---
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]