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16. Vl Ana1LinA Koordinaten und Matrixdarstellung $3.35   Add to cart

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16. Vl Ana1LinA Koordinaten und Matrixdarstellung

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Dies ist eine vollständige Vorlesungsmitschrift zur 16. Vorlesung des Moduls Analysis I und lineare Algebra für Ingenieurswissenschaften. In dieser Vorlesung werden die Matrixdarstellung linearer Abbildungen, die Koordinatendarstellung und der Basiswechsel von Matrizen thematisiert.

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  • January 30, 2023
  • 6
  • 2022/2023
  • Class notes
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Se Vein u dimensionalen Ik Vektorraum und B 5,5 bi eine Basis
JederVektor EE V hateine eindeutige darstellung EnEtabit anti
E k heißtkoordinatenrektordesVektors V bezüglich
E G derBasis B


Definition Koordinatenabbildung


Die Abbildungkg V K E
EINE KEKE
Bsp 1 K KEI tanzt azzzlao.an.az E 4
B 2
1 z z plz 3 57 7
knipst Ed

Sei V ein u dimensionalen Ik Vektorraum und B b bi eineBasis von V

Dann ist die Koordinatenabbildung kg V 1k linear und bijektiv
Die Inverse kj 1k V ist
kill Gbitacht taub
Beweis

Seien T.EE V E abi taz 5 t anti
E ß 5 tßbt Babi

VIE HatB 5 hatB b t lantß bi

hat KEI
Etütfüßf
E abi ta 5 t anbi.AE
XaEtJaEt Aanrn AE etwa
I Efta

, E abitanzt t anti
kj kda kj E antitakt anti F kjokg

id knlkjlfp
f
kglaiita.br
also ist hi
anti
die Umkehrabbildung von
Kiki
Krs
ich




Basiswechsel im Vektorraum
V
Seien zum Vektorraum Vene Basen gegeben B 5 _In B 5 bi
Symmetrieachse




EEIG.IE 3 koki i i a
es
gibt eine nx n Matrix F TBB
Eiko mitkg ok TBB F

Tage ist Transformationsmatrix diedieKoordinaten bzgl B in die
Koordinaten
bzgl B transformiert
Tg TBB T


Was steht in Tage drin
Exemplarisch 1 Spalte Uni kj e kg kj Y kg 115 07 tobi

kg b koordinaten des 1 Vektors
der Basis B in der
Basis B

7 Esseien B 5 bi und B bi bi zweiBasen von V

Die te Spalte der MatrixTag
j
enthalte
Kpf Dann
gilt
KBlr
I KB.li
IEEE
TBBi TBg Es

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