

}
{} {}

\inputexercise
{}
{

Let $K$ be a field and $I$ an index set. Show that
\mathdisp {K^I \defeq \operatorname{Maps} \, (I, K)} { }
with pointwise addition and scalar multiplication is a $K$-vector space.

}
{} {}

\inputexercise
{}
{

Let $K$ be a field and let
\mathdisp {\begin{matrix} a _{ 1 1 } x _1 + a _{ 1 2 } x _2 + \cdots + a _{ 1 n } x _{ n } & = & 0 \\ a _{ 2 1 } x _1 + a _{ 2 2 } x _2 + \cdots + a _{ 2 n } x _{ n } & = & 0 \\ \vdots & \vdots & \vdots \\ a _{ m 1 } x _1 + a _{ m 2 } x _2 + \cdots + a _{ m n } x _{ n } & = & 0 \end{matrix}} { }
be a system of linear equations over $K$. Show that the set of all solutions of the system is a subspace of $K^n$. How is this solution space related to the solution spaces of the individual equations?

}
{} {}

\inputexercise
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{

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

}
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\inputexercise
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{

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

}
{} {}

\zwischenueberschrift{Hand-in-exercises}

\inputexercise
{3}
{

Compute over the complex numbers the following product of matrices
\mathdisp {\begin{pmatrix} 3-2 { \mathrm i} & 1+5 { \mathrm i} & 0 \\ 7 { \mathrm i} & 2+ { \mathrm i} & 4- { \mathrm i} \end{pmatrix} \begin{pmatrix} 1-2 { \mathrm i} & - { \mathrm i} \\ 3-4 { \mathrm i} & 2+3 { \mathrm i} \\ 5-7 { \mathrm i} & 2- { \mathrm i} \end{pmatrix}} { . }

}
{} {}

\inputexercise
{4}
{

We consider the matrix
\mathdisp {M= \begin{pmatrix} 0 & a & b & c \\ 0 & 0 & d & e \\ 0 & 0 & 0 & f \\ 0 & 0 & 0 & 0 \end{pmatrix}} { }
over a field $K$. Show that the fourth power of $M$ is $0$, that is
\mathdisp {M^4=MMMM =0} { . }

}
{} {}

\inputexercise
{3}
{

Let $K$ be a field and let $V$ be a $K$-vector space. Show that the following properties hold (where \mathlk{\lambda \in K}{} and \mathlk{v \in V}{}). \aufzaehlungvier{We have $0v=0$. }{We have $\lambda 0=0$. }{We have$(-1) v= -v$. }{If
\mathl{\lambda \neq 0}{} and
\mathl{v \neq 0}{} then
\mathl{\lambda v \neq 0}{.} }

}
{} {}

\inputexercise
{3}
{

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.

}
{} {}