# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 22

    Exercises

### Exercise

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

### Exercise *

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

### Exercise

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

### Exercise

Let $M$ be a $m\times n$- matrix. Show that the matrix product $Me_{j}$ of $M$ with the $j$-th standard vector (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 (regarded as a row vector)?

### Exercise

Let
    $D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$
be a diagonal matrix, and $M$ an $n\times n$-matrix. Describe $DM$ and $MD$.

### Exercise

Let
    $D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$
be a diagonal matrix, and $c={\begin{pmatrix}c_{1}\\\vdots \\c_{n}\end{pmatrix}}$ an $n$-tuple over a field $K$, and let $x={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}$ be a tuple of variables. What is specific about the system of linear equations
    $Dx=c\,,$
and how can you solve it?

### Exercise

Compute the product of matrices
    ${\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}},$
according to the two possible parantheses.
For the following statement we will get soon a simpler proof via the relation between matrices and linear mappings.

### Exercise

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

### Exercise

Show that the matrix multiplication of square matrices is, in general, not commutative.
For a matrix $M$, we denote by $M^{n}$ the $n$-th fold composition (matrix multiplication) of $M$ with itself. $M^{n}$ is called the $n$-th power of the matrix.

### Exercise

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

### Exercise

Out of the resources $R_{1},R_{2}$ and $R_{3}$, several commodities $P_{1},P_{2},P_{3},P_{4}$ are produced. The following table shows, how much of the resources is needed to produce the commodities (always in suitable units).
a) Establish a matrix, which computes, applied to a four-tuple of commodities, the required resources.
b) The following table shows how much of each commodity shall be produced in a month.
What resources are necessary?
c) The following table shows how much of each resource is delivered on a certain day.
Which tuples of commodities can be produced from this without waste?

### Exercise

Determine (approximately) the coordinates of the sketched point (the side length of a box represents a unit).

### Exercise

Draw the following points in the Cartesian plane $\mathbb {R} ^{2}$.
    $(3,-7),\,(-1,-2),\,(0,5),\,(4,4),\,(4,5),\,(-3,0),\,(0,0).$

### Exercise

Let a point $P=(x,y)$ be given in the plane $\mathbb {R} ^{2}$. Sketch the points
    $(-x,y),\,(x,-y),\,(-x,-y),\,(3x,3y),(-2x,-2y).$

### Exercise

Let a point $P=(x,y)$ be given in the plane $\mathbb {R} ^{2}$. Sketch the set of all points
    $(cx,cy),\,c\in \mathbb {R} .$

### Exercise

Draw two points $P$ and $Q$ in the Cartesian plane $\mathbb {R} ^{2}$ and add them.

### Exercise

Show that the product space $K^{n}$, for a field $K$, is, with componentwise addition and scalar multiplication, the properties
* $r(su)=(rs)u$,
* $r(u+v)=ru+rv$,
* $(r+s)u=ru+su$,
    $1u=u\,,$
hold.

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

### 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

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 $K$ 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

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

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 $K$ 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
    $C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}\mathbb {R} \right\}}\,$
be the set of all real Cauchy sequences. Show that $C$ is a linear subspace of the space of all sequences
    $F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}\mathbb {R} \right\}}\,.$

### 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.
    Hand-in-exercises

### Exercise (3 marks)

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

### Exercise (3 marks)

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

### Exercise (4 marks)

Let $n\in \mathbb {N}$. Find and prove a formula for the $n$-th power of the matrix
    ${\begin{pmatrix}a&b\\0&c\end{pmatrix}}.$

### Exercise (2 marks)

Find, appart from the matrices ${\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ and ${\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}$, four more matrices $M$ fulfilling the property $M^{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}$.

### 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 $s\in K$).
* 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 (3 marks)

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.