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11. Vl Ana1LinA Basis und Dimension $3.21   Add to cart

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11. Vl Ana1LinA Basis und Dimension

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Dies ist eine vollständige Vorlesungsmitschrift zur 11. Vorlesung des Moduls Analysis I und lineare Algebra für Ingenieurswissenschaften. In dieser Vorlesung werden Basis und Dimension von Vektorräumen, lineare Unabhängigkeit und Koordinatenvektoren thematisiert.

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  • January 30, 2023
  • 6
  • 2022/2023
  • Class notes
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Bsp III bilden ein Erzeugendensystem der IR


5 II 0 1 E 2E 2E 3E to E
4E 5E 2 E
Definition lineare Unabhängigkeit
Abhängigkeit

Gegeben seieine endliche Menge von Vektoren E E
linear knabhängig wenn die Gleichung
i E
A Ft tara ö
erfülltist für Aia jo
linear Abhängig wenn sie nich linearAbhängigsind
i E
Dh es gibt koeffizienten An Ja sodass

A Ft tara Ö
undmindestens ein AjtÖ

Bsp 1 E II ist linear abhängig denn

e E e E HE
2 I Ti sindlinear unabhängig denn


EI EI 7 7 0
E E E 72 0
t.AE 0


3 II E sindlinear abhängig

4 1 I M I

, 4 ü ist linear unabhängig
für E Ö
Ö ist linear abhängig denn 15 ö ö



5 hm Vektorraum
der Polynome sin polz 1 plz Z P C z

Lin unabhängig


Jop Xp KPa Gypolynom Nulvektor
derPolynome
muss für alle
ZEE gelten
2 B Zoo t

A OA A linear unabhängig



6 generell gilt 1z z z sind linear unabhängig
I Az 7,7 Anz O Nallpolynom

O O z 0 zit z

A O A AE An Koeffizientenvergleich


I Definition Basis

Sei Ve Ö ein IK Vektorraum


Eine Vektoren
menge E heißtBasis von Vwenngilt
i dieVektoren GE E sindein Erzeugendensystem von V d h

kspa.EE ü

i die Vektoren
E sindlinear unabhängig
D h eine Basis isteinlinearunabhängiges Erzeugendensystem

für k Ö ist die Leere Menge eine Basis
Sprachlich Basis Singular
Basen Plural

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