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

Exercises

### Exercise

Compute the following product of matrices

${\displaystyle {\begin{pmatrix}Z&E&I&L&E\\R&E&I&H&E\\H&O&R&I&Z\\O&N&T&A&L\end{pmatrix}}\cdot {\begin{pmatrix}S&E&I\\P&V&K\\A&E&A\\L&R&A\\T&T&L\end{pmatrix}}.}$

### Exercise *

Compute, over the complex numbers, the following product of matrices

${\displaystyle {\begin{pmatrix}2-{\mathrm {i} }&-1-3{\mathrm {i} }&-1\\{\mathrm {i} }&0&4-2{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\1-{\mathrm {i} }\\2+5{\mathrm {i} }\end{pmatrix}}.}$

### Exercise

Determine the product of matrices

${\displaystyle e_{i}\circ e_{j},}$

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

### Exercise

Let ${\displaystyle {}M}$ be a ${\displaystyle {}m\times n}$- matrix. Show that the matrix product ${\displaystyle {}Me_{j}}$ of ${\displaystyle {}M}$ with the ${\displaystyle {}j}$-th standard vector (regarded as column vector), is the ${\displaystyle {}j}$-th column of ${\displaystyle {}M}$. What is ${\displaystyle {}e_{i}M}$, where ${\displaystyle {}e_{i}}$ is the ${\displaystyle {}i}$-th standard vector (regarded as a row vector)?

### Exercise

Let

${\displaystyle {}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,}$

be a diagonal matrix, and ${\displaystyle {}M}$ an ${\displaystyle {}n\times n}$-matrix. Describe ${\displaystyle {}DM}$ and ${\displaystyle {}MD}$.

### Exercise

Let

${\displaystyle {}D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,}$

be a diagonal matrix, and ${\displaystyle {}c={\begin{pmatrix}c_{1}\\\vdots \\c_{n}\end{pmatrix}}}$ an ${\displaystyle {}n}$-tuple over a field ${\displaystyle {}K}$, and let ${\displaystyle {}x={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}$ be a tuple of variables. What is specific about the system of linear equations

${\displaystyle {}Dx=c\,,}$

and how can you solve it?

### Exercise

Compute the product of matrices

${\displaystyle {\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}},}$

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 ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}A}$ be an ${\displaystyle {}m\times n}$-matrix, ${\displaystyle {}B}$ an ${\displaystyle {}n\times p}$-matrix and ${\displaystyle {}C}$ a ${\displaystyle {}p\times r}$-matrix over ${\displaystyle {}K}$. Show that ${\displaystyle {}(AB)C=A(BC)}$.

### Exercise

Show that the matrix multiplication of square matrices is, in general, not commutative.

For a matrix ${\displaystyle {}M}$, we denote by ${\displaystyle {}M^{n}}$ the ${\displaystyle {}n}$-th fold composition (matrix multiplication) of ${\displaystyle {}M}$ with itself. ${\displaystyle {}M^{n}}$ is called the ${\displaystyle {}n}$-th power of the matrix.

### Exercise

Compute, for the matrix

${\displaystyle {}M={\begin{pmatrix}2&4&6\\1&3&5\\0&1&2\end{pmatrix}}\,,}$

the powers

${\displaystyle M^{i},\,i=1,2,3,4.}$

### Exercise

Out of the resources ${\displaystyle {}R_{1},R_{2}}$ and ${\displaystyle {}R_{3}}$, several commodities ${\displaystyle {}P_{1},P_{2},P_{3},P_{4}}$ are produced. The following table shows, how much of the resources is needed to produce the commodities (always in suitable units).

${\displaystyle {}R_{1}}$ ${\displaystyle {}R_{2}}$ ${\displaystyle {}R_{3}}$
${\displaystyle {}P_{1}}$ ${\displaystyle {}6}$ ${\displaystyle {}2}$ ${\displaystyle {}3}$
${\displaystyle {}P_{2}}$ ${\displaystyle {}4}$ ${\displaystyle {}1}$ ${\displaystyle {}2}$
${\displaystyle {}P_{3}}$ ${\displaystyle {}0}$ ${\displaystyle {}5}$ ${\displaystyle {}2}$
${\displaystyle {}P_{4}}$ ${\displaystyle {}2}$ ${\displaystyle {}1}$ ${\displaystyle {}5}$

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.

 ${\displaystyle {}P_{1}}$ ${\displaystyle {}P_{2}}$ ${\displaystyle {}P_{3}}$ ${\displaystyle {}P_{4}}$ ${\displaystyle {}6}$ ${\displaystyle {}4}$ ${\displaystyle {}7}$ ${\displaystyle {}5}$

What resources are necessary?

c) The following table shows how much of each resource is delivered on a certain day.

 ${\displaystyle {}R_{1}}$ ${\displaystyle {}R_{2}}$ ${\displaystyle {}R_{3}}$ ${\displaystyle {}12}$ ${\displaystyle {}9}$ ${\displaystyle {}13}$

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 ${\displaystyle {}\mathbb {R} ^{2}}$.

${\displaystyle (3,-7),\,(-1,-2),\,(0,5),\,(4,4),\,(4,5),\,(-3,0),\,(0,0).}$

### Exercise

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

${\displaystyle (-x,y),\,(x,-y),\,(-x,-y),\,(3x,3y),(-2x,-2y).}$

### Exercise

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

${\displaystyle (cx,cy),\,c\in \mathbb {R} .}$

### Exercise

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

### Exercise

Show that the product space ${\displaystyle {}K^{n}}$, for a field ${\displaystyle {}K}$, is, with componentwise addition and scalar multiplication, the properties

1. ${\displaystyle {}r(su)=(rs)u}$,
2. ${\displaystyle {}r(u+v)=ru+rv}$,
3. ${\displaystyle {}(r+s)u=ru+su}$,
4. ${\displaystyle {}1u=u\,,}$

hold.

### Exercise

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

${\displaystyle V\times W}$

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

### Exercise

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

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

### Exercise

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

### Exercise

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

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

### Exercise

Let ${\displaystyle {}K}$ be a field and let

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

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

### Exercise

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

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

which fulfill the condition

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

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

### Exercise

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

### Exercise

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

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

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

### Exercise

Let

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

be the set of all real Cauchy sequences. Show that ${\displaystyle {}C}$ is a linear subspace of the space of all sequences

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

### Exercise

Show that the subset

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

is a linear subspace.

### Exercise

Show that the subset

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

is a linear subspace.

### Exercise

Show that the subset

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

is not a linear subspace.

Hand-in-exercises

### Exercise (3 marks)

Compute, over the complex numbers, the following product of matrices

${\displaystyle {\begin{pmatrix}3-2{\mathrm {i} }&1+5{\mathrm {i} }&0\\7{\mathrm {i} }&2+{\mathrm {i} }&4-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1-2{\mathrm {i} }&-{\mathrm {i} }\\3-4{\mathrm {i} }&2+3{\mathrm {i} }\\5-7{\mathrm {i} }&2-{\mathrm {i} }\end{pmatrix}}.}$

### Exercise (3 marks)

We consider the matrix

${\displaystyle {}M={\begin{pmatrix}0&a&b&c\\0&0&d&e\\0&0&0&f\\0&0&0&0\end{pmatrix}}\,}$

over a field ${\displaystyle {}K}$. Show that the fourth power of ${\displaystyle {}M}$ is ${\displaystyle {}0}$, that is

${\displaystyle {}M^{4}=MMMM=0\,.}$

### Exercise (4 marks)

Let ${\displaystyle {}n\in \mathbb {N} }$. Find and prove a formula for the ${\displaystyle {}n}$-th power of the matrix

${\displaystyle {\begin{pmatrix}a&b\\0&c\end{pmatrix}}.}$

### Exercise (2 marks)

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

### Exercise (3 marks)

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

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

### Exercise (3 marks)

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