From 651843dbef3b6065f21c02469aa62e43b4d5f02a Mon Sep 17 00:00:00 2001 From: Jan Meyer Date: Sat, 25 Apr 2026 20:57:48 +0200 Subject: [PATCH] vault backup: 2026-04-25 20:57:48 --- .obsidian/community-plugins.json | 15 - .../plugins/obsidian-file-color/data.json | 2 +- 11 Homework/Mathe II/Homework 2.pdf | 1615 +++++++++++++++++ 11 Homework/Mathe II/Homework 2.typ | 74 + 30 Library/Hausaufgabe02.pdf | Bin 0 -> 268218 bytes Untitled.base | 3 + 6 files changed, 1693 insertions(+), 16 deletions(-) create mode 100644 11 Homework/Mathe II/Homework 2.pdf create mode 100644 11 Homework/Mathe II/Homework 2.typ create mode 100644 30 Library/Hausaufgabe02.pdf create mode 100644 Untitled.base diff --git a/.obsidian/community-plugins.json b/.obsidian/community-plugins.json index a37479b..e69de29 100644 --- a/.obsidian/community-plugins.json +++ b/.obsidian/community-plugins.json @@ -1,15 +0,0 @@ -[ - "obsidian-minimal-settings", - "edit-in-neovim", - "obsidian-git", - "better-export-pdf", - "templater-obsidian", - "highlightr-plugin", - "obsidian-file-color", - "darlal-switcher-plus", - "dataview", - "obsidian-vimrc-support", - "emoji-shortcodes", - "typst-mate", - "inline-math" -] \ No newline at end of file diff --git a/.obsidian/plugins/obsidian-file-color/data.json b/.obsidian/plugins/obsidian-file-color/data.json index aeb5ea2..587ba26 100644 --- a/.obsidian/plugins/obsidian-file-color/data.json +++ b/.obsidian/plugins/obsidian-file-color/data.json @@ -1,5 +1,5 @@ { - "cascadeColors": true, + "cascadeColors": false, "colorBackground": false, "palette": [], "fileColors": [] diff --git a/11 Homework/Mathe II/Homework 2.pdf b/11 Homework/Mathe II/Homework 2.pdf new file mode 100644 index 0000000..202f92c --- /dev/null +++ b/11 Homework/Mathe II/Homework 2.pdf @@ -0,0 +1,1615 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 2 + /Kids [103 0 R 105 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 6 0 R + /Count 2 +>> +endobj + +3 0 obj +<< + 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+--- + +=== (b) $f(x) = x^6 / (x^4 + 3x^2 + 2)$ + +Polynomdivision: +$ x^6 / (x^4 + 3x^2 + 2) = x^2 - 3 + (7x^2 + 6) / ((x^2 + 1)(x^2 + 2)) $ + +Partialbruchzerlegung des Restes: +$ (7x^2 + 6) / ((x^2 + 1)(x^2 + 2)) = 8 / (x^2 + 2) - 1 / (x^2 + 1) $ + +Stammfunktion: +$ F(x) = 1/3 x^3 - 3x - arctan(x) + 4 sqrt(2) arctan(x / sqrt(2)) + C $ + +#pagebreak() +== Problem 3: Eigenwerte + +=== (a) Berechnung der Eigenwerte + +#let id = "I" +#let det = "det" + +==== (i) $A_1 = mat(3, -1, 0; 3, -1, 0; 4, -1, -2)$ +Charakteristisches Polynom (Entwicklung nach der 3. Spalte): +$ p(lambda) = (-2-lambda) det mat(3-lambda, -1; 3, -1-lambda) $ +$ p(lambda) = (-2-lambda) (lambda^2 - 2lambda) = -lambda(lambda - 2)(lambda + 2) $ +*Eigenwerte:* $lambda_1 = 0, lambda_2 = 2, lambda_3 = -2$ + +==== (ii) $A_2 = mat(1, 0, -1; 1, 0, 2; 1, 0, 1)$ +Charakteristisches Polynom (Entwicklung nach der 2. Spalte): +$ p(lambda) = (-lambda) det mat(1-lambda, -1; 1, 1-lambda) $ +$ p(lambda) = -lambda (lambda^2 - 2lambda + 2) $ +Nullstellen via $p q$-Formel: $lambda = 1 plus.minus sqrt(-1)$ +*Eigenwerte:* $lambda_1 = 0, lambda_2 = 1 + i, lambda_3 = 1 - i$ + +--- + +=== (b) Beweis: Eigenwerte von $A B$ und $B A$ + +Sei $lambda != 0$ ein Eigenwert von $A B$. Dann existiert ein Eigenvektor $v != 0$, sodass: +$ (A B) v = lambda v $ + +Multiplikation von links mit $B$: +$ B(A B v) = B(lambda v) $ +$ (B A)(B v) = lambda (B v) $ + +Da $lambda != 0$ und $v != 0$, ist $B v != 0$. +Somit ist $B v$ ein Eigenvektor von $B A$ zum Eigenwert $lambda$. $square$ diff --git a/30 Library/Hausaufgabe02.pdf b/30 Library/Hausaufgabe02.pdf new file mode 100644 index 0000000000000000000000000000000000000000..220eb767389d23f99101d10d908039faff9f6168 GIT binary patch literal 268218 zcmcG#V~{9Kvo1Qey~eg}+qP}no;9{@+n!lt+qP}rwRPg$^SwLH{&8rMJH-u z?QG(RPbX?^;A|pnVq|A*0>#S<<>c&WVqgR1zCNU>Wp}`e<~vh++cUOWPpL~%LS83v zl2SlZlHCMqS^_UwwQ6b2K{SzW==FmLz$hj;U)i5{+)SJ36_i^fIJzv&_3ynBc=@<%O>KJMj7 zd+xI4jiAVS{dM6jQCrJugtpXpcaypkuqzxC<19{B;iANf1@HSHP}ET3fcX{2_F$OK zrmnHQo*Y=NZpnI1?Wa_0yMHGvfx}b#wHwyeLsCE$_VCe=p(^q<$zzk>Lsju~7=6@E zVafI;bSd6TmC{OVYA_}X5*xzI)1o!nx_iBDC|!6W5<+G^EXM)5x71KKu+e!JJURz& z--_MsalPA`HT6*GxHqyheSri8l+bl=9(i9x)xLs<{+t*j-*FAt3rZ 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