ជំពូក១ :  ប្រព័ន្ធលីនេអ៊ែរ (Linear System)

October 2nd, 2020

From Wikipedia:

ក្នុងលំហ $\mathbb {R}^{3}$ ប្លង់ $P_{2}$ ( ព័ណ៌ខៀវខាងលើ ) មិនមែនជាលំហវ៉ិចទ័ររងទេ តែវាជាលំហអាហ្វីនរង។ ទិសដៅរបស់វា គឺប្លង់ $P_{1}$ ( ព័ណ៌បៃតងនៅខាងក្រោម ) ដែលជាលំហវ៉ិចទ័ររង។ ទោះបីវ៉ិចទ័រ $\vec{a}$ និង $\vec{b}$ ស្ថិតក្នុង $P_{2}$ តែផលសងរបស់វា គឺជា វ៉ិចទ័រប្រាប់ទីតាំង their difference is a displacement vector: $\vec{a} - \vec{b} \notin P_{2}$ តែ $\vec{a} - \vec{b} \in P_{1}$.

ក្នុងគណិតវិទ្យា លំហអាហ្វីន (affine space) គឺជាទម្រង់ធរណីមាត្រមួយ ដែលកំណត់បានភាពទូទៅនៃលក្ខណៈរបស់ លំហអឺគ្លីដ (Euclidean spaces) តាមវិធីមួយដែលមិនអាស្រ័យទៅនឹងគោលគំនិតទាំងឡាយណាទាក់ទងនឹងចំងាយ និងរង្វាស់មុំ តែវារក្សាបានលក្ខណៈទាំងឡាយណា ដែលទាក់ទងនឹងភាពស្របគ្នា និងសមាមាត្រនៃប្រវែងរបស់អង្កត់ស្របគ្នា។

នៅក្នុងលំហអាហ្វីន គេមិនអាចធ្វើការបែងចែកភាពខុសគ្នានៃចំណុចគល់តម្រុយទាំងឡាយបានទេ។ ដូច្នេះ មិនមានវ៉ិចទ័រណាមួយ ដែលមានចំណុចគល់នៅនឹងថ្កល់នោះទេ ហើយក៏គ្មានវ៉ិចទ័រណាមួយ អាចភ្ជាប់បានតែមួយរបៀបគត់ទៅនឹងចំណុចមួយនោះដែរ។ ជំនួសមកវិញ ក្នុងលំហអាហ្វីន មានការបម្លាស់ទីតាំងនៃវ៉ិចទ័រ ដែលគេហៅថា វ៉ិចទ័របម្លែងកិល ឬត្រឹមតែ បម្លែងកិល រវាងពីរចំណុចក្នុងលំហ។ In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.[1] Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. We see that

  សេចក្ដីផ្ដើមអំពីប្រព័ន្ធសមីការលីនេអ៊ែរ

Linear algebra is an essential tool for all branches of mathematics, in particular when it is about the modeling and then numerically solving problems from various fields: physical or mechanical sciences, life sciences, chemistry, economics, engineering sciences ...
Linear systems intervene through their applications in many contexts, because they form the computational basis of linear algebra. They also allow to treat a good part of the theory of linear algebra in finite dimension. This is why this course begins with a study of linear equations and their solution.
The aim of this chapter is essentially practical: it is a question of solving linear systems. The theoretical part will be reviewed and proven in the ‍‌Matrices chapter.

  និយមន័យ និងលក្ខណៈបឋមៗ

ចំណាំ : បើ $\mathcal{E}$ ជាលំហអាហ្វីន តាមទិសដៅ $\vect{\E}$ នោះធាតុរបស់ $\mathcal{E}$ ហៅថា ចំណុច ហើយគេកំណត់សរសេរដោយអក្សរធំ $A,B, C,D,M,N,P$, $\cdots$។ ចំណែកឯធាតុរបស់លំហ $\vect{\E}$ ហៅថា វ៉ិចទ័រ ហើយកំណត់សរសេរដោយអក្សរតូច មានសញ្ញាព្រួញនៅខាងលើ គឺ $\vec{i}, \vec{j}, \vec{k}, \vec{u}, \vec{v}, \cdots $។

ចំណាំ: ដោយសារបែបនេះហើយ ទើបធាតុរបស់លំហវ៉ិចទ័រកំណត់លើកាយ $\mathbb{R}$ នៅក្នុងបរិបទខ្លះ ត្រូវបានចាត់ទុកថាជា ចំណុច ឬក៏ជា វ៉ិចទ័រ។

  វិធានគណនា

យក $\mathcal{E}$ ជាលំហអាហ្វីនភ្ជាប់ទៅនឹងលំហវ៉ិចទ័រ $\vect{\E}$ ។ ដូច្នេះគេបាន

1. $\forall M, N \in \mathcal{E},\ \vect{AM}= \vect{AN}\Longrightarrow M = N$
2. $\forall A, B \in \mathcal{E},\ \vect{AB} =\vect{0} \Longleftrightarrow A=B$
3. $\forall A, B \in \mathcal{E},\ \vect{BA} = -\vect{AB}$
4. $\forall A, B, C \in \mathcal{E},\ \vect{BC} = \vect{AC} - \vect{AB}$
5. ចំពោះគ្រប់ចំណុច $A\in \mathcal{E}$ និងគ្រប់វ៉ិចទ័រ $\vect{u} \in \vect\E$ នោះមានចំណុចតែមួយគត់ $M\in \mathcal{E}$ ដែល $\vect{AM} = \vect{u}$
6. បើ $A$ និង $B$ គឺជាចំណុចពីររបស់ $\mathcal{E}$ នោះមានវ៉ិចទ័រតែមួយគត់ $\vect{u}\in \vect\E$ ដែល $\vect{AB} = \vect{u}$

[Show proof]

  លក្ខណៈរបស់ប្រលេឡូក្រាម

សំណើ. ឧបមា $A, B, C$ និង $D$ គឺជាចំណុចបួននៅក្នុងលំហអាហ្វីន $\mathcal E$។ ដូច្នេះ លក្ខណៈខាងក្រោមសមមូលគ្នា៖

1. $\vect{AB} = \vect{CD}$
2. $\vect{AC} = \vect{BD}$
3. $\vect{AB} + \vect{AC} = \vect{AD}$

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Motivation

This is the last part in our series exploring how to get new topological spaces from old ones. For now, at least. We mentioned the definition of the product topology for a finite product way back in Example 2.3 .6 in the lecture notes concerning bases of topologies, but we did not do anything with it at the time. In that same section we also discussed the following basis on $\mathbb{R}^{2}$ that we were eventually able to show generates the usual topology on $\mathbb{R}^{2}$ : \[ \mathcal{B} = \left\{(a, b) \times(c, d) \subseteq \mathbb{R}^{2}:\; a < b, c < d\right\} \] This is, essentially, how (finite) product topologies work in general, as we will see shortly. We will also explore a new way of analyzing topological properties themselves. We have already seen that all the topological properties we care about are preserved by homeomorphisms, and that some are preserved under weaker maps like continuous surjections (recall that the image of a dense set under a continuous function is dense in the range of the function). We also saw that some topological properties are hereditary (like Hausdorffness and second countability) while some are not (like separability). In this section we will explore another way of analyzing properties, by asking whether they are preserved by finite products. Also, while reading this section, note that we are specifically talking about finite products of topological spaces, and any time we refer to a product of spaces the reader should assume we mean a finite product. Infinite products are substantially more complicated, and we will deal with them later in the course. Finite products are actually quite straightforward.

2. Finite product topologies

Definition 2.1. Let $(X, \mathcal{T}_1)$ and $(Y, \mathcal{T}_2)$ be topological spaces. The product topology on $X \times Y$ is the topology generated by the basis \[ \mathcal{B}_{X\times Y} = \big\{U \times V: U \in \mathcal{T}_1, V \in \mathcal{T}_2\big\} \] More generally if $\left(X_{1}, \mathcal{T}_{1}\right), \ldots,\left(X_{n}, \mathcal{T}_{n}\right)$ are topological spaces, the product topology on \[ \prod_{i=1}^{n} X_{i} = X_{1} \times \cdots \times X_{n} \] is the topology generated by the basis \[ \mathcal{B}_{\prod_{i=1}^{n} X_{i}} = \big\{U_{1} \times U_{2} \times \cdots \times U_{n} \mid \; U_{i} \in \mathcal{T}_{i}, \; \text{ for all } i=1, \ldots, n\big\} \]

Simple as that. This definition is what you should want it to be, more or less. It is natural to expect that products of sets that are open in each coordinate should be open in the product. That alone does not give you a topology, it turns out, but it does give you a basis. So you generate a topology from it, and the result is the product topology. As you would expect, bases play nicely with this definition as well, as shown by the following easy proposition.

Proposition 2.2. Let $(X, \mathcal{T}_1)$ and $(Y, \mathcal{T}_2)$ be topological spaces, and let $\mathcal{B}_{X}$ and $\mathcal{B}_{Y}$ be bases on $X$ and $Y$ that generate $\mathcal{T}_1$ and $\mathcal{T}_2$, respectively. Then \[ \mathcal{B}=\left\{U \times V: U \in \mathcal{B}_{X}, V \in \mathcal{B}_{Y}\right\} \] is a basis for the product topology on $X \times Y$

Proof. $\mathcal{B}_{X} \subseteq \mathcal{T}$ and $\mathcal{B}_{Y} \subseteq \mathcal{U},$ and so every element of $\mathcal{B}$ is open in the product topology.
Now fix an open set $U$ in the product topology, and some point $(x, y) \in U .$ We need to find an element $B \in \mathcal{B}$ such that $x \in B \subseteq U .$ By definition of the product topology, there must be some $U_{X} \in \mathcal{T}$ and $U_{Y} \in \mathcal{U}$ such that $(x, y) \in U_{X} \times U_{Y} \subseteq U .$ Using the fact that $\mathcal{B}_{X}$ and $\mathcal{B}_{Y}$ are bases, find sets $B_{X} \in \mathcal{B}_{X}$ and $B_{Y} \in \mathcal{B}_{Y}$ such that $x \in B_{X} \subseteq U_{X}$ and $y \in B_{Y} \subseteq U_{Y} .$ But then we have: \[ (x, y) \in B_{X} \times B_{Y} \subseteq U_{X} \times U_{Y} \subseteq U \] so $B=B_{X} \times B_{Y}$ is the set we were looking for. Of course, this fact generalizes to larger finite products and the proof is similarly straightforward.

Barycentre de deux points

Définition 1: On appelle barycentre des points \(A\) et \(B\) ( ou \(A\) et \(B\) deux points du plan ou de I'espace ) affectés respectivement des coefficients \(\alpha, \beta\) ( ou \(\alpha, \beta\) sont des réels tels que \(\alpha+\beta \neq 0\) l'unique point \(G\) tel que \(\alpha \overrightarrow{G A}+\beta \overrightarrow{G B}=\overrightarrow{0}\)

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