From e94aff9ca5d9b37e66c537944cc74a361a5075c8 Mon Sep 17 00:00:00 2001 From: Jan Meyer Date: Sun, 31 May 2026 17:48:53 +0200 Subject: [PATCH] vault backup: 2026-05-31 17:48:53 --- 11 Homework/Mathe II/Homework 7.pdf | 1575 +++++++++++++++++++++++++++ 11 Homework/Mathe II/Homework 7.typ | 52 + 40 Extras/OOP/die_einen_da | 2 +- 3 files changed, 1628 insertions(+), 1 deletion(-) create mode 100644 11 Homework/Mathe II/Homework 7.pdf create mode 100644 11 Homework/Mathe II/Homework 7.typ diff --git a/11 Homework/Mathe II/Homework 7.pdf b/11 Homework/Mathe II/Homework 7.pdf new file mode 100644 index 0000000..ac40712 --- /dev/null +++ b/11 Homework/Mathe II/Homework 7.pdf @@ -0,0 +1,1575 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 2 + /Kids [100 0 R 102 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 6 0 R + /Count 2 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 6 0 R + /First 4 0 R + /Last 5 0 R + /Count -2 + /Title (Problem 1) + /Dest 96 0 R +>> 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Für die positive Definitheit prüfen wir die Hauptminoren nach Sylvester: +- $D_1 = 1 > 0$ +- $D_2 = det mat(1, 0; 0, 1) = 1 > 0$ +- $D_3 = 1 dot (5 - 4) = 1 > 0$ +- $D_4 = 3 dot D_3 = 3 > 0$ +Da alle Hauptminoren positiv sind, ist $A$ positiv definit und definiert ein Skalarprodukt. + +== (b) +Ansatz für die Projektion $p$: $p = X c$ mit $X = (u_1, u_2)$ und $c = vec(c_1, c_2)$. +Normalengleichung: $X^T A X c = X^T A v$. + +Berechnung der Gramschen Matrix $X^T A X$: +$ X^T A X = mat(u_1^T A u_1, u_1^T A u_2; u_2^T A u_1, u_2^T A u_2) = mat(4, 1; 1, 2) $ + +Berechnung der rechten Seite $X^T A v$: +$ A v = mat(1, 0, 0, 0; 0, 1, -2, 0; 0, -2, 5, 0; 0, 0, 0, 3) mat(2; 2; 2; 2) = mat(2; -2; 6; 6) $ +$ X^T A v = mat(u_1^T A v; u_2^T A v) = mat(8; 0) $ + +Lösen des LGS: +$ mat(4, 1; 1, 2) mat(c_1; c_2) = mat(8; 0) $ +Aus der 2. Zeile folgt $c_1 = -2 c_2$. Eingesetzt in die 1. Zeile ergibt $4(-2 c_2) + c_2 = 8 <=> -7 c_2 = 8 <=> c_2 = -8/7$. Damit ist $c_1 = 16/7$. + +Einsetzen in $p$: +$ p = c_1 u_1 + c_2 u_2 = 16/7 mat(1; 0; 0; 1) - 8/7 mat(1; 1; 0; 0) = mat(8/7; -8/7; 0; 16/7) $ diff --git a/40 Extras/OOP/die_einen_da b/40 Extras/OOP/die_einen_da index 3af3ede..d4f69c4 160000 --- a/40 Extras/OOP/die_einen_da +++ b/40 Extras/OOP/die_einen_da @@ -1 +1 @@ -Subproject commit 3af3edea6ffcfbe6aa68892523a07765d8538fab +Subproject commit d4f69c43a38f660df6aca7ad6976b4edef95c724