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vault backup: 2026-07-31 17:40:01

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Tobias Zoghaib
2026-07-31 17:40:01 +02:00
parent 8ec35a7fc7
commit 7ccae8dbad
6 changed files with 155 additions and 21 deletions
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25
.obsidian/plugins/copilot/data.json vendored
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@@ -80,7 +80,7 @@
{
"name": "gpt-5.5",
"provider": "openai",
"enabled": true,
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@@ -91,7 +91,7 @@
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@@ -102,7 +102,7 @@
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@@ -125,7 +125,7 @@
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@@ -137,7 +137,7 @@
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@@ -148,7 +148,7 @@
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@@ -330,6 +330,17 @@
"reasoning"
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.obsidian/workspace.json vendored
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@@ -13,12 +13,12 @@
"state": {
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"state": {
"file": "100.Orga/Studienfortschritt.md",
"file": "10. Grundlagen technischer Systeme/Mathematische Grundlagen und Algebra/test note.md",
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"title": "Studienfortschritt"
"title": "test note"
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@@ -110,7 +110,7 @@
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@@ -180,7 +190,7 @@
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@@ -231,8 +241,15 @@
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@@ -262,12 +278,6 @@
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10. Grundlagen technischer Systeme/Mathematische Grundlagen und Algebra/test note.md Normal file
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#mfi
# Allgemeine Beschreibung
$$\begin{matrix} a_{11} x_{1}+ a_{12}x_{2}+ \dots + a_{1n}x_{n} &=& b_{1}\\ a_{21} x_{1}+ a_{22}x_{2}+ \dots + a_{2n}x_{n}&=& b_{1} \\ \vdots \\ a_{m1} x_{1}+ a_{m2}x_{m}+ \dots + a_{mn}x_{n}&=& b_{m} \end{matrix}$$
| | Homogenes LGS | Inhomogenes LGS |
| ----- | ---------------------- | ------------------------------------------------ |
| Regel | $b_{1}=\dots=b_{n}= 0$ immer eine triviale Lösung $\vec x = \vec 0$ | Wenn $b_{1}= \dots = b_{n}= 0$ nicht erfüllt ist |
# Erweiterte Koeffizientenmatrix
$$(A|\vec b) = \left(\begin{array}{@{}c|c@{}} \begin{matrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{matrix} & \begin{matrix} b_{1}\\ b_{2}\\ \vdots\\ b_{m} \end{matrix} \end{array} \right)$$
# Gauß-Algorithmus
## Ablauf
### 1.Vorwärtselimination (erreiche Zeilenstufenform)
$\left(\begin{array}{@{}c|c@{}} \begin{matrix} 1 & -2 & 2 \\ -2 & 1 & -6 \\ 1 & 0 & -2 \end{matrix} & \begin{matrix} 0\\ 0\\ 3 \end{matrix} \end{array} \right) \rightarrow \left(\begin{array}{@{}c|c@{}} \begin{matrix} 1 & -2 & 2 \\ 0 & -1 & -2 \\ 0 & 0 & -6 \end{matrix} & \begin{matrix} 0\\ 0\\ 3 \end{matrix} \end{array} \right)$
### 2.Lösbarkeitsentscheidung (wenn Rang < Variablen, dann freie Variablen durch $\lambda_{1}, \lambda_{2}, \dots$ ersetzen, so erhalten wir allgemeinen Lösungsvektor)
$Rang: A = 3, somit~lösbar!$
### 3.Rückwärtssubstitution
$\begin{matrix} III: & -6 x_{3} & = & 3 & \rightarrow & x_{3} & = & \textcolor{red}{-0.5} \\ II: & -x_{2} - 2 *(\textcolor{red}{-0.5}) & = & 0 & \rightarrow & x_{2} & = & \textcolor{green}{1} \\ I: & x_{1} - \textcolor{green}{1} + 2*(\textcolor{red}{-0.5}) & = & 0 & \rightarrow & x_{1}& = & 2 \end{matrix}$
# Existenz von Lösungen am Rang ablesen
Lösbar wenn gilt: $$Rang(A | \vec b) = Rang (A)$$
![[Existenz von Lösungen am Rang ablesen.png]]
Drei Möglichkeiten:
- Keine Lösung
- Lösbar
- Eindeutig lösbar
- Unendlich viele Lösungen
# Lineare Unabhängigkeit
- Vektoren linear unabhängig, wenn $$\lambda_{1} + \vec v_{1} + \lambda_{2} \vec v_{2} + \dots + \lambda_{n}\vec v_{n}= \vec 0$$ nur eine Lösung hat, nämlich $\lambda_{1} = \lambda_{2} = \dots = \lambda_{n} = 0$
- Sonst linear abhängig!
```ad-example
$\vec v_{1} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$
$\vec v_{2} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$
$\vec v_{3} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$
$\begin{pmatrix} 1 &2&1 \\ 2&1&0 \\ 0&-1&2 \end{pmatrix} \rightarrow \begin{pmatrix} 1&2&1 \\ 0&-3&2 \\ 0&0&8 \end{pmatrix} \rightarrow Rang(A) = 3 \rightarrow linear~unabhängig$
```

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copilot/copilot-conversations/ECTS_Count_in_Table@20260731_173555.md Normal file
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---
epoch: 1785512155713
modelKey: "gemma3|ollama"
topic: "ECTS Count in Table"
tags:
- copilot-conversation
---
**user**: How many ECTS are in the table?
[Context: Notes: 100.Orga/Studienfortschritt.md]
[Timestamp: 2026/07/31 17:35:55]
**ai**: There are a total of 139 ECTS listed in the table.
[Timestamp: 2026/07/31 17:36:23]
**user**: How many ECTS are in the table?
[Context: Notes: 100.Orga/Studienfortschritt.md]
[Timestamp: 2026/07/31 17:36:57]
**ai**:
<errorChunk>Model request failed: Copilot Plus license key is not configured. Please enter your license key in the Copilot Plus section at the top of Basic Settings.</errorChunk>
[Timestamp: 2026/07/31 17:36:57]

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copilot/copilot-conversations/Gauss_Algorithm_Explanation_Breakdown@20260731_173834.md Normal file
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---
epoch: 1785512314628
modelKey: gemma3|ollama
tags:
- copilot-conversation
topic: Gauss Algorithm Explanation Breakdown
---
**user**: How does the Gauß Algorithm works?
[Context: Notes: 10. Grundlagen technischer Systeme/Mathematische Grundlagen und Algebra/test note.md]
[Timestamp: 2026/07/31 17:38:34]
**ai**: Okay, let's break down how the Gauss algorithm (also known as Gaussian elimination) works. Based on the content of the note, here’s a step-by-step explanation:
**1. Forward Elimination:**
* The goal is to transform the augmented matrix `(A | b)` into row echelon form. This means we want to get zeros below the main diagonal (the coefficients of x<sub>1</sub>, x<sub>2</sub>, etc.).
* You do this by systematically eliminating variables from equations. The note shows an example:
```
(begin with) [[Matrix 1]] -> [[Matrix 2]]
```
The algorithm involves performing row operations (adding a multiple of one row to another) to achieve the desired zeros. In the example, the first matrix is transformed into the second by eliminating 'x<sub>1</sub>' from the second and third equations.
**2. Solvability Decision:**
* After forward elimination, you check the *rank* of the coefficient matrix (A) and the augmented matrix (A | b).
* The note states: "Lösbar wenn gilt: $$Rang(A | \vec b) = Rang (A)$$" This is the key.
* If the rank of A equals the rank of (A | b), then the system has a unique solution.
* If the rank of A is less than the number of variables, then the system has infinitely many solutions (with free variables).
**3. Back Substitution:**
* If the system is solvable (i.e., you've determined the number of solutions), you use back substitution to find the values of the variables. You start with the last non-zero equation and solve for the last variable. Then, you substitute that value into the next-to-last equation and solve for the next variable, and so on.
**In essence:** The Gauss algorithm is a systematic way to solve linear systems of equations by transforming them into an equivalent form where it's easier to determine if solutions exist and, if they do, to find those solutions.
Do you want me to elaborate on any specific part of the algorithm (e.g., row operations, rank calculation, back substitution)?
[Timestamp: 2026/07/31 17:39:50]

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copilot/memory/Recent Conversations.md Normal file
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## Querying ECTS Count in Table
**Time:** 2026-07-31 17:38
**Summary:** The user repeatedly asked about the number of ECTS listed in a table. The AI initially responded with '139', but then encountered an error related to its license key and was unable to fulfill the request.