OpenLLM-France/wikimedia huggingface dataset:
# Determinant/Field/Recursively/Multilinearity/No proof/Section

We want to show that the recursively defined determinant is a "multilinear“ and "alternating“ mapping, where we identify
    $\operatorname {Mat} _{n}(K)\cong (K^{n})^{n}\,,$
so a matrix is identified with the $n$-tuple of the rows of the matrix. We consider a matrix as a tuple of columns
    ${\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}$
where the entries $v_{i}$ are row vectors of length $n$.

## Theorem

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. Then the determinant
    $\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$
is multilinear. This means that for every $k\in \{1,\ldots ,n\}$, and for every choice of $n-1$ vectors $v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}$, and for any $u,w\in K^{n}$, the identity
    $\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u+w\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}+\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\w\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,$
holds, and for $s\in K$, the identity
    $\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,$

### Proof

$\Box$

## Theorem

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. Then the determinant
    $\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$
* If in $M$ two rows are identical, then $\det M=0$. This means that the determinant is alternating.
* If we exchange two rows in $M$, then the determinant changes with factor $-1$.

### Proof

$\Box$

## Theorem

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix
$K$
* We have $\det M\neq 0$.
* The rows of $M$ are linearly independent.
* $M$ is invertible.
* We have $\operatorname {rk} \,M=n$.

### Proof

The relation between rank, invertibility and linear independence was proven in fact. Suppose now that the rows are linearly dependent. After exchanging rows, we may assume that $v_{n}=\sum _{i=1}^{n-1}s_{i}v_{i}$. Then, due to fact and fact, we get
    $\det M=\det {\begin{pmatrix}v_{1}\\\vdots \\v_{n-1}\\\sum _{i=1}^{n-1}s_{i}v_{i}\end{pmatrix}}=\sum _{i=1}^{n-1}s_{i}\det {\begin{pmatrix}v_{1}\\\vdots \\v_{n-1}\\v_{i}\end{pmatrix}}=0\,.$
Now suppose that the rows are linearly independent. Then, by exchanging of rows, scaling and addition of a row to another row, we can transform the matrix successively into the identity matrix. During these manipulations, the determinant is multiplied with some factor $\neq 0$. Since the determinant of the identity matrix is $1$, the determinant of the initial matrix is $\neq 0$.
$\Box$
# Determinant/Field/Recursively/Multiplication theorem/No proof/Section

We discuss without proofs further important theorems about the determinant. The proofs rely on a systematic account of the properties which are characteristic for the determinant, namely the properties multilinear and alternating. By these properties, together with the condition that the determinant of the identity matrix is $1$, the determinant is already determined.

## Theorem

Let $K$ denote a field, and $n\in \mathbb {N} _{+}$. Then for matrices $A,B\in \operatorname {Mat} _{n}(K)$, the relation
    $\det {\left(A\circ B\right)}=\det A\cdot \det B\,$

### Proof

$\Box$

## Definition

Let $K$ be a field, and let $M=(a_{ij})_{ij}$ be an $m\times n$-matrix over $K$. Then the $n\times m$-matrix
    ${M^{\text{tr}}}={\left(b_{ij}\right)}_{ij}{\text{ with }}b_{ij}:=a_{ji}$
$M$
The transposed matrix arises by interchanging the role of the rows and the columns. For example, we have
    ${{\begin{pmatrix}t&n&o&d\\r&s&s&x\\a&p&e&y\end{pmatrix}}^{\text{tr}}}={\begin{pmatrix}t&r&a\\n&s&p\\o&s&e\\d&x&y\end{pmatrix}}\,.$

## Theorem

Let $K$ denote a field, and let $M$ denote an $n\times n$-matrix over $K$. Then
    $\det M=\det {M^{\text{tr}}}\,.$

### Proof

$\Box$
This implies that we can compute the determinant also by expanding with respect to the rows, as the following statement shows.

## Corollary

Let $K$ be a field, and let $M={\left(a_{ij}\right)}_{ij}$ be an $m\times n$-matrix over $K$. For $i,j\in \{1,\ldots ,n\}$, let $M_{ij}$ be the matrix which arises from $M$, by leaving out the $i$-th row and the $j$-th column. Then (for $n\geq 2$ and for every fixed $i$ and $j$)
    $\det M=\sum _{i=1}^{n}(-1)^{i+j}a_{ij}\det M_{ij}=\sum _{j=1}^{n}(-1)^{i+j}a_{ij}\det M_{ij}\,.$

### Proof

For $j=1$, the first equation is the recursive definition of the determinant. From that statement, the case $i=1$ follows, due to fact. By exchanging columns and rows, the statement follows in full generality, see exercise.
$\Box$
# Determinant/Diagonal block matrix/Left lower block 0/Exercise

Let $M$ be the following square matrix
    $M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,$
where $A$ and $D$ are square matrices. Prove that $\det M=\det A\cdot \det D$.
# Determinant/Expansion for arbitrary row or column/Exercise

Prove that you can expand the determinant according to each row and each column.
# Determinant/C/Compute/1+i 3-2i 5/i 1 3-i/2i -4-i 2+i/Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}1+{\mathrm {i} }&3-2{\mathrm {i} }&5\\{\mathrm {i} }&1&3-{\mathrm {i} }\\2{\mathrm {i} }&-4-{\mathrm {i} }&2+{\mathrm {i} }\end{pmatrix}}.$
# Determinant/Field/Recursively/Introduction/Section
## Definition

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix over $K$ with entries $a_{ij}$. For $i\in \{1,\ldots ,n\}$, let $M_{i}$ denote the $(n-1)\times (n-1)$-matrix, which arises from $M$, when we remove the first column and the $i$-th row. Then one defines recursively the determinant of $M$ by
    $\det M={\begin{cases}a_{11}\,,&{\text{ for }}n=1\,,\\\sum _{i=1}^{n}(-1)^{i+1}a_{i1}\det M_{i}&{\text{ for }}n\geq 2\,.\end{cases}}\,$
The determinant is only defined for square matrices. For small $n$, the determinant can be computed easily.

## Example

For a $2\times 2$-matrix
    $M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,,$
we have
    $\det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad-cb.$

## Example

For a $3\times 3$-matrix $M={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}$, we have
    $\det {\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}.$
This is called the rule of Sarrus.

## Lemma

For an upper triangular matrix
    $M={\begin{pmatrix}b_{1}&\ast &\cdots &\cdots &\ast \\0&b_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&b_{n-1}&\ast \\0&\cdots &\cdots &0&b_{n}\end{pmatrix}}\,,$
    $\det M=b_{1}b_{2}\cdots b_{n-1}b_{n}\,.$
identity matrix we get
$\det E_{n}=1$

### Proof

This follows with a simple induction directly from the recursive definition of the determinant.
$\Box$
# Arch/quiz4

Read the question, select the best answer by clicking the circle by it.
When finished, click the "Submit" button at the bottom.
# Arch/quiz6
## Drum Beat Music - Sugarplum

Listen closely for the instrument, which (Piano, Drums, or Voices) is on which track. Read the following questions and select the best answer. When you are finished making your selection, click the "Submit" button.
# Arch/quiz8
#### Music Colour

Listen to the music tracks closely for the colour that is sung about, indicate in the following which colour is in which track
When you are finished making your selection, click the "Submit" button.
# Arch/quiz2
## Knowledge Check 1

Read the following questions and select the best answer. When you are finished making your selection, click the "Submit" button.
# Arch/quiz5
## Security: Identity Theft

| statue Whose statue is here statue Aberdeen .. Auchtermuchty .. Dundee .. New York .. | Answers .. Jimmy |
# ArcGIS/Clip Tool

How to get to the clip tool: Data Management Tools -> Raster -> Raster Processing -> Clip
Clip is used to cut out a piece of one feature class using one or more features in another feature class as a "cookie cutter". This tools is helpful in creating a new feature class. The clip features can be points, lines, and polygons. Attributes values from the input feature classes are also transferred to the resulting clip output. The clipped area is specified either by a rectangular shape using minimum and maximum x- and y-coordinates or by using an output file.You can save your output to BMP, GIF, GRID, IMG, JPEG, JPEG 2000, PNG, TIFF, or any geodatabase raster dataset.
Example: If you had a map and only wanted a certain area of the map, you would be able to use the clip tool to cut out that area from the map.
# Arch/quiz1

On the internet open in a Search Images window and enter the below town followed by the word RED WHITE GREEN YELLOW BLUE BLACK
| What colour are ... | COLOUR |
| GRASS | .. |
| CUSTARD | .. |
| COCA COLA | .. |
| DIANE KEATON | .. |
| MICKEY MOUSE | .. |
# Arch/quiz3

On the internet open in a Search Images window and enter the below text followed by either .. RED YELLOW BLUE ORANGE
| Type EURO2008 then .. | COLOUR |
| Spain | .. |
| Holland | .. |
| France | .. |
| Poland | .. |
| Sweden | .. |
| Italy | .. |
Treasure Hunt arch/start :
# ArcGIS/Buffer Tool

Here is where information about the buffer tool goes.
# Archaeological Field Methods
## Basic Introduction

Archaeology is a vast subject area, but this page aims to give an overview on the practical field methods in archaeology. It is hoped that it will become a handbook for archaeologists.

## What is archaeology?

Archaeology has been defined as the "study of The word archaeology comes from ancient Greek, meaning "the study of the beginning."

## What are field archaeology methods?

Introduction to non evasive archeological surveying. This involves the use of a magnetometer which test the conductivity of an area. Spikes from the lower end of you're base line typically indicate that you have found an archeological site and is a good place to begin your search. For instance say you have a general base line at 50 miligaughs and either gradually or all at once it jumps to 100 chances are you just found a habitation site from possibly 30 years ago or who knows maybe 30,000 ybp. This is where you begin other field tactics.

## Project management and Planning excavation

In archaeological excavation English Heritage produced a model of scientific management which is referred to as Management of Archaeological Projects MAP. This document gives the archaeologist clearly defined phases to follow in the organisation of their excavation work. There are five phases: Phase 1 Project Planning Phase 2 Fieldwork Phase 3 Assessment of potential for analysis Phase 4 Analysis and report preparation Phase 5 Dissemination Reference Andrews, G 1991 Management of archaeological projects, London, English Heritage.

## Post-excavation work

Drewett P.L, 1999, Field Archaeology: An Introduction, UCL Press Gamble C, 2004, Archaeology: The Basics, Routledge.

## External Links

* Google Map of the Ancient Ruins of Africa
* Google Map of the Ancient Ruins of Asia
* Google Map of the Ancient Ruins of Europe
* Google Map of the Ancient Ruins of South and Central America
* Google Map of the Ruins of The United States
* Google Map of Aztec, Egyptian, Inca, Mayan & Chinese Pyramids
* Google Map of Ancient Lost Cities
# Christian Thought From the Eighteenth to the Twentieth Century/Russian-Orthodox Imperialism

The page is under construction..
          "The ...". (...)
To get the bigger picture we want to study several short overviews and then some representatives of ... of the eighteenth century. Please go to each of the below web sites and study each of these essays extensively. Make notes while you read. You will need your notes later to write an essay.
Write an essay about the following topic:
          "Which specifics ...?"
The essay should have at least 2000 words. Click here to submit your essay.
From left to right: Archbishop Gavriil Petrov, Archbishop Stefan Yavorskiy, Archbishop Feofan Prokopovich.
<< back to syllabus
Units: 1 | 2 | 3 | 4 | 5
# Christian apologetics

Christian apologetics is the field of study concerned with the systematic defense of Christianity"; hence this project focuses on developing learning materials that defend the integrity of the Bible.

## Resources

* Defense of Creation
* Defense of the Resurrection
# Christian Anarchism
## Course Introduction

This course (in rudimentary stub-form at this time) is designed to be a basic introduction to Christian Anarchism. It would be appropriate for clergy, seminarians, college students and activists.
   Christian anarchism is a movement in political theology that claims Christianity is fundamentally anarchistic.It is an important movement. It is grounded in the belief that there is only one source of authority to which Christians are ultimately answerable, the authority of God as embodied in the teachings of Jesus, and thus rejects the idea that human governments have ultimate authority over human societies. Christian anarchists denounce the state as they claim it is violent, deceitful and, when glorified, idolatrous. Christian anarchists hold that the proper relationship between God and people is the "Reign of God" in which human relationships would be characterized by divided authority, servant leadership, and universal compassion rather than the hierarchical, authoritarian structures normally attributed to religion.
   More than any other Bible source, the Sermon on the Mount is used as the basis for Christian anarchism. Most Christian anarchists are pacifists and reject the use of violence, such as war. Leo Tolstoy's The Kingdom of God Is Within You is often regarded as a key text for modern Christian anarchism.

## Course Structure
### Section 1: Introduction
#### Definitions: what is Anarchism? What is Christianity? What is Christian Anarchism?

Anarchism which aligns itself with socially libertarian ideals in the Bible, and the belief in the abolition of the state, which includes all unnecessary hierarchies.

#### Political theory and taxonomy
### Section 3: Practical Application/Orthopraxy
#### Anarchism and Ethics
#### Religious vs. Non-Religious Anarchism
## More information
### Videos

* Coming soon

### Audio

* Coming soon

### Books

* Coming soon

### Web

* Coming soon
# Christian Thought From the Eighteenth to the Twentieth Century/Roman Catholic Globalization

This page is under construction..
          "The missionary Church opposed this state of affairs from the beginning, and nearly everything positive that was done for the benefit of the indigenous peoples resulted from the call and clamor of the missionaries. The fact remained, however, that widespread injustice was extremely difficult to uproot ... Even more important than Bartolome de Las Casas was the Bishop of Nicaragua, Antonio de Valdeviso, who ultimately suffered martyrdom for his defense of the Indian". (Enrique Dussel, in: A History of the Church in Latin America)
* How did the moral philosophy towards indigenous people that was taught by the "School of Salamanca" influence the global spread of Catholicism? What about slavery in that context?
* Were Roman Catholic missionaries in the Americas and in Asia accomplices of the European nations and their colonial agenda?
To get the bigger picture we want to study several short overviews and then some representatives of ... the eighteenth century. Please go to each of the below web sites and study each of these essays extensively. Make notes while you read. You will need your notes later to write an essay.
* Jeff Mirus: Jesuits in 18th Century Tibet
* John E. Worth: Spanish Missions
* Marian T. Horvat: Ven. Antonio Margil of Jesus: Apostle of New Spain
* Adriaan C. van Oss: Catholic Colonialism: A Parish History of Guatemala, 1524-1821
Write an essay about the following topic:
          "Which ...?"
The essay should have at least 2000 words. Click here to submit your essay.
From left to right: Archbishop Diego del Corro (Santiago, Chile), Pope Benedictus (Benoit) XIV, Matteo Ricci (on the left, Jesuit in China) and Paul Xu Guanggi (Chinese convert).
<< back to syllabus
Units: 1 | 2 | 3 | 4 | 5
# Christian Thought From the Eighteenth to the Twentieth Century

MA-Level
Overview
In this course we study Christian thought of the eighteenth, nineteenth, and twentieth century. Space and time allow only to pick a few representatives of each era that seem typical for their time and movement. The course is divided into three modules with each five units. Module 1 covers the eighteenth century, module 2 the nineteenth, and module 3 the twentieth century. The course participants are encouraged to research wider and deeper on their own .
Primary learning objectives for this course:
* Logical, systematic, and competent active participation in online academic debate
* Writing/publication skill for online encyclopedias (wikis)
* Writing/publication skill for research essays
You will meet the objectives listed above through a combination of the following activities in this course:
* Attend all synchronous tlk.io discussions.
* Publish two Wikipedia articles. (See: Kill, M: Teaching Digital Rhetoric: Wikipedia, p.389ff.)
* Publish or publicly answer/discuss three Quora questions.
* Write five major essays during the course that will be peer-reviewed.
Please read the essays "Colonialism", "Enlightenment", and "Liberalism" at SEP.
Please also read the essay "Theological Liberalism" at theopedia.
Recommended text:
* James C Livingston: Modern Christian Thought, The Enlightenment And the Nineteenth Century. 2006, 2nd ed. at Alban Books, ISBN 080063795X.
This course will be delivered entirely online. You will use your Wikiversity account to login to the course. Every page of this course has a "tandem" discussion page that we will use to debate about form and content of the course page. Your favorite button should be the "edit" button of the discussion page.
Module 1: The Eighteenth Century
* Unit 1: Revivalism in the Americas
* Unit 2: Methodism in England
* Unit 3: Pietism in Germany
* Unit 4: Roman Catholic Globalization
* Unit 5: Russian-Orthodox Imperialism
Module 2: The Nineteenth Century
Module 3: The Twentieth Century
# Creating Dynamic Lessons/Introduction

Go Back or Go to the next part of the lesson: Part 2: Intro Practice.
Lesson Introduction
What is the point of this lesson?
Learners will use online tools to be able to design a dynamic, standards-oriented digital unit. The result will be a unit that contains online instruction, online practice, and online assessment.
If you complete this lesson, what will you be able to do?
    o Learner will create a written, digital lesson that includes all of the elements from the checklist assessment.
    o Learners will be able to locate online resources for dynamic, digital lesson introductions.
    o Learners will be able to design a lesson that contains an attention-grabbing introduction for the learner.
    o Learners will be able to design a lesson that contains a portion of the lesson where students can digitally practice what they know and receive feedback.
    o Learners will be able to implement checklist assessments into a lesson.
At the end of this lesson, you will use this checklist to assess your lessons and other people's lessons.
Directions:Check the box that contains the answer to your question.
| Question | Yes | No |
| 1. Did you include an attention-getter in your lesson? | 1 | 0 |
| 2. Did you introduce your lesson and its main goals after the attention-getter? | 1 | 0 |
| 3. After the introduction, did you include the instructional phase of your lesson that teaches the lesson's main goals? | 1 | 0 |
| 4. Did you include some examples in the instruction? | 1 | 0 |
| 5. After the lesson's instruction, did you offer some methods for the student to practice the lesson's goals? | 1 | 0 |
| 6. At the end of the lesson, did you include an assessment to check your student's learning? | 1 | 0 |
| ..........................................................................................................................................................................Total | ---- | --- |
Now, onto the "meat and potatoes" of the lesson! Go Back or Go to the next part of the lesson: Part 2: Intro Practice.

## All Pages in the Lesson
### Creating Dynamic Lessons

* Part 1: Introduction
* Part 2: Intro_Practice
* Part 3: The Lesson
* Part 4: Practice
* Part 5: Practice the Practice Phase
* Part 6: Practicing Creating a Lesson
* Part 7: Assessment Overview
* Part 8: Assessment
* Part 9: Summary
# Creating Dynamic Lessons/Lesson

Go Back or Go to the next part of the lesson: Part 4: Practice.
Digital Lessons
This (Press Ctrl on your keyboard and then click the link) digital lesson was created using Windows Movie Maker and
published using youtube.com, or you can read it in plain text below.
You can present digital lessons in many ways. A movie or podcast could be good options! The lesson phase should contain the instruction parts of the lesson.
You could use direct instruction that contains a lecture, which can be presented in several different ways. The lesson phase could also contain a combination of explanations, examples, and non-examples.
You not want to skip the lesson phase and go directly to the assessment phase. The lesson is what the students will be tested
over later.
Here are some ways you may choose to teach your students the lesson:
* Type out lesson for student to read
* Record lesson as a podcast for student to hear
* Create a slide show presentation
* Create a video with still pictures and text to view and/or hear
* Film yourself teaching the lesson
* Film various scenarios, actors/actresses, and student while having a voice narrator
* Use a video animation website to create a lesson
* Use a presentation website to create a lesson
* Use a combination of the above methods!
More, more, more! Please click "Discuss" tab at the top of this page to view examples of the previous methods. Please reflect on your own methods and research on the internet, and add two examples of your own. Explore the other posted examples. Who knows what other awesome sites might be available! If you need to start a new method category, go for it!
Go Back or Go to the next part of the lesson: Part 4: Practice.

## All Pages in the Lesson
### Creating Dynamic Lessons

* Part 1: Introduction
* Part 2: Intro_Practice
* Part 3: The Lesson
* Part 4: Practice
* Part 5: Practice the Practice Phase
* Part 6: Practicing Creating a Lesson
* Part 7: Assessment Overview
* Part 8: Assessment
* Part 9: Summary
# Creating Dynamic Lessons/Creating Lesson

<Go Back or Go to the next part of the lesson: Part 7: Assessment Overview.
So what are the steps of a lesson again? ReviewA lesson should contain these elements in the following order:
* Attention-Getter
* Introduction
* Lesson Content
* Lesson Practice
* Lesson Assessment
Assignment Challenge!You need to assess what your students have learned. The next page will discuss how to create a lesson's assessment. However, it's time to practice creating the first four steps of a lesson. Challenge Accepted?Select the "Discuss" tab at the top of the page. Design a digital lesson to teach vocabulary that describes how you will--
* Grab the learner's attention
* Introduce the lesson's goals and objectives
* Select the lesson's content
* Engage the learner in a practice activity
How can you use a variety of digital resources to make your lesson POP?
Go Back or Go to the next part of the lesson: Part 7: Assessment Overview.

## All Pages in the Lesson
### Creating Dynamic Lessons

* Part 1: Introduction
* Part 2: Intro_Practice
* Part 3: The Lesson
* Part 4: Practice
* Part 5: Practice the Practice Phase
* Part 6: Practicing Creating a Lesson
* Part 7: Assessment Overview
* Part 8: Assessment
* Part 9: Summary