OpenLLM-France/wikimedia huggingface dataset:
# Matrix/2305/Canonical decomposition/Exercise

We consider the matrix
    ${\begin{pmatrix}2&3\\0&5\end{pmatrix}},$
and the three decompositions
    ${\begin{pmatrix}2&3\\0&5\end{pmatrix}}={\begin{pmatrix}2&0\\0&5\end{pmatrix}}+{\begin{pmatrix}0&3\\0&0\end{pmatrix}}\,,$
    ${\begin{pmatrix}2&3\\0&5\end{pmatrix}}={\begin{pmatrix}2&3\\0&5\end{pmatrix}}+{\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,$
and
    ${\begin{pmatrix}2&0\\0&5\end{pmatrix}}={\begin{pmatrix}2&0\\0&5\end{pmatrix}}+{\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,.$
Which one is (or are) the canonical additive decomposition in the sense of fact (and with respect to which basis)?
# Jordan matrix/Eigenvector/One-dimensional/Exercise

Let $M$ be a Jordan matrix to the eigenvalue $\lambda$. Show that the eigenspace of $M$ to the eigenvalue $\lambda$ is one-dimensional, and that there exist no further eigenvectors.
# Upper triangular matrix/44/Jordan normal form/1/Exercise

Determine a basis such that linear mapping given by
    $M={\begin{pmatrix}4&-6&0&-1\\0&5&2&7\\0&0&4&4\\0&0&0&5\end{pmatrix}}\,$
is in Jordan normal form with respect to this basis.
# Positive characteristic/2x2-Jordan-matrix/Finite order/Exercise

Let $K$ be a field of positive characteristic $p>0$. Show that the matrix
    ${\begin{pmatrix}1&1\\0&1\end{pmatrix}}$
has finite order $p$.
# Endomorphism/Trigonalizable/Canonical additive decomposition/Fact/Proof/Projection/Exercise

Describe the direct sum decomposition of the $p_{i}$ with respect to the generalized eigenspaces in the proof of fact.
# Upper triangular matrix/4x4/Kernel one-dimensional/Exercise

Determine, for the real matrix
    $M={\begin{pmatrix}3&5&2&4\\0&3&6&-1\\0&0&3&2\\0&0&0&3\end{pmatrix}}\,,$
its Jordan normal form (we do not ask for a basis).
# Matrix/-2 2 7 0 -2 -6 0 0 -2/Jordan/Exercise

Suppose that the linear mapping
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$
is given by the matrix
    ${\begin{pmatrix}-2&2&7\\0&-2&-6\\0&0&-2\end{pmatrix}}$
with respect to the standard basis. Find a basis such that $\varphi$ is described by the matrix
    ${\begin{pmatrix}-2&1&0\\0&-2&1\\0&0&-2\end{pmatrix}}$
with respect to the new basis.
# Jordan matrix/Nilpotent/Kernel sequence/Flag/Exercise

Let $M$ be a nilpotent $n\times n$-Jordan matrix. Show, that the kernels $\operatorname {kern} M^{i}$ form a flag in $K^{n}$.
# Endomorphism/Finite order/C/Characteristic polynomial/Exercise

Let $\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional $\mathbb {C}$-vector space $V$, and suppose that $\varphi$ has finite order. Show that the characteristic polynomial of $\varphi$ divides a polynomial of the form ${\left(X^{n_{1}}-1\right)}\cdots {\left(X^{n_{r}}-1\right)}$. Show also that the characteristic polynomial does not, in general, divide a polynomial of the form $X^{n}-1$.
# Endomorphism/Finite order/R/Characteristic polynomial/Exercise

Let $\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional $\mathbb {R}$-vector space $V$, and suppose that $\varphi$ has finite order. Show that the characteristic polynomial of $\varphi$ divides a polynomial of the form ${\left(X^{n_{1}}-1\right)}\cdots {\left(X^{n_{r}}-1\right)}$.
# Endomorphism/Finite order/Minimal polynomial/Exercise

Let $\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional $K$-vector space $V$. Show that $\varphi$ has finite order if and only if the minimal polynomial of $\varphi$ is a divisor of $X^{n}-1$ for some $n\in \mathbb {N} _{+}$.
# Quarter rotation/Diagonalization over C/Exercise

The matrix
    $M={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\,$
describes a quarter-turn in the real plane. Diagonalize this matrix over the complex numbers.
# Jordan-Block/2/Powers/Induction/Exercise

Let
    $M={\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}\,$
with $\lambda \in K$. Show by induction that
    $M^{n}={\begin{pmatrix}\lambda ^{n}&n\lambda ^{n-1}\\0&\lambda ^{n}\end{pmatrix}}\,$
holds.
# Jordan-Block/2/Powers/Induction/Exercise/Solution

For
    $n=0,1\,,$
the statement is clear. Suppose now that the statement is already proven for some $n$. Then
    ${\begin{aligned}M^{n+1}&=M^{n}\circ M\\&={\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}^{n}\circ {\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}\\&={\begin{pmatrix}\lambda ^{n}&n\lambda ^{n-1}\\0&\lambda ^{n}\end{pmatrix}}\circ {\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}\\&={\begin{pmatrix}\lambda ^{n+1}&\lambda ^{n}+n\lambda ^{n-1}\cdot \lambda \\0&\lambda ^{n+1}\end{pmatrix}}\\&={\begin{pmatrix}\lambda ^{n+1}&(n+1)\lambda ^{n}\\0&\lambda ^{n+1}\end{pmatrix}}.\end{aligned}}$
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Trigonalizable/Canonical additive decomposition/Fact/Factreferencenumber

Theorem 28.1
# Jordan normal form/Canonical additive decomposition/Exercise

Let $M$ be an $n\times n$-matrix in Jordan normal form, and let $D$ denote the corresponding diagonal matrix. Show that the canonical additive decomposition in the sense of fact is
    $M=D+(M-D)\,.$
# Jordan normal form/Product/Exercise

Show that the product of two matrices in Jordan normal form is, in general, not in Jordan normal form.
# Jordan normal form/One eigenvalue/Block number/Exercise

Let $M$ be a matrix in Jordan normal form, where only one eigenvalue occurs. Show that the number of the Jordan blocks in $M$ equals the dimension of the eigenspace.
# Jordan normal form/One eigenvalue/Minimal polynomial/Exercise

Let $M$ be an $n\times n$-matrix with Jordan blocks $J_{1},\ldots ,J_{k}$, and suppose that all diagonal entries are $\lambda$. Determine the minimal polynomial of $M$.
# Jordan matrix/Minimal polynomial/Exercise

Let $M$ be an $n\times n$-Jordan matrix to the eigenvalue $\lambda$. Determine the minimal polynomial of $M$.
# Diagonal matrix/No homothety/Does not commute/Exercise

Let
    $D={\begin{pmatrix}a_{11}&0&\cdots &\cdots &0\\0&a_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{n-1\,n-1}&0\\0&\cdots &\cdots &0&a_{nn}\end{pmatrix}}\,.$
Show that $D$ commutes with all $n\times n$-matrices $M$ if and only if all its diagonal entries coincide.
# Addition in natural space/Ear and Finger/Exercise

Lay the connecting arrow from your left ear to the right small finger of the person in front of you, in a parallel way, at the tip of the nose of your left neighbor. What is the result?
# Affine spaces/Product/Basis/Exercise

Let $E$ and $F$ be affine spaces over the field $K$, and let an affine basis $P_{1},\ldots ,P_{n}$ of $E$ and an affine basis $Q_{1},\ldots ,Q_{m}$ of $F$ be given. Show that
    $(P_{1},Q_{1}),\,(P_{1},Q_{2}),\ldots ,(P_{1},Q_{m}),\,(P_{2},Q_{1}),\,(P_{3},Q_{1}),\ldots ,(P_{n},Q_{1})$
is an affine basis of the product space $E\times F$.
# Affine spaces/Product space/Exercise

Let $E$ and $F$ be affine spaces over the field $K$. Show that the product space $E\times F$ is also an affine space.
# Affine space/Product with K/Standard space/Basis/Exercise

Let $V$ be a $K$-vector space. We consider the set
    $E={\left\{(v,1)\mid v\in V\right\}}\subset V\times K\,,$
which is an affine space over $V$.
a) Show that the points $P_{i}=(v_{i},1)$, $i=1,\ldots ,n$, form an affine basis of $E$ if and only if the $P_{i}$ (considered as vectors in $V\times K$) form a vector space basis of $V\times K$.
b) Show that, in this case, for a point $P\in E$, the barycentric coordinates of $P$ with respect to $P_{1},\ldots ,P_{n}$ equal the coordinates of $P$ with respect to the vector space basis $P_{1},\ldots ,P_{n}$.
# Linear mapping/Kernel and fiber/3/Exercise

Let
    $\varphi \colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{2},{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}\longmapsto {\begin{pmatrix}6x-5y-3z+8w\\x+5y+4z-2w\end{pmatrix}}.$
Determine, for the set
    $E={\left\{Q\in \mathbb {R} ^{4}\mid \varphi (Q)={\begin{pmatrix}-3\\-7\end{pmatrix}}\right\}}\,,$
a description using a starting point and a translating space.
# Time/Affine line/Translation/Exercise

Time is an affine line over $\mathbb {R}$. Lay the connecting arrow from the moment of your first milk tooth to the moment of your school enrollment at the moment now. What is the result?
# Affine space/Basic rules/Exercise

Let $E$ be an affine space over the $K$-vector space $V$. Show the following identities in $V$.
* ${\overrightarrow {PP}}=0$ for $P\in E$.
* ${\overrightarrow {PQ}}=-{\overrightarrow {QP}}$ for $P,Q\in E$.
* ${\overrightarrow {PQ}}+{\overrightarrow {QR}}={\overrightarrow {PR}}$ for $P,Q,R\in E$.
# Linear algebra (Osnabrück 2024-2025)/Affine space/Basic rules/Exercise/Exercisereferencenumber

Exercise 29.10
# Points/Natural space/No basis/Imagination/Exercise

Imagine four points in the natural $3$-space such that they do not form an affine basis of the space, but such that each three of these points form an affine basis in an affine plane.
# Affine basis/Natural space/Imagination/Exercise

Imagine four points in the natural $3$-space such that they form an affine basis of the space.
# Additive mixing/Favorite color/Barycentric coordinates/Exercise

What are the barycentric coordinates of your favorite color in additive mixing?
# Vector space/Points/Barycentric combination/Exercise

Let $V$ be a vector space over $K$, which we consider as an affine space. Let $\sum _{i\in I}a_{i}v_{i}$ with $v_{i}\in V$, $a_{i}\in K$ and $\sum _{i\in I}a_{i}=1$ denote a barycentric combination. Show that the point defined by this barycentric combination in affine space equals the vector sum $\sum _{i\in I}a_{i}v_{i}$.
# Barycentric coordinates/Point difference/Exercise

Let $E$ be an affine space over the $K$-vector space $V$. Let $P_{i}$, $i\in I$, and $P,Q$ denote points in $E$, and let $\sum _{i\in I}a_{i}P_{i}$ be a barycentric combination. Show that
    $P-Q+\sum _{i\in I}a_{i}P_{i}={\overrightarrow {QP}}+{\left(\sum _{i\in I}a_{i}P_{i}\right)}\,$
holds, where the left-hand expression is a barycentric combination.
# Vector space/Translating vector/Difference/Exercise

Let $V$ be a $K$-vector space, which we consider as an affine space over itself. Let $P,Q\in V$ be points. Show ${\overrightarrow {PQ}}=Q-P$.
# Barycentric coordinates/Point representation/Exercise

Let $P$ be a point in an affine space $E$ over $V$. Show that the following expressions are barycentric combinations for $P$ (let $Q\in E$ and $v\in V$).
* $P$.
* $P+Q-Q$.
* $(P+v)-(Q+v)+Q$.
# Barycentric coordinates/Point representation/Exercise/Solution

* We have
    $P+{\overrightarrow {PP}}=P+0=P\,.$
* We have
    $P+{\overrightarrow {PP}}+{\overrightarrow {PQ}}-{\overrightarrow {PQ}}=P+0=P\,.$
* We have
    ${\begin{aligned}P+{\overrightarrow {P,P+v}}-{\overrightarrow {P,Q+v}}+{\overrightarrow {PQ}}&=P+v+{\overrightarrow {Q+v,P}}+{\overrightarrow {PQ}}\\&=P+v+{\overrightarrow {Q+v,Q}}\\&=P+v-v\\&=P.\end{aligned}}$
# Affine space/Product with K/Realization/Exercise

Let $E$ be a nonempty affine space over a $K$-vector space $V$. Let $P\in E$ be a fixed point, and let
    $\theta \colon V\longrightarrow E,v\longmapsto P+v,$
be the corresponding bijection. Using this bijection, we identify $E$ with
    $E'={\left\{(v,1)\in V\times K\mid v\in V\right\}}\,$
via the mapping
    $\varphi \colon E\longrightarrow E',P\longmapsto (\theta ^{-1}(P),1).$
a) Show that $E'$ is an affine subspace of $V\times K$, with translation space $V\times 0$.
b) Show that
    $\varphi (Q+v)=\varphi (Q)+v\,$
holds for all $Q\in E$.
# Empty space/Affine space/Exercise

Show that the empty set is an affine space over any $K$-vector space $V$.
# Wikiphilosophers/Meaning of life/tropwine

Life is merely a repetitive, endless sequence of chemical manipulations that aim to create order from disorder or, in other words, to reduce the entropy inside of itself. In order to do so life must intake energy from its surroundings while evading death from natural factors such as predation. Life is capable of creating copies of itself, splinters of the chemical reaction that partake in the same quest all over again.
Due to self-reinforcing imperfections in the chemistry of life, processes such as evolution are able to take place. In this complexity increases: the development of more complex social behaviours, of larger brains, of more complex internal structures and better physical adaptions to its environment (such as fur or camouflage) are just a few of the consequences of evolution inherently driven by natural selection.
Life, being a system of chemical reactions, is a consequence of the way our universe is structured. The "laws" that make up what is physically possible and impossible in this universe have shaped life to what it is today. Life arose on this planet due to the conditions and molecules that were present on it at the time. All it took was for the first self replicating chemical machine to come into existence for life to diverge and evolve.
Life has no meaning, meaning would imply a creator of which I believe there is none. The only meaning life has is derived from the nature of the universe itself, which is a question for the metaphysicians.
To put it simply, life is a consequence of the way the universe is.

## Footnotes

* ↑ DNA transcription has inherent errors, when these errors occur it is known as a genetic mutation, one of the key drivers of natural selection. Given 4 billion years of evolution the perfection of this process would most most likely had occurred if it was not disbeneficial to any organism that would have adopted it.
# Motivation and emotion/Assessment/Topic/Checklist

* Title and sub-title: Use the correct title and sub-title; use sentence casing
* Headings: Create a heading structure with 3 to 6 top-level headings between the Overview and Conclusion; may include sub-headings
* Overview: Outlining an engaging scenario, key points, and focus questions
* Key points: At least 3 dot points per section, covering theory, research, and application
* Figure: Display and cite (e.g., see Figure 1) at least one relevant figure with a caption
* Learning feature: Include at least one learning feature (case study, embedded wiki links, table, and/or quiz)
* References: Provide at least 6 APA style citations and references to key peer-reviewed sources
* Resources: Provide at least 2 internal (wiki) links (See also) and at least 2 external links (External links)
* User page: Create a Wikiversity user page with information about yourself
* Social contribution: Summarise at least 3 types of social contribution on your user page
For more information, see the topic development author guidelines and marking criteria.
# Line equation/Parameter form/1/Exercise

Determine the parameter form for the line given by the equation
    $4x+7y=3\,$
in $\mathbb {Q} ^{2}$.
# Motivation and emotion/Assessment/Quizzes
## Overview

* Weight: 25%
* Due: Fortnightly — see schedule
* Tasks
** 6 equally-weighted, 10-question, 10-minute, multiple-choice, online quizzes.
** One quiz per module.
** Based on theories and research as described by the Reeve (2018) textbook and discussed in lectures and tutorials.
* Follow the instructions

## Marking and feedback

* Marks will be available immediately after each quiz is submitted
* Automated feedback will be provided to indicate correct answers
* Once the feedback screen has been closed, this information will no longer be available
* Follow up if you don't understand the feedback

## Extensions and late submissions

* Extension requests require an Assignment Extension Form to be submitted via UCLearn) with appropriate documentary evidence
* No late submissions are accepted for this exercise

## Learning outcomes

How the unit's learning outcomes are addressed by this assessment exercise:
| Learning outcome | Assessment task |
| Identify the major principles of motivation and emotion | The quizzes test understanding of key theories and research in the field of motivation and emotion as emphasised in the textbook, lectures, and tutorials. |

## Graduate attributes

How the unit's graduate attributes are addressed by this assessment exercise:
| Graduate attribute | Assessment task |
| Be professional — up-to-date knowledge and skills | Develop a broad understanding of current psychological theory and research about motivation and emotion. |

## Instructions

* Attempts:
** Quizzes can be attempted any time before the due date
** Each quiz can be attempted once
** Missed quizzes receive 0
* Content: Quizzes consist of 10 randomly selected multiple-choice questions from a test bank designed to assess knowledge of content covered in the corresponding textbook chapters, lectures, and tutorials
* Time limit: 10 minutes
* Academic integrity:
** Complete independently, with no assistance from others or genAI
** Study materials including the Reeve (2018) textbook, lecture/study notes, and internet resources are permitted.
* Bug bounty (bonus marks): Email the unit convener if you identify a quiz error or possible improvement. Accepted revisions earn bonus quiz marks.

## Schedule

The recommended schedule for regular completion of the quizzes is:
| Module | Week | Quiz | Chapter | Due Mon 9am |
| | 00 | 0: Practice | | |
| 1 | 01-02 | 1: Introduction | 01 - 02 | Week 04 |
| 2 | 03-04 | 2: Needs | 03 - 06 | Week 06 |
| 3 | 05-06 | 3: Goals and self | 07 - 11 | Week 08 |
| 4 | 07-09 | 4: Emotion | 12 - 13 | Week 11 |
| 5 | 10-11 | 5: Individual emotions | 14, 16 | Week 13 |
| 6 | 12-13 | 6: Growth and interventions | 15, 17 | Week 15 |
Due dates
* 1 - Week 04 Mon 9am 19 Aug 2024
* 2 - Week 06 Mon 9am 02 Sep 2024
* 3 - Week 08 Mon 9am 16 Sep 2024
* 4 - Week 11 Mon 9am 07 Oct 2024
* 5 - Week 13 Mon 9am 21 Oct 2024
* 6 - Week 15 Mon 9am 04 Nov 2024
# Vector space/Affine subspace/Arbitrary starting point/Exercise

Let $V$ be a vector space, $U\subseteq V$ a linear subspace, and let $E=P+U$ be an affine subspace. Show that, for every point $Q\in E$, we can also write $E=Q+U$.
# Vector space/Affine subspace/Linear subspace and 0/Exercise

Let $V$ be a vector space, and let $E\subseteq V$ denote an affine subspace. Show that $E$ is a linear subspace of $V$ if and only if $E$ contains $0$.
# Linear mapping/Kernel and fiber/1/Exercise

Let
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ,{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto 4x-6y+9z.$
Determine, for the set
    $E={\left\{Q\in \mathbb {R} ^{3}\mid \varphi (Q)=5\right\}}\,,$
a description using a starting point and a translation space.
# Linear mapping/Kernel and fiber/2/Exercise

Let
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}7x+y-3z\\4x+5y\end{pmatrix}}.$
Determine, for the set
    $E={\left\{Q\in \mathbb {R} ^{3}\mid \varphi (Q)={\begin{pmatrix}4\\-2\end{pmatrix}}\right\}}\,,$
a description using a starting point and a translating space.
# Space/Planed/Point on two planes/1/Exercise

We consider the three planes $E,F,G$ in $\mathbb {Q} ^{3}$, given by the following equations.
    $E={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 5x-4y+3z=2\right\}}\,,$
    $F={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 7x-5y+6z=3\right\}}\,,$
    $G={\left\{(x,y,z)\in \mathbb {Q} ^{3}\mid 2x-y+4z=5\right\}}\,.$
Determine all points in $E\cap F\setminus E\cap F\cap G$.
# Polynomial/Bounded degree/Value/Affine space/Exercise

Let $d\in \mathbb {N} _{+}$, and let $K$ denote a field. Let $n$ different elements $a_{1},\ldots ,a_{n}\in K$ and $n$ elements $b_{1},\ldots ,b_{n}\in K$ be given. Show that the set $E$ of all polynomials $P$ of degree at most $d$, satisfying
    $P(a_{i})=b_{i}\,$
for $i=1,\ldots ,n$, is an affine subspace of $K[X]_{\leq d}$. What is the corresponding linear subspace? What can we say about the dimension of $E$, when is $E$ empty?
# Point set/R^3/Affinely independent/1/Exercise

Check whether the points
    ${\begin{pmatrix}5\\4\\7\end{pmatrix}},\,{\begin{pmatrix}-2\\1\\6\end{pmatrix}},\,{\begin{pmatrix}3\\-9\\4\end{pmatrix}},\,{\begin{pmatrix}-8\\8\\3\end{pmatrix}}$
in $\mathbb {R} ^{3}$ are affinely independent.
# Affine mapping/Fixed points/Identity/Exercise

Let $E$ be an affine space of dimension $n$, and let
    $\psi \colon E\longrightarrow E$
denote an affine mapping. Let $P_{1},\ldots ,P_{n+1}\in E$ be affinely independent points, and suppose that they are also fixed points of $\psi$. Show that $\psi$ is the identity.
# Affine space/Affine plane/Description as preimage/2/Exercise

Describe the affine plane
    $E={\left\{{\begin{pmatrix}5\\6\\-2\end{pmatrix}}+s{\begin{pmatrix}3\\-4\\8\end{pmatrix}}+t{\begin{pmatrix}5\\4\\7\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$
as the preimage over $1$ of an affine mapping $\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.
# Affine spaces/Affine mapping/Affine subspaces/Exercise

Let
    $\varphi \colon E\longrightarrow F$
be an affine-linear mapping between the affine spaces $E$ and $F$ over $K$. Show that, for every affine subspace $G\subseteq F$, the preimage $\varphi ^{-1}(G)$ is an affine subspace of $E$.
# Affine mapping/Fibers parallel/Exercise

Let
    $\varphi \colon E\longrightarrow F$
be an affine-linear mapping between the affine spaces $E$ and $F$ over $K$. Show that the preimages $\varphi ^{-1}(Q)$ for all $Q\in F$ are parallel to each other.
# Affine mapping/Affine line/Points/Image/1/Exercise

Determine, by a drawing, the image point of $P$ under the affine mapping $\varphi$ given by $\varphi (P_{i})=Q_{i}$.
# R to R/Affine-linear/Barycentric combination/Exercise

Let $\psi \colon \mathbb {R} \rightarrow \mathbb {R}$ be a mapping of the form
    $\psi (x)=ax+b\,$
with certain $a,b\in \mathbb {R}$. Show directly that $\psi$ is compatible with barycentric combinations.
# Affine space/Finite point family/Affine basis/Characterization/Exercise

Let $E$ be an affine space over the $K$-vector space $V$, and let $P_{1},\ldots ,P_{n}$ be a finite family of points in $E$. Show that the following statements are equivalent.
* The points form an affine basis of $E$.
* The points form a minimal affine generating system of $E$.
* The points are maximally affinely independent.
# Motivation and emotion/Tutorials/Review

Tutorial 12: Review This is the twelth tutorial for the motivation and emotion unit of study.

## Overview

This tutorial provides active review of key learnings from the unit.

## Nuggets of truth

Baleen whales (such as the Blue Whale which is the largest animal on earth) eats the tiniest food – plankton (see Figure 1).
How does it grow so big from such tiny food?
A baleen whale sucks in large volumes of sea water and uses its baleen to filter out plankton for consumption.
Learners are similar to baleen whales. After being exposed to a large volume of content, what really matters is what has been filtered out and extracted by learners.
Thus, this exercise seeks to crystallise and share the:
* pearls of wisdom
* nuggets of truth
* flashes of insight
* a-ha moments
* take-home messages etc.
that have been sparked by your engagement with the motivation and emotion unit.
* What are the greatest insights or best ideas about motivation and emotion you've acquired through this unit?
* What was the take-home message of the chapter and presentation you worked on?
* What have you learnt from the unit that you've put into practice in your own life?
* What aspect(s) of motivation and emotion are you inspired to learning more about in future?
* Share your ideas via Wikiversity

## Multimedia presentation

* Have you read the multimedia presentation guidelines?
* Have you watched the how to record a multimedia presentation and publish it online?
* What ideas do you have?
** Format?
** Platform?
** Content?
* FAQ
** Image attributions
** References
* What questions do you have?

## Recording

* Tutorial 12 (2023) (there was no recording in 2024)
# Affine mapping/One point suffices/Exercise

Let $K$ be a field, and let $E$ and $F$ denote affine spaces over the vector spaces $V$ and $W$, respectively. Let a mapping
    $\psi \colon E\longrightarrow F,$
a linear mapping
    $\varphi \colon V\longrightarrow W,$
and a point $P\in E$ be given such that
    $\psi (P+v)=\psi (P)+\varphi (v)\,$
holds for all $v\in V$. Show that $\psi$ is affine-linear.