OpenLLM-France/wikimedia huggingface dataset:
# 3D Modelling/Mathematics
## Introduction

This page about 3D Modelling in Mathematics can be displayed as Wiki2Reveal slides. Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides. The following aspects of 3D Modelling in Mathematics are considered in detail:
* (1) Image of 3D models,
* (2) Animation of 3D models,
* (3) STL models with the option to rotate the 3D object manually.

## Example

The following example of Klein bottleis a STL model that can be used to visualize surface in Mathematics. The in Wikiversity learning resources can use 3D models to support spatial representation of

## Animation

The 3D models can be used in conjunction with other options of visualization, like GIF animation showing the time evolution of the 3D surface.

## Learning Tasks / Activities

Identify different type of visualizations and compare them in terms of
* benefits for the learner,
* workload to create those visualizations,
* use of 3D objects instead of 3D visualization (e.g. 3D printing)
* benefits of Open Educational Resources for application in digital learning environments

## Target Group

The target groups for 3D Modelling/Mathematics of the learning resource are:
* Bachelor/Master students mathematics,
* Bachelor/Master students mathematics education

## Learning Tasks / Activities

Identify different type of visualizations and compare them in terms of
* benefits for the learner,
* workload to create those visualizations,
* benefits of Open Educational Resources for application in digital learning environments
# Vector space/Basis/Exchange lemma/Fact/Proof

We show first that the new family is a generating system. Because of
    $w=\sum _{i=1}^{n}s_{i}v_{i}\,$
and $s_{k}\neq 0$, we can express the vector $v_{k}$ as
    $v_{k}={\frac {1}{s_{k}}}w-\sum _{i=1}^{k-1}{\frac {s_{i}}{s_{k}}}v_{i}-\sum _{i=k+1}^{n}{\frac {s_{i}}{s_{k}}}v_{i}\,.$
Let $u\in V$ be given. Then, we can write
    ${\begin{aligned}u&=\sum _{i=1}^{n}t_{i}v_{i}\\&=\sum _{i=1}^{k-1}t_{i}v_{i}+t_{k}v_{k}+\sum _{i=k+1}^{n}t_{i}v_{i}\\&=\sum _{i=1}^{k-1}t_{i}v_{i}+t_{k}{\left({\frac {1}{s_{k}}}w-\sum _{i=1}^{k-1}{\frac {s_{i}}{s_{k}}}v_{i}-\sum _{i=k+1}^{n}{\frac {s_{i}}{s_{k}}}v_{i}\right)}+\sum _{i=k+1}^{n}t_{i}v_{i}\\&=\sum _{i=1}^{k-1}{\left(t_{i}-t_{k}{\frac {s_{i}}{s_{k}}}\right)}v_{i}+{\frac {t_{k}}{s_{k}}}w+\sum _{i=k+1}^{n}{\left(t_{i}-t_{k}{\frac {s_{i}}{s_{k}}}\right)}v_{i}.\end{aligned}}$
To show the linear independence, we may assume $k=1$ to simplify the notation. Let
    $t_{1}w+\sum _{i=2}^{n}t_{i}v_{i}=0\,$
be a representation of $0$. Then
    $0=t_{1}w+\sum _{i=2}^{n}t_{i}v_{i}=t_{1}{\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}+\sum _{i=2}^{n}t_{i}v_{i}=t_{1}s_{1}v_{1}+\sum _{i=2}^{n}{\left(t_{1}s_{i}+t_{i}\right)}v_{i}\,.$
From the linear independence of the original family, we deduce $t_{1}s_{1}=0$. Because of $s_{1}\neq 0$, we get $t_{1}=0$. Therefore, $\sum _{i=2}^{n}t_{i}v_{i}=0$, and hence $t_{i}=0$ for all $i$.
# Basis exchange theorem/1/Example

We consider the standard basis $e_{1},e_{2},e_{3}$ of $K^{3}$ and the two linearly independent vectors $u_{1}={\begin{pmatrix}3\\2\\1\end{pmatrix}}$ and $u_{2}={\begin{pmatrix}5\\4\\2\end{pmatrix}}$. We want to extend this family to a basis, using the standard basis and according to the inductive method described in the proof of the basis exchange theorem. We first consider
    $u_{1}=3e_{1}+2e_{2}+e_{3}\,.$
Since no coefficient is $0$, we can extend $u_{1}$ with any two standard vectors to obtain a basis. We work with the new basis
    $u_{1},e_{1},e_{2}.$
In a second step, we would like to include $u_{2}$. We have
    $u_{2}={\begin{pmatrix}5\\4\\2\end{pmatrix}}=2{\begin{pmatrix}3\\2\\1\end{pmatrix}}-e_{1}=2u_{1}-e_{1}+0e_{2}\,.$
According to the proof, we have to get rid of $e_{1}$, as its coefficient is $\neq 0$ in this equation (we can not get rid of $e_{2}$). The new basis is, therefore,
    $u_{1},u_{2},e_{2}.$
# Standard space/K^n/Dimension n/Fact/Proof

The standard basis $e_{i}$, $i=1,\ldots ,n$, consists of $n$ vectors; hence, the dimension is $n$.
# Complex numbers/Real/Two-dimensional/Example

The complex numbers form a two-dimensional real vector space; a basis is $1$ and ${\mathrm {i} }$.
# Vector space/Linear subspace/Dimensions/Compare/Fact/Proof

Set $n=\dim _{K}{\left(V\right)}$. Every linearly independent family in $U$ is also linearly independent in $V$. Therefore, due to the basis exchange theorem, every linearly independent family in $U$ has length $\leq n$. Suppose that $k\leq n$ has the property that there exists a linearly independent family with $k$ vectors in $U$ but no such family with $k+1$ vectors. Let ${\mathfrak {u}}=u_{1},\ldots ,u_{k}$ be such a family. This is then a maximal linearly independent family in $U$. Therefore, due to fact, it is a basis of $U$.
# Vector space/Subspaces of K^123/Example

Let $K$ be a field. It is easy to get an overview over the linear subspaces of $K^{n}$, as the dimension of a linear subspace equals $k$ with $0\leq k\leq n$, due to fact. For $n=0$, there is only the null space itself; for $n=1$, there is the null space and $K$ itself. For $n=2$, there is the null space, the whole plane $K^{2}$, and the one-dimensional lines through the origin. Every line $G$ has the form
    $G=Kv={\left\{sv\mid s\in K\right\}}\,,$
with a vector $v\neq 0$. Two vectors different from $0$ define the same line if and only if they are linearly dependent. For $n=3$, there is the null space, the whole space $K^{3}$, the one-dimensional lines through the origin, and the two-dimensional planes through the origin.
# Vector space/Basis complement/Fact/Proof

Let $b_{1},\ldots ,b_{n}$ be a basis of $V$. Due to the basis exchange theorem, there are $n-k$ vectors from the basis ${\mathfrak {b}}$ that, together with the given vectors $u_{1},\ldots ,u_{k}$, form a basis of $V$.
# K^3/215/137/412/Basis/Exercise

Prove that in $\mathbb {R} ^{3}$, the three vectors
    ${\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}$
form a basis.
# Linearly independent/Simple properties/Fact/Proof/Exercise

Let $K$ be a field, let $V$ be a $K$-vector space, and let $v_{i}$, $i\in I$, be a family of vectors in $V$. Prove the following facts.
* If the family is linearly independent, then for each subset $J\subseteq I$, also the family $v_{i}$ , $i\in J$ is linearly independent.
* The empty family is linearly independent.
* If the family contains the null vector, then it is not linearly independent.
* If a vector appears several times in the family, then the family is not linearly independent.
* A vector $v$ is linearly independent if and only if $v\neq 0$.
* Two vectors $v$ and $u$ are linearly independent if and only if $u$ is not a scalar multiple of $v$, and vice versa.
# Family of vectors/Scaling/Generating system and linearly independent/Exercise

Let $K$ be a field, let $V$ be a $K$-vector space, and let $v_{i}$, $i\in I$, be a family of vectors in $V$. Let $\lambda _{i}$, $i\in I$, be a family of elements $\neq 0$ in $K$. Prove that the family $v_{i}$, $i\in I$, is linearly independent (a system of generators of $V$, a basis of $V$), if and only if the same holds for the family $\lambda _{i}v_{i}$, $i\in I$.
# Vector space/Addition in coordinate system/Independence of basis/Exercise

Let $V$ be a $K$-vector space, let $v_{1},\ldots ,v_{n}$ be a basis of $V$, and let
    $\psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V,{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n},$
be the corresponding bijective mapping in the sense of remark. Show that this mapping transforms the componentwise addition in $K^{n}$ into the vector addition in $V$, that is,
    $\psi _{\mathfrak {v}}{\left({\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\right)}=\psi _{\mathfrak {v}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+\psi _{\mathfrak {v}}{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\,$
holds.
# Ordered field/Sequences/Zero sequences as linear subspace/1 over n and 1 over n^2/Exercise

Let $K$ be an ordered field, and let
    $V=K^{\mathbb {N} _{+}}\,$
be the vector space of all sequences in $K$ (with componentwise addition and scalar multiplication).
a) Show that (without using theorems about convergent sequences), the set of zero sequences, that is,
    $U={\left\{(x_{n})_{n\in \mathbb {N} _{+}}\mid (x_{n})_{n\in \mathbb {N} _{+}}{\text{ converges to 0}}\right\}}\,$
is a $K$-linear subspace of $V$.
b) Are the sequences
    $(1/n)_{n\in \mathbb {N} _{+}}{\text{ and }}(1/n^{2})_{n\in \mathbb {N} _{+}}$
linearly independent in $V$?
# Matrices/Standard matrices/Basis/Exercise

Show that, for the space of all $m\times n$-matrices $\operatorname {Mat} _{m\times n}(K)$, the matrices $E_{ij}$ that have in position $(i,j)$ the entry $1$, and elsewhere $0$, form a basis.
# Polynomial ring/Basis/Linear form product/Exercise

Let $\mathbb {R} [X]$ be the polynomial ring over $\mathbb {R}$. We set $P_{0}=1$, and, for $n\in \mathbb {N} _{+}$, we set
    $P_{n}=(X-1)(X-2)\cdots (X-n)\,.$
Show that $P_{n}$, $n\in \mathbb {N}$, is a basis of $\mathbb {R} [X]$.
# Field extension/Q in R/Basis/Exercise

Let $\mathbb {Q} ^{n}$ be the $n$-dimensional standard vector space over $\mathbb {Q}$, and let $v_{1},\ldots ,v_{n}\in \mathbb {Q} ^{n}$ be a family of vectors. Prove that this family is a $\mathbb {Q}$-basis of $\mathbb {Q} ^{n}$ if and only if the same family, considered as a family in $\mathbb {R} ^{n}$, is an $\mathbb {R}$-basis of $\mathbb {R} ^{n}$.
# Linear system/3 variables/Solution space is (3,2,-5)/Exercise

Let $K$ be a field. Find a linear system of equations in three variables whose solution space is exactly
    ${\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.$
# Q^2/Non-trivial representation of 0/1/Exercise

Find, for the vectors
    ${\begin{pmatrix}4\\5\end{pmatrix}},\,{\begin{pmatrix}1\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\7\end{pmatrix}}$
in $\mathbb {Q} ^{2}$, a non-trivial representation of the zero vector.
# Q^3/Non-trivial representation of 0/1/Exercise

Find, for the vectors
    ${\begin{pmatrix}7\\-5\\3\end{pmatrix}},\,{\begin{pmatrix}-4\\1\\-6\end{pmatrix}},\,{\begin{pmatrix}2\\8\\0\end{pmatrix}},\,{\begin{pmatrix}5\\-5\\8\end{pmatrix}}$
in $\mathbb {Q} ^{3}$, a non-trivial representation of the zero vector.
# Theorem of Hamel/Examples/Exercise

What does the Theorem of Hamel mean in the following examples?
* The real numbers $\mathbb {R}$ as a $\mathbb {Q}$-vector space.
* The set of all real seqeunces
    $\mathbb {R} ^{\mathbb {N} }={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid x_{n}\in \mathbb {R} \right\}}\,.$
* The set of all continuous functions from $\mathbb {R}$ to $\mathbb {R}$.
# Symmetric matrix/Definition

An $n\times n$-matrix
    $M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$
is called symmetric if $a_{ij}=a_{ji}$ holds for all $i,j$.
# Base change/Standard basis in R^4/Example/Exercise

Consider the standard basis $e_{1},e_{2},e_{3},e_{4}$ in $\mathbb {R} ^{4}$ and the three vectors
    ${\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.$
Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector, as shown in the base exchange theorem. Can one take any standard vector?
# Linear system/Basis completion/1/Exercise

We consider the linear equations
    $9x-8y+7z-8u+4v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,3y+7z-4u+6v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2z+5u+7v=0\,,$
over $\mathbb {R}$.
* Determine a basis ${\mathfrak {b}}_{1}$ of the solution space of this linear system.
* Extend the basis ${\mathfrak {b}}_{1}$ to a basis ${\mathfrak {b}}_{2}$ of the solution space of the linear system consisting of the first two equations.
* Extend the basis ${\mathfrak {b}}_{2}$ to a basis ${\mathfrak {b}}_{3}$ of the solution space of the linear system consisting of the first equation alone.
* Extend the basis ${\mathfrak {b}}_{3}$ to a basis ${\mathfrak {b}}_{4}$ of the total space $\mathbb {R} ^{5}$.
# Basic multiplication table/Last digit/Vector space dimension/Exercise

We consider the last digit in the basic multiplication table as a family of $9$-tuples of length $9$, that is, the row vectors in the matrix
    ${\begin{pmatrix}1&2&3&4&5&6&7&8&9\\2&4&6&8&0&2&4&6&8\\3&6&9&2&5&8&1&4&7\\4&8&2&6&0&4&8&2&6\\5&0&5&0&5&0&5&0&5\\6&2&8&4&0&6&2&8&4\\7&4&1&8&5&2&9&6&3\\8&6&4&2&0&8&6&4&2\\9&8&7&6&5&4&3&2&1\end{pmatrix}}.$
What is the dimension of the linear subspace in $\mathbb {R} ^{9}$ generated by these tuples?
# Polynomial/Zeroes in -1,0,1/Degree up to 6/Dimension/Exercise

Show that the set of all real polynomials of degree $\leq 6$ that have a zero at $-1$, at $0$ and at $1$, is a finite-dimensional subspace of $\mathbb {R} [X]$. Determine the dimension of this vector space.
# Linear system/Basis completion/2/Exercise

We consider the linear equations
    $8x-3y+5z+7u+6v=0\,,$
    $9x+2y+\,\,z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,7y-\,\,z+4u\,\,\,\,\,\,\,\,\,\,\,\,\,\,=0\,,$
over $\mathbb {R}$.
* Determine a basis ${\mathfrak {b}}_{1}$ of the solution space of this linear system.
* Extend the basis ${\mathfrak {b}}_{1}$ to a basis ${\mathfrak {b}}_{2}$ of the solution space of the linear system consisting of the first two equations.
* Extend the basis ${\mathfrak {b}}_{2}$ to a basis ${\mathfrak {b}}_{3}$ of the solution space of the linear system consisting of the first equation alone.
* Extend the basis ${\mathfrak {b}}_{3}$ to a basis ${\mathfrak {b}}_{4}$ of the total space $\mathbb {R} ^{5}$.
# Vector space/Product/Dimension/Exercise

Let $K$ be a field, and let $V$ and $W$ be two finite-dimensional $K$-vector spaces with
    $\dim _{K}{\left(V\right)}=n\,$
and
    $\dim _{K}{\left(W\right)}=m\,.$
What is the dimension of the Cartesian product $V\times W$?
# Polynomials/Zeroes at -2,3/Degree at most 4/Dimension/Exercise

Show that the set of all real polynomials of degree $\leq 4$ that have a zero for $-2$ and for $3$, forms a finite-dimensional linear subspace in $\mathbb {R} [X]$. Determine its dimension.
# Complex vector space/Finite-dimensional/Real basis/Exercise

Let $V$ be a finite-dimensional vector space over the complex numbers, and let $v_{1},\ldots ,v_{n}$ be a basis of $V$. Prove that the family of vectors
    $v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}$
forms a basis for $V$, considered as a real vector space.
# Linear system/Dimension/2/Exercise

a) Determine the dimension of the solution space of the linear system
    $2x+5y+7z+4u-3v+2w=0\,$
    $4x+9y+6z+5u-v+w=0\,$
    $7x+8y-3z+u+3v+3w=0\,$
    $-x+6y+16z+8u-7v=0\,$
in the variables $x,y,z,u,v,w$.
b) What is the dimension of the solution space if we consider the system in the variables $x,y,z,u,v,w,r,s$?
# R as Q Vector space/Q to Q else zero/Exercise

Consider the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
that sends a rational number $q\in \mathbb {Q}$ to $q$, and all the irrational numbers to $0$. Is this a linear map? Is it compatible with multiplication by a scalar?
# Real plane/Linear mapping/Not injective/Rational restriction/Exercise

Give an example for a linear mapping
    $\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$
that is not injective but such that its restriction
    $\mathbb {Q} ^{2}\longrightarrow \mathbb {R} ^{2}$
is injective.
# Telephone company/Change/Exercise

The telephone companies $A,B$ and $C$ compete for a market, where the market customers in a year $j$ are given by the customers-tuple $K_{j}=(a_{j},b_{j},c_{j})$ (where $a_{j}$ is the number of customers of $A$ in the year $j$ etc.). There are customers passing from one provider to another one during a year.
* The customers of $A$ remain for $80\%$ with $A$, while $10\%$ of them goes to $B$, and the same percentage goes to $C$.
* The customers of $B$ remain for $70\%$ with $B$, while $10\%$ of them goes to $A$, and $20\%$ goes to $C$.
* The customers of $C$ remain for $50\%$ with $C$, while $20\%$ of them goes to $A$, and $30\%$ goes to $B$.
a) Determine the linear map (i.e. the matrix) that expresses the customers-tuple $K_{j+1}$ with respect to $K_{j}$.
b) Which customers-tuple arises from the customers-tuple $(12000,10000,8000)$ within one year?
c) Which customers-tuple arises from the customers-tuple $(10000,0,0)$ in four years?
# Newspaper/Subscription/Movement of customers/Matrix/Exercise

The newspapers $A,B$ and $C$ sell subscriptions, and they compete in a local market with $100000$ customers. Within a year, one can observe the following movements.
* The subscribers of $A$ stick with a percentage of $80\%$ to $A$, $10\%$ switch to $B$, $5\%$ switch to $C$ and $5\%$ become nonreaders.
* The subscribers of $B$ stick with a percentage of $60\%$ to $B$, $10\%$ switch to $A$, $20\%$ switch to $C$ and $10\%$ become nonreaders.
* The subscribers of $C$ stick with a percentage of $70\%$ to $C$, nobody switches to $A$, $10\%$ switch to $B$ and $20\%$ become nonreaders.
* Among the nonreaders, $10\%$ subscribe to $A,B$ or $C$, the rest remains nonreaders.
a) Establish the matrix that describes the movement of customers within a year.
b) In a certain year, each of the three newspapers has $20000$ subscribers and there are $40000$ nonreaders. How does the distribution look like after a year?
c) The three newspapers expand to another city, where there are no newspapers at all so far, but also $100000$ potential customers. How many subscribers does each newspaper have (and how many nonreaders) after three years, if the same movements hold in the new city?
# Population/Reproduction and extinction/Exercise

An animal population consists of babies (first year), freshers (second year), rockers (third year), mature ones (fourth year), and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year $j$ is given by a $5$-tuple $B_{j}=(b_{1,j},b_{2,j},b_{3,j},b_{4,j},b_{5,j})$.
During a year, $7/8$ of the babies become freshers, $9/10$ of the freshers become rockers, $5/6$ of the rockers become mature ones, and $2/3$ of the mature ones reach the fifth year.
Babies and freshers can not reproduce yet, then they reach sexual maturity, and $10$ rockers generate $5$ new pets, and $10$ of the mature ones generate $8$ new babies, and the babies are born one year later.
a) Determine the linear map (i.e., the matrix) that expresses the total stock $B_{j+1}$ with respect to the stock $B_{j}$.
b) What will happen to the stock $(200,150,100,100,50)$ in the next year?
c) What will happen to the stock $(0,0,100,0,0)$ in five years?
# Endomorphism/Lines/Axes/Exercise

We consider the vectors $u={\begin{pmatrix}8\\7\\4\end{pmatrix}}$ and $v={\begin{pmatrix}6\\3\\9\end{pmatrix}}$ in $\mathbb {R} ^{3}$. Determine the dimension of the vector space, consisting of all linear mappings
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}$
that send $u$ to the $x$-axis and $v$ to the $y$-axis. Describe the corresponding linear subspace of the space of matrices with respect to suitable chosen bases.
# Endomorphism/Line/To axes/No vector space/Exercise

We consider the line $G=\mathbb {R} {\begin{pmatrix}-9\\8\\3\end{pmatrix}}\subseteq \mathbb {R} ^{3}$. Let $M\subseteq \operatorname {Hom} _{\mathbb {R} }{\left(\mathbb {R} ^{3},\mathbb {R} ^{2}\right)}$ be the set of all linear mappings that send this line to the axes of coordinates. Show that $M$ is not a linear subspace of the space of homomorphisms.
# Linear projection/Idempotent/22/Determine/Exercise

a) Show that the $2\times 2$-matrices
    ${\begin{pmatrix}{\frac {1\pm {\sqrt {1-4bc}}}{2}}&b\\c&{\frac {1\mp {\sqrt {1-4bc}}}{2}}\end{pmatrix}}$
describe projections. Here, $b,c\in K$ are such that a square root ${\sqrt {1-4bc}}$ exists.
b) Determine all $2\times 2$-matrices
    ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$
that describe a projection.
# Matrices/2/U to T/Describe with system of equations/1/Exercise

Let
    $U=\langle {\begin{pmatrix}4\\-7\end{pmatrix}}\rangle \subseteq K^{2}\,$
and
    $T=\langle {\begin{pmatrix}-2\\5\end{pmatrix}}\rangle \subseteq K^{2}\,.$
a) Describe the linear subspace $W$ of the space of all $2\times 2$-matrices that map the linear subspace $U$ to the linear subspace $T$, as the solution space of a linear system.
b) Describe $W$ by an eliminated system.
c) Determine the dimension of $W$.
# Matrices/3/U to T/Describe with system of equations/1/Exercise

Let
    $U=\langle {\begin{pmatrix}3\\4\\8\end{pmatrix}},\,{\begin{pmatrix}2\\1\\-1\end{pmatrix}}\rangle \subseteq K^{3}\,$
and
    $T=\langle {\begin{pmatrix}7\\1\\3\end{pmatrix}},\,{\begin{pmatrix}5\\4\\2\end{pmatrix}}\rangle \subseteq K^{3}\,.$
a) Describe the linear subspace $W$ of the space of all $3\times 3$-matrices that map the linear subspace $U$ into the linear subspace $T$, as a solution space of a linear system.
b) Describe $W$ by an eliminated linear system.
c) Determine the dimension of $W$.
# Matrices/3/U to T/Describe with system of equations/2/Exercise

Let
    $U=\langle {\begin{pmatrix}-2\\5\\-4\end{pmatrix}},\,{\begin{pmatrix}3\\4\\6\end{pmatrix}}\rangle \subseteq K^{3}\,$
and
    $T=\langle {\begin{pmatrix}6\\3\\4\end{pmatrix}},\,{\begin{pmatrix}1\\5\\-8\end{pmatrix}}\rangle \subseteq K^{3}\,.$
a) Describe the linear subspace $W$ of the space of all $3\times 3$-matrices that map the linear subspace $U$ into the linear subspace $T$, as a solution space of a linear system.
b) Describe $W$ by an eliminated linear system.
c) Determine the dimension of $W$.
# Determinant/Laplace-matrix/Complete graph/Exercise

We consider the matrix
    ${\begin{pmatrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{pmatrix}}.$
* Compute the determinant of $M$.
* Determine the determinant for every matrix that arises when we remove from $M$ a row and a column.
# Sarrus-determinant/4x4/Properties/Exercise

The Sarrusminant of a $n\times n$-matrix is computed by repeating the first $n-1$ columns of the matrix in the same order behind the matrix, and then by adding up the $n$ products of the diagonals and subtracting the $n$ products of the antidiagonals. We restrict to the case $n=4$. That is, for a matrix
    $M={\begin{pmatrix}a&b&c&d\\e&f&g&h\\l&m&n&m\\q&r&s&t\end{pmatrix}}\,,$
we consider
    ${\begin{pmatrix}a&b&c&d&a&b&c\\e&f&g&h&e&f&g\\l&m&n&m&l&m&n\\q&r&s&t&q&r&s\end{pmatrix}}$
and the Sarrusminant is
    $\operatorname {Sar} (M)=afnt+bgpq+chlr+dems-qmgd-rnha-speb-tlfc\,.$
* Show that the mapping is multilinear (in the rows of the matrix).
    $\Psi \colon \operatorname {Mat} _{4}(K)\longrightarrow K,M\longmapsto \operatorname {Sar} (M),$
* Show that, for $4\times 4$-matrices that contain a zero-row, the Sarrusminant is $0$.
* Show that, for $4\times 4$-matrices that contain a zero-column, the Sarrusminant is $0$.
* Show that, for an upper triangular matrix, the Sarrusminant is the product of the diagonal elements.
* Show that the Sarrusminant is not alternating.
* Give an example for an invertible matrix, where the Sarrusminant equals $0$.
* Give an example for a not-invertible matrix, where the Sarrusminant equals $1$.
# Decimal sequence/Primary school/Mapping/Properties/Exercise

We consider the mapping
    $f\colon \mathbb {N} \longrightarrow \mathbb {N}$
that is described in exercise (the natural numbers are given as finite sequences in the decimal system).
* Is $f$ increasing?
* Is $f$ surjective?
* Is $f$ injective?
* Does $f$ have a fixed point?
# Polynomial ring/Field/One variable/Product of linear factors/Divisors/Exercise

Let $K$ be a field, and let $K[X]$ be the polynomial ring over $K$, and let $P\in K[X]$ be a polynomial that has a factorization into linear factors. Let $T$ be a divisor of $P$. Show that $T$ has also a factorization into linear factors, and that the multiplicity of every linear factor $X-a$ in $T$ is bounded from above by its multiplicity in $P$.