From 80dce2009ef6bd00c2cb984cf8d60950c69e8c13 Mon Sep 17 00:00:00 2001 From: alexander Date: Fri, 20 Feb 2026 22:59:11 +0100 Subject: [PATCH] =?UTF-8?q?Started=20Adding=20Gau=C3=9F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- src/cheatsheets/LinearAlgebra.typ | 58 ++++++++++++++++++++++++++++-- src/images/linAlg/Gauss1.jpg | Bin 0 -> 77983 bytes src/images/linAlg/Gauss1a.jpg | Bin 0 -> 60011 bytes src/images/linAlg/Gauss2.jpg | Bin 0 -> 57319 bytes src/images/linAlg/Gauss3.jpg | Bin 0 -> 76361 bytes 5 files changed, 56 insertions(+), 2 deletions(-) create mode 100644 src/images/linAlg/Gauss1.jpg create mode 100644 src/images/linAlg/Gauss1a.jpg create mode 100644 src/images/linAlg/Gauss2.jpg create mode 100644 src/images/linAlg/Gauss3.jpg diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ index 8a063da..73d44f9 100644 --- a/src/cheatsheets/LinearAlgebra.typ +++ b/src/cheatsheets/LinearAlgebra.typ @@ -228,6 +228,8 @@ #SeperatorLine + Normalweiße alle Abbildung/Matrizen in Kannoischer Basis $hat(e)_1 = vec(1, 0, dots.v), hat(e)_2 = vec(0, 1, dots.v), ...$ + ] #colbreak() @@ -295,6 +297,7 @@ *Rang* $op("Rang") f := dim op("Bild") f$ + - Bei Matrizen: \ $Rang(f) <= min(n, m) equiv min("#Spalten", "#Zeilen")$ *Nullraum/Kern:* \ $kern(Phi) := {v in V | Phi(v) = 0}$ @@ -307,14 +310,65 @@ #linebreak() $"Wenn" dim(V) = dim(Bild(A)) "oder" dim(kern(A)) = 0 \ <=> A "bijektiv" <=> "invertierbar"$ + + #SeperatorLine + + - Homogense Lineares Gleichungsystem: $A ve(x) = ve(0) $ Lösungsmenge: $LL = kern(A)$, immer: $ve(0) in L$ \ + + - In-Homogense LGS: $A ve(x) = ve(b) $ + + #SeperatorLine + + *Gaußalgorithmus* + + #grid(columns: (auto, 1fr), + row-gutter: 1mm, + column-gutter: 2mm, + + image("../images/linAlg/Gauss1a.jpg", width: 2cm), + [ + Gleichungssystem: $A ve(x) = b$ \ + $A in KK^(m times n)$ + In Zeilenstufenform Bringen, Operationen: + - Zeile $dot lambda$ ($x in CC$ mit $dot i$) + - Zeile vertauschen + - Zeile $+$ Zeile + ], + ) + + + + #grid(columns: (auto, 1fr), + row-gutter: 1mm, + column-gutter: 2mm, + image("../images/linAlg/Gauss2.jpg", width: 2cm), + [ + *Eindeutige Lösung* $-->$ Normale Rückeinsetzung + + $n equiv "#Spalten" equiv dim ve(x)$ + + $Rang(A) = n$ + ], + + image("../images/linAlg/Gauss1.jpg", width: 2cm), + [ + *Nullzeile*: + Pro Nullzeile eine frei Var $t, s, ...$ + + ], + + image("../images/linAlg/Gauss3.jpg", width: 2cm), + [ + *Wiederspruch*: Keine Lösung + ] + ) ] - #colbreak() // Matrix Basics #bgBlock(fill: colorMatrix)[ #subHeading(fill: colorMatrix)[Matrix Basics] - Matrix $equiv$ Linera Abbildung \ + Linera Abbildung $equiv$ EINER eindeutige Matrix \ - Sclar/Matrix: $lambda dot A$ - Matrix/Matrix: $A + B$ diff --git a/src/images/linAlg/Gauss1.jpg b/src/images/linAlg/Gauss1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..170aea7b2ec402d0bad06c9cf1e19dba962d8a25 GIT binary patch literal 77983 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