OpenLLM-France/wikimedia huggingface dataset:
# Direct product/Base change/1/Exercise

Let $V$ and $W$ be finite-dimensional $K$-vector spaces. Let ${\mathfrak {v}}=v_{1},\ldots ,v_{n}$ and ${\mathfrak {u}}=u_{1},\ldots ,u_{n}$ be bases of $V$, and ${\mathfrak {w}}=w_{1},\ldots ,w_{m}$ and ${\mathfrak {z}}=z_{1},\ldots ,z_{m}$ bases of $W$. Let $M_{\mathfrak {u}}^{\mathfrak {v}}$ and $M_{\mathfrak {z}}^{\mathfrak {w}}$ be the transformation matrices. What is the transformation matrix for the base change from the basis $(v_{1},0),\ldots ,(v_{n},0),(0,w_{1}),\ldots ,(0,w_{m})$ to the basis $(u_{1},0),\ldots ,(u_{n},0),(0,z_{1}),\ldots ,(0,z_{m})$ of the product space $V\times W$?
# R to R/Even and odd/Direct sum/Exercise

Let $V=\operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}$ be the vector space of all functions from $\mathbb {R}$ to $\mathbb {R}$. Show that there exists a direct sum decomposition
    $V=G\oplus U\,,$
where $G$ denotes the linear subspace of all even functions, and $U$ denotes the linear subspace of all odd functions.
# Linear subspaces/Same dimension/Common direct complement/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and let $U_{1},U_{2}\subseteq V$ denote linear subspaces of the same dimension. Show that $U_{1}$ and $U_{2}$ have a common direct complement.
# Matrix space/Diagonal matrices/Direct sum/Exercise

Show that the space of $n\times n$-matrices over a field $K$ is the direct sum of the space of the diagonal matrices, the space of the upper triangular matrices with zero diagonal, and the space of the lower triangular matrices with zero diagonal.
# Vector space/Direct sum/Linear subspace not/Exercise

Suppose that the vector space $V$ is the direct sum of the linear subspaces $V_{1}$ and $V_{2}$. Show that a linear subspace $U\subseteq V$ is not necessarily the direct sum of the linear subspaces $U\cap V_{1}$ and $U\cap V_{2}$.
# Linear mapping/0/Exercise

Let
    $\varphi \colon V\longrightarrow W$
be a linear mapping between the $K$-vector spaces $V$ and $W$. Show that $\varphi (0)=0$ holds.
# Linear mapping/0/Exercise/Solution

Due to the additivity of a linear mapping, we have
    $\varphi (0)=\varphi (0+0)=\varphi (0)+\varphi (0)\,.$
Addition with the negative element of $\varphi (0)$ that is, with $-\varphi (0)$, yields
    $0=\varphi (0)\,.$
# Linear mapping/Negative vector/Exercise

Let
    $\varphi \colon V\longrightarrow W$
denote a linear mapping between the $K$-vector spaces $V$ and $W$. Let $v\in V$. Show that $\varphi (-v)=-\varphi (v)$ holds.
# Proportionality/Gold price/Exercise

The price of an ounce of gold is $1100$ €.
a) What is the price for seven ounces of gold?
b) How much gold do we get for $10000$ €?
# Proportionality/Pedestrian/Exercise

Five pedestrians walk a certain distance in 35 minutes. The next day, seven pedestrians walk the same distance with the same speed. How long does it take for them?
# Matrix/Linear mapping/Exercise

Let $M={\left(a_{ij}\right)}_{1\leq i\leq m,\,1\leq j\leq n}$ be an $m\times n$-matrix over a field $K$. Show that the corresponding mapping
    $K^{n}\longrightarrow K^{m},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto M{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}s_{j}\\\sum _{j=1}^{n}a_{2j}s_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}s_{j}\end{pmatrix}},$
is linear.
# Linear subspaces/Same dimension/Automorphism/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and let $U_{1},U_{2}\subseteq V$ denote linear subspaces of the same dimension. Show that there exists an $K$-automorphism
    $\varphi \colon V\longrightarrow V$
such that
    $\varphi (U_{1})=U_{2}\,.$
# Vector spaces/Group homomorphism/Q-linear/Exercise

Let $V$ and $W$ $\mathbb {Q}$-vector spaces, and let
    $\varphi \colon V\longrightarrow W$
be a group homomorphism. Show that $\varphi$ is already $\mathbb {Q}$-linear.
# Q/Addition/Linear mapping/Matrix/Exercise

Show that the addition
    $+\colon \mathbb {Q} ^{2}=\mathbb {Q} \times \mathbb {Q} \longrightarrow \mathbb {Q}$
is a linear mapping. How does its matrix with respect to the standard basis look like?
# Vector space/Finite family/Linearity of substitution/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{1},\ldots ,v_{n}$ be a family of vectors in $V$. Show that the mapping
    $\varphi \colon K^{n}\longrightarrow V,(s_{1},\ldots ,s_{n})\longmapsto \sum _{i=1}^{n}s_{i}v_{i},$
is linear.
# Linear mappings/Product mapping/Linear/Exercise

Let $K$ be a field. For $i\in \{1,\ldots ,n\}$, let $K$-vector spaces $V_{i}$ and $W_{i}$, and linear mappings
    $\varphi _{i}\colon V_{i}\longrightarrow W_{i}$
be given. Show that the product mapping
    $\varphi =\varphi _{1}\times \varphi _{2}\times \cdots \times \varphi _{n}\colon V_{1}\times V_{2}\times \cdots \times V_{n}\longrightarrow W_{1}\times W_{2}\times \cdots \times W_{n},\left(v_{1},\,v_{2},\,\ldots ,\,v_{n}\right)\longmapsto \left(\varphi _{1}(v_{1}),\,\varphi _{2}(v_{2}),\,\ldots ,\,\varphi _{n}(v_{n})\right),$
is also a linear mapping between the product spaces.
# Linear mapping/Determined on generating set/No mapping/Exercise

Let $K$ be a field, and let $V$ and $W$ be $K$-vector spaces. Let $v_{1},\ldots ,v_{n}$ be a system of generators for $V$, and let $w_{1},\ldots ,w_{n}$ be a family of vectors in $W$.
a) Prove that there is at most one linear map
    $\varphi \colon V\longrightarrow W$
such that $\varphi (v_{i})=w_{i}$ for all $i$.
b) Give an example of such a situation, where there is no linear mapping with $\varphi (v_{i})=w_{i}$ for all $i$.
# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet x

Other exercises

### Exercise

Let $K$ be a field and let $A$ be a $m\times n$-matrix and $B$ a $n\times p$-matrix over $K$. Prove the following relationships concerning the rank
    $\operatorname {rk} \,AB\leq \operatorname {rk} \,A{\text{ and }}\operatorname {rk} \,AB\leq \operatorname {rk} \,B.$
Prove that equality on the left occurs if $B$ is invertible, and equality on the right occurs if $A$ is invertible. Give an example of non-invertible matrices $A$ and $B$ such that equality on the left and on the right occurs.

### Exercise

Prove that the series
    $\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}$
converges with sum equal to $1$.

### Exercise

Examine for each of the following subsets $M\subseteq \mathbb {Q}$ the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.
* $\{2,-3,-4,5,6,-1,1\}$,
* $\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}$,
* $]-5,2]$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}$,
* $\mathbb {Q} _{-}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 4\right\}}$,
* ${\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.

### Exercise

Explain why the factorial function is continuous.
# Vector space/I to K/Contravariance in index set/Exercise

Let $K$ be a field, and let $S$ and $T$ be sets. Show that a mapping
    $\psi \colon S\longrightarrow T$
defines a linear mapping
    $\operatorname {Map} \,{\left(T,K\right)}\longrightarrow \operatorname {Map} \,{\left(S,K\right)},\varphi \longmapsto \varphi \circ \psi .$
# Vector space/I to K/Direct/Co- and contravariance in index set/Exercise

Let $K$ be a field, and let $S$ and $T$ denote sets. Let
    $\psi \colon S\longrightarrow T$
be a mapping.
a) Show that, by $e_{s}\mapsto e_{\psi (s)}$, a linear mapping
    $K^{(S)}\longrightarrow K^{(T)}$
is determined.
b) Suppose now that $\psi$ has also the property that all its fibers are finite. Show that this defines a linear mapping
    $K^{(T)}\longrightarrow K^{(S)},\varphi \longmapsto \varphi \circ \psi .$
# Vector space/Mapping set to K/Disjoint union/Exercise

Let $K$ be a field, and let $I$ denote an index set, together with a partition
    $I=I_{1}\uplus I_{2}\,.$
Show that there exists a natural isomorphism
    $\operatorname {Map} \,{\left(I,K\right)}\cong \operatorname {Map} \,{\left(I_{1},K\right)}\oplus \operatorname {Map} \,{\left(I_{2},K\right)}\,.$
# Finite field/Number of linear mappings/Exercise

Let $K$ be a finite field with $q$ elements. Determine the number of linear mappings
    $\varphi \colon K^{n}\longrightarrow K^{m}.$
# Ribbon around earth/One meter/Animal/Exercise

Around the Earth along the equator, a ribbon is placed. However, the ribbon is one meter longer than the equator, so that it is lifted uniformly all around to be tense. Which of the following creatures can run/fly/swim/dance under it?
* An amoeba.
* An ant.
* A tit.
* A flounder.
* A boa constrictor.
* A guinea pig.
* A boa constrictor that has swallowed a guinea pig.
* A very good limbo dancer.
# Matrix multiplication/Composition/Fact/Proof/Exercise

Let $B$ be an $n\times p$-Matrix, and $A$ be an $m\times n$-matrix. Let
    $K^{p}{\stackrel {B}{\longrightarrow }}K^{n}{\stackrel {A}{\longrightarrow }}K^{m}$
denote the corresponding linear mappings. Show that the matrix product $A\circ B$ describes the composition of the linear mappings.
# Linear mapping/Determination on basis/Scaling/Proof/Exercise

Complete the proof of the theorem on determination on basis, by proving the compatibility with the scalar multiplication.
# Are all conspiracy theories nonsense?

Nowadays, there are quite a lot of conspiracy theories. From the moon landing never happening to the assassination of John F. Kennedy involving multiple people to 9/11 being an inside job. But are all conspiracy theories nonsense? Could certain conspiracy theories be true? Should we trust the generally accepted stories, often promoted by governments? Or should we think for ourselves and personally investigate the truth?

## All conspiracy theories are nonsense
### Pro

* Pro Conspiracy theories are theories based on people's inventions to explain things they do not understand.
** Objection There are also scientists, such as physicians, who believe in conspiracy theories. You can assume that their research is not based on nonsense. They must have made well-considered decisions, especially considering that some physicians, for instance, have been suspended after coming forward with their ideas.

### Con

* Con The Watergate scandal, before it became clear what exactly had happened, was also considered a conspiracy theory at first. President Nixon kept insisting that he was not involved in the break-in at the Democratic National Committee headquarters, but it later turned out that he actually was. This scandal turned out not to be a nonsensical conspiracy theory but the actual truth.
** Objection Generally, however, most conspiracies are proven false by an overwhelming amount of evidence. It's very possible that the Watergate Scandal may have only been deemed a conspiracy if the public had all of the evidence. Without any evidence the theory would be nonsensical.
* Con Adherents of doctrines that contradict what a particular religious community considers to be fundamental beliefs were once regarded as heretics. For example, Galileo's opinions were met with opposition within the Catholic Church. In 1616, during the Galileo affair, the Inquisition declared heliocentrism to be "formally heretical". Furthermore, according to the Apology of Socrates, Socrates was charged with, among other things, "not believing in the gods in whom the city believes." Yet, both of these individuals introduced ideas that have significantly influenced our understanding of the world.
# Linear mapping/Plane/Image and preimage/1/Exercise

We consider the linear mapping $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ given by the matrix ${\begin{pmatrix}8&4\\5&9\end{pmatrix}}$.
* Determine the image of the line defined by the equation
    $5x-7y=0\,.$
* Determine the preimage of the line given by the equation
    $6x-11y=0\,.$
# Linear subspace/Realization as kernel of a linear mapping/Exercise

Let $K$ be a field, and let $V$ denote a finite-dimensional $K$-vector space. Let $U\subseteq V$ be a linear subspace. Show that there exists a $K$-vector space $W$ and a surjective $K$-linear mapping
    $\varphi \colon V\longrightarrow W$
such that $U=\operatorname {kern} \varphi$ holds.
# Projection image/Point/Preimage/Exercise

Let a linear mapping $\varphi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{2}$ as in example be given, in order to represent three-dimensional shapes in the plane. Imagine the preimage for a point $P\in \mathbb {R} ^{2}$! How do the corresponding line equations look like? What points $Q\in \mathbb {R} ^{3}$ have the same image point as the corner $(1,1,1)$ of the unit cube?
# Linear mapping/Kernel and image/Direct sum/Exercise

Let $A={\begin{pmatrix}3&1&5\\-1&0&2\end{pmatrix}}$, and let
    $f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},x\longmapsto A\cdot x$
denote the corresponding linear mapping.
* Determine a basis and the dimension of $\operatorname {kern} f$, and of $\operatorname {Im} f$.
* Find a linear subspace $V\subseteq \mathbb {R} ^{3}$ such that holds.
    $\mathbb {R} ^{3}=V\oplus \operatorname {kern} f\,$
* Does there also exist a linear subspace $U\subset \mathbb {R} ^{2}$, $U\neq 0$, such that $\mathbb {R} ^{2}=U\oplus \operatorname {Im} f$?
# Linear mapping/Criterion for trivial intersection of image and kernel/Exercise

Let $V$ be a finite-dimensional $K$-vector space, and $f\colon V\rightarrow V$ be an endomorphism. Show that the following statements are equivalent.
* $\operatorname {kern} f=\operatorname {kern} {\left(f\circ f\right)}$,
* $\operatorname {kern} f\cap \operatorname {Im} f=\{0\}$,
* $V=\operatorname {kern} f\oplus \operatorname {Im} f$,
* $\operatorname {Im} f=\operatorname {Im} {\left(f\circ f\right)}$.
# Mapping/x^2-1/Image and preimage/No linear subspaces/Exercise

We consider the mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}-1.$
Show that for this mapping, neither the image of a linear subspace $U\subseteq \mathbb {R}$ must be a linear subspace, nor the preimage of a linear subspace must be a linear subspace.
# Linear mapping/Graph/Sketch/Exercise

How does the graph of a linear mapping
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$
    $g\colon \mathbb {R} \longrightarrow \mathbb {R} ^{2},$
    $h\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R}$
look like? How can you see in a sketch of the graph the kernel of the map?
# Linear mapping/Shape/Translation/Image/Exercise

Let $F\subseteq \mathbb {R} ^{n}$ be a "geometric shape“, for example, a circle or a rectangle in the plane. Let
    $\theta _{w}\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n},x\longmapsto x+w,$
be the translation with the vector $w$, and let $F'=\theta _{w}(F)$ denote the translated shape. Let
    $\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$
be a linear mapping. Show that the image $\varphi (F')$ arises from the image $\varphi (F)$ by a translation.
# Linear mapping/Standard basis/(5 1 2 3)/Matrix with respect to 1 4 and 4 2/Exercise

Let $\varphi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be the linear mapping determined by the matrix $M={\begin{pmatrix}5&1\\2&3\end{pmatrix}}$ (with respect to the standard basis). Determine the describing matrix of $\varphi$ with respect to the basis ${\begin{pmatrix}1\\4\end{pmatrix}}$ and ${\begin{pmatrix}4\\2\end{pmatrix}}$.
# Database Applications

A database application is a computer program whose primary purpose is retrieving information from a computerized database. Example database applications include LibreOffice Base and Microsoft Access.

## Resources

* Introduction
* Tables
* Queries
* Advanced Queries
* Forms
* Reports

## See Also

* Databases
* Database Management
* Wikipedia: Database application
* Wikipedia: LibreOffice Base
* Wikipedia: Microsoft Access
# Base change/Transformation matrix/R^4 to R^3/Exercise

We consider the families of vectors
    ${\mathfrak {u}}={\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {v}}={\begin{pmatrix}1\\0\\1\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\end{pmatrix}}$
in $\mathbb {R} ^{4}$ and in $\mathbb {R} ^{3}$. We denote the standard bases by ${\mathfrak {e}}_{4}$ and ${\mathfrak {e}}_{3}$. We consider the linear mapping
    $f\colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{3}$
given by the matrix
    $A={\begin{pmatrix}-2&3&2&3\\-3&5&0&1\\-1&2&-2&-2\end{pmatrix}}\,$
with respect to the standard bases. Determine the describing matrices of $f$ with respect to the bases
a) ${\mathfrak {e}}_{4}$ and ${\mathfrak {v}}$,
b) ${\mathfrak {u}}$ and ${\mathfrak {e}}_{3}$,
c) ${\mathfrak {u}}$ and ${\mathfrak {v}}$.
# Dimension formula/Linear subspaces and linear mapping/Exercise

Prove fact using fact.
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Dimension formula/Fact/Factreferencenumber

Theorem 11.5
# Linear algebra (Osnabrück 2024-2025)/Linear subspace/Sum and intersection/Dimension/Fact/Factreferencenumber

Theorem 9.7
# Real vector space/Finite-dimensional/As group isomorphic/Exercise

Let $V$ and $W$ be finite-dimensional real vector spaces, both not the zero-space. Are these spaces isomorphic as commutative groups?
# Base change/Three bases/Composition/Identity/Exercise

Prove fact with the help of fact and example.
# Vector space/Finite-dimensional/Injective surjective bijective/Basis equivalence/Exercise

Prove fact using fact and exercise.
# Vector space/Translation/Definition

Let $K$ be a field, and let $V$ be a $K$-vector space. For a vector $v\in V$, the mapping
    $V\longrightarrow V,x\longmapsto x+v,$
is called the translation by the vector $v$.
# Linear algebra (Osnabrück 2024-2025)/Vector space/Dimension n and n vectors/Equivalences/Fact/Factreferencenumber

Corollary 8.10
# Linear algebra (Osnabrück 2024-2025)/Vector space/Finite dimensional/Injective surjective bijective/Fact/Factreferencenumber

Corollary 11.9
# Linear algebra (Osnabrück 2024-2025)/Base change/Three bases/Composition/Fact/Factreferencenumber

Lemma 9.5
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Matrix/Composition/Fact/Factreferencenumber

Lemma 11.10
# Linear algebra (Osnabrück 2024-2025)/Vector space/Finite family/Properties of substitution/Exercise/Exercisereferencenumber

Exercise 10.23
# Linear algebra (Osnabrück 2024-2025)/Identity/Matrix to basis/Transformation matrix/Example/Examplereferencenumber

Example 10.13
# Linear algebra (Osnabrück 2024-2025)/R^3/Projection image/Linear mapping/Example/Examplereferencenumber

Example 10.11
# Linear mapping/Mulled wine/Price and calories/Exercise

On the real vector space $G=\mathbb {R} ^{4}$ of mulled wines, we consider the two linear maps
    $\pi \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 8z+9n+5r+s,$
and
    $\kappa \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 2z+n+4r+8s.$
We consider $\pi$ as the price function, and $\kappa$ as the caloric function. Determine a basis for $\operatorname {ker} {\left(\pi \right)}$, one for $\operatorname {ker} {\left(\kappa \right)}$ and one for $\operatorname {ker} {\left(\pi \times \kappa \right)}$.
# Matrix/C/2+5i 1-2i 3-4i 6-2i/Invertible/Solution for 54+72i 0/Exercise

a) Determine if the complex matrix
    $M={\begin{pmatrix}2+5{\mathrm {i} }&1-2{\mathrm {i} }\\3-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,$
is invertible.
b) Find a solution to the inhomogeneous linear system of equations
    $M{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}54+72{\mathrm {i} }\\0\end{pmatrix}}\,.$
# Does an individual have a responsibility to maintain a community?

Why should we be entitled to follow a set guideline to exist in a society?

## An individual has a responsibility to maintain a community
### Pro

* Pro If an individual is able to survive without the help of the community for a prolonged time, especially in regards to food, clothing, transportation, housing, tools, medicine, land ownership, cattle ownership, personal security, entertainment or companionship, he does not have a responsibility. However, as only very few individuals are able to achieve this, society is the natural habitat of humans.

### Con

* Con To our knowledge we have free will, and therefore should be entitled to use our free will in a way that is not inhibited by society.
** Objection The existence of free will, if anything, reinforces our responsibilities to our community, assuming said freedom also establishes moral responsibility. Assuming there are true moral facts (objective or subjective), then we must exercise our free will in accordance with our moral obligations, including maintaining the community.
** Objection This argument commits the informal logical fallacy petitio principii by basing its conclusion, "we do not inhibit a responsibility to maintain a community", on an assumption, "we have free will", that is in need of as much proof or demonstration as the conclusion itself. Since no such proof or demonstration is presented in favor of its premise, we necessarily have to conclude that no such proof or demonstration has been presented for its conclusion, either, establishing the conclusion as the premise is established, as an unjustified assumption.
# Rust

Rust is a programming language that runs fast, prevents segfaults, and guarantees thread safety.

## Features

* zero-cost abstractions
* move semantics
* guaranteed memory safety
* threads without data races
* trait-based generics
* pattern matching
* type inference
* minimal runtime
* efficient C bindings
* no garbage collector
* runs on bare metal
* ownerships and borrows

## Prerequisites

* Basic knowledge of computer science
* Basic knowledge on any programming language
* A computer with a Rust-enabled IDE (VisualRust, intellij-rust, etc.)
* Understanding of static types

## Resources

Open-Source
* https://github.com/rust-unofficial/awesome-rust
* https://github.com/ctjhoa/rust-learning
Online tutors Online learning sites
* https://rustbyexample.com
* https://doc.rust-lang.org/book/
* https://learnxinyminutes.com/docs/rust/
* https://learning-rust.github.io/
* https://stevedonovan.github.io/rust-gentle-intro/
* https://www.ralfj.de/projects/rust-101/main.html
* https://siciarz.net/24-days-rust-conclusion-2016/
Online content instructor Bold textOnline Code Environment
* https://play.rust-lang.org
* https://repl.it/languages/rust
* https://tio.run/#rust
Online Community
* https://www.reddit.com/r/rust/
* https://www.rust-lang.org/
* https://discord.com/invite/rust-lang-community
Good Programming Patterns
* https://pascalhertleif.de/artikel/good-practices-for-writing-rust-libraries/
# Motivation and emotion/Book/2021/Freedom and motivation

* Multimedia presentation (3 min)

## Definitions

[Provide more detail]

### What is freedom?

https://en.wikipedia.org/wiki/Freedom

### What is motivation?

https://en.wikipedia.org/wiki/Motivation

## Working notes

It could be useful to have a temporary section for working notes during the topic development and chapter drafting. This section will be ignored when the topic development is marked, but remove it before finalising the book chapter.

## Conclusion

The Conclusion is arguably the most important section. It should be possible for someone to read only the Overview and the Conclusion and still get a good idea of the topic.
* Suggestions for this section:
* What is the answer to the question in the sub-title (based on psychological theory and research)?
* What are the answers to the focus questions?
* What are the practical, take-home messages?
# Linear algebra (Osnabrück 2024-2025)/Space/Plane equation/Example for sets/Example/Examplereferencenumber

Example 1.1
# Linear algebra (Osnabrück 2024-2025)/Matrix/R/Mapping/Example/Examplereferencenumber

Example 2.8
# Linear algebra (Osnabrück 2024-2025)/Mapping/Composition/Associativity/Fact/Factreferencenumber

Lemma 2.11