9.5. Endliche Vektorräume


Die beiden Eigenschaften "linear unabhängig" und "maximal" stehen in keinerlei Beziehung zu einander. So ist z.B. in 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3845@ :

Wir interessieren uns nun für Sequenzen, die beide Eigenschaften gleichzeitig aufweisen. Sie stellen ein grundlegendes Konzept der Linearen Algebra dar.
 
Definition:  Es sei V ein Vektorraum.

Eine Sequenz v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ von Vektoren aus V heißt eine (endliche) Basis von V, falls sie linear unabhängig und maximal ist.

Der Vektorraum V heißt endlich, falls V eine (endliche) Basis besitzt.
 

Beachte:

Durch die beiden vorangegangenen Abschnitte liegen bereits viele Beispiele und Ergebnisse zu Basen vor. Wir tragen zunächst die wichtigsten zusammen.
 
Beispiel:
  1. e 1 ,, e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaaaaa@3C52@ ist eine Basis des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ , die Standardbasis (oder: kanonische Basis) des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ . Insbesondere also ist n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ ein endlicher Vektorraum.
     
  2. 1,X,, X n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacYcacaWGybGaaiilaiablAciljaacYcacaWGybWaaWbaaSqabeaacaWGUbaaaaaa@3CB3@ ist eine Basis, die Standardbasis des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaaaaa@41BE@ . Also ist n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaaaaa@41BE@ ein endlicher Vektorraum.
     
  3. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecuaaa@409E@ ist kein endlicher Vektorraum, denn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecuaaa@409E@ lässt keine maximalen Sequenzen zu, also auch keine Basen.
     
  4. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyybIymaaa@3765@ ist Basis von    VV={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlaadAfacaaMf8Uaeyi1HSTaaGzbVlaadAfacqGH9aqpcaGG7bGaaGimaiaac2haaaa@4267@ . Der Nullraum ist also ein endlicher Vektorraum mit MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyybIymaaa@3765@ als einziger Basis.

Die Rolle der leeren Sequenz, bzw. des Nullraums ist in 4. geklärt. Im Weiteren sei daher stets V{0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiabgcMi5kaacUhacaaIWaGaaiyFaaaa@3B48@ vorausgesetzt. Direkt übertragen lassen sich auch die folgenden Eigenschaften:

 
Bemerkung:  Es sei V ein Vektorraum, v 1 ,, v k ,xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4bGaeyicI4SaamOvaaaa@4087@ , α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyiyIKRaaGimaaaa@3A0C@ , dann gilt:
  1. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ Basis von V v 1 ,,α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqycaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4829@ Basis von V.
  2. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ Basis von V v 1 ,, v i ,, v j + v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@4E2D@ Basis von V.
  3. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ Basis von V v 1 ,, v i ,, v j +α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@4FCC@ Basis von V.

Mit dieser Bemerkung lassen sich also, wie bei der Maximalität und der linearen Unabhängigkeit auch, Basisnachweise durch Zurückführen auf Referenzsequenzen durchführen. So ist z.B. X 2 , X 2 +X,X+3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiaacYcacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiwaiaacYcacaWGybGaey4kaSIaaG4maaaa@3F27@ auf Grund der folgenden Äquivalenzen eine Basis von 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaaikdaaaaaaa@4187@ :

X 2 , X 2 +X,X+3  ist Basis von  2     (Subtrahiere den ersten Vektor vom zweiten.) X 2 ,X,X+3  ist eine Basis von  2     (Subtrahiere vom dritten Vektor den zweiten.) X 2 ,X,3  ist eine Basis von  2     (Dividiere den dritten Vektor durch 3.) X 2 ,X,1  ist eine Basis von  2     (Ändere die Reihenfolge der Vektoren.) 1,X, X 2   ist eine Basis von  2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuWaaaaabaaabaGaamiwamaaCaaaleqabaGaaGOmaaaakiaacYcacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiwaiaacYcacaWGybGaey4kaSIaaG4maiaabMgacaqGZbGaaeiDaiaabccacaqGcbGaaeyyaiaabohacaqGPbGaae4CaiaabccacaqG2bGaae4Baiaab6gatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=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LriqnaaCaaaleqabaGaaGOmaaaaaOqaaiaabsmacaqGUbGaaeizaiaabwgacaqGYbGaaeyzaiaabccacaqGKbGaaeyAaiaabwgacaqGGaGaaeOuaiaabwgacaqGPbGaaeiAaiaabwgacaqGUbGaaeOzaiaab+gacaqGSbGaae4zaiaabwgacaqGGaGaaeizaiaabwgacaqGYbGaaeiiaiaabAfacaqGLbGaae4AaiaabshacaqGVbGaaeOCaiaabwgacaqGUbGaaeOlaaqaaiabgsDiBlaaywW7aeaacaaIXaGaaiilaiaadIfacaGGSaGaamiwamaaCaaaleqabaGaaGOmaaaakiaabMgacaqGZbGaaeiDaiaabccacaqGLbGaaeyAaiaab6gacaqGLbGaaeiiaiaabkeacaqGHbGaae4CaiaabMgacaqGZbGaaeiiaiaabAhacaqGVbGaaeOBaiab=LriqnaaCaaaleqabaGaaGOmaaaaaOqaaaaaaaa@4E35@

 

In einigen speziellen Situationen kann man bei einem Basistest auf die Überprüfung der Maximalität verzichten:
 
Bemerkung: 
  1. In jedem Vektorraum V gilt:
    v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ ist Basis von < v 1 ,, v k > v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iaaywW7cqGHuhY2caaMf8UaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@4A84@ ist linear unabhängig.
     
  2. In n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ gilt:
    x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3C78@ ist Basis des n x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaOGaaGzbVlabgsDiBlaaywW7caWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaa@448A@ ist linear unabhängig. 
     

Beweis:

Zu 1.:  Es ist nur zu beachten, dass v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ stets maximal in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E87@ ist.

Zu 2.:  Diese Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ " ist klar!

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3845@ ": Wir gehen indirekt vor. Angenommen x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3C78@ ist keine Basis, d.h. gemäß Voraussetzung: x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3C78@ ist nicht maximal, es gibt also ein x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AFD@ , so dass x< x 1 ,, x n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgMGiplabgYda8iaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadIhadaWgaaWcbaGaamOBaaqabaGccqGH+aGpaaa@4111@ . Mit x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3C78@ ist nach einem Ergebnis in Teil 3 dann aber auch x 1 ,, x n ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaakiaacYcacaWG4baaaa@3E2F@ linear unabhängig, d.h. also: es gibt in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ eine linear unabhängige Sequenz der Länge n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaigdaaaa@387C@ ! - Widerspruch.

 

Auch die folgende Bemerkung ist i.w. eine Übertragung bereits gesicherter Erkenntnisse. Mit ihr lassen sich in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ sehr leicht "zu lange" und "zu kurze" Sequenzen als Basen ausschließen. Die eigentliche Bedeutung dieser Aussage liegt aber in dem genannten Zusatz; er bereitet nämlich bereits die Einführung des Dimensionsbegriffs vor.
 
Bemerkung:  Für alle x 1 ,, x k n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@4093@ gilt:
x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ Basis des n k=n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaOGaaGzbVlabgsDiBlaaywW7caWGRbGaeyypa0JaamOBaaaa@40E7@ .

Alle Basen des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ haben somit n Elemente, insbesondere also stets gleich viele!

Beweis:  Da x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ sowohl linear unabhängig als auch maximal in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ ist, hat man kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgsMiJkaad6gaaaa@3984@ und kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgwMiZkaad6gaaaa@3995@ , also k=n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaad6gaaaa@38D5@ .

 

Die besondere Bedeutung die den Basen eines Vektorraums zukommt liegt u.a. an folgendem Verhalten:
 
Bemerkung und Bezeichnung:  Es sei v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ eine Basis von V.
Dann gibt es zu jedem xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3948@ genau einen Vektor α n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3B9F@ , so dass x= α 1 v 1 ++ α n v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWG2bWaaSbaaSqaaiaad6gaaeqaaaaa@4433@ .

α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@378B@ heißt der zu x gehörige Koordinatenvektor bzgl. der Basis v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ .

Beweis:

xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3948@ gegeben. Da v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ maximal ist, gibt es wenigstens ein α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@378B@ der genannten Art. Angenommen x besitzt eine zweite Darstellung x= α 1 v 1 ++ α n v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iqbeg7aHzaafaWaaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiqbeg7aHzaafaWaaSbaaSqaaiaad6gaaeqaaOGaamODamaaBaaaleaacaWGUbaabeaaaaa@444B@ , mit einem von α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@378B@ verschiedenen Vektor α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqySdeMbauaaaaa@3797@ . Insbesondere ist dann
 

α 1 v 1 ++ α n v n = α 1 v 1 ++ α n v n ( α 1 α 1 ) v 1 ++( α n α n ) v n =0 α 1 α 1 == α n α n =0,  da   v 1 ,, v n   linear unabhängig α= α   - Widerspruch. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AE80@

Wir üben den neuen Begriff an einigen Beispielen. Man beachte dabei, dass ein α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@378B@ bereits dann der Koordinatenvektor zu x bzgl. v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ ist, wenn die Gleichung x= α 1 v 1 ++ α n v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWG2bWaaSbaaSqaaiaad6gaaeqaaaaa@4433@ gültig ist.
 
Beispiel:
  1. ( 1 1 ),( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaaaa@3CB5@ ist eine Basis von 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3845@ .

    ( 4 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGinaaqaaiabgkHiTiaaiodaaaaacaGLOaGaayzkaaaaaa@39EA@ ist der Koordinatenvektor von ( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaaaaa@39E6@ , denn ( 1 2 )=4( 1 1 )3( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaGaeyypa0JaaGinamaabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgkHiTiaaiodadaqadaqaauaabeqaceaaaeaacaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaaaaa@436D@ .

    ( 1 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaicdaaaaacaGLOaGaayzkaaaaaa@38F7@ ist der Koordinatenvektor von ( 1 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaaaaa@38F8@ , denn ( 1 1 )=1( 1 1 )+0( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaeyypa0JaaGymamaabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgUcaRiaaicdadaqadaqaauaabeqaceaaaeaacaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaaaaa@426E@ .

  2. 1,X, X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacYcacaWGybGaaiilaiaadIfadaahaaWcbeqaaiaaikdaaaaaaa@3AAA@ ist eine Basis von 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaaikdaaaaaaa@4187@ .

    ( 2 7 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaeyOeI0IaaGOmaaqaaiaaiEdaaeaacaaIZaaaaaGaayjkaiaawMcaaaaa@3AAB@ ist der Koordinatenvektor von 2+7X+3 X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaGOmaiabgUcaRiaaiEdacaWGybGaey4kaSIaaG4maiaadIfadaahaaWcbeqaaiaaikdaaaaaaa@3D7A@ .

    Allgemein gilt für die Basis 1,X,, X n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacYcacaWGybGaaiilaiablAciljaacYcacaWGybWaaWbaaSqabeaacaWGUbaaaaaa@3CB3@ des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaaaaa@41BE@ :

    ( α 0 α n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaeqySde2aaSbaaSqaaiaaicdaaeqaaaGcbaGaeSO7I0eabaGaeqySde2aaSbaaSqaaiaad6gaaeqaaaaaaOGaayjkaiaawMcaaaaa@3EC9@ ist der Koordinatenvektor von α 0 + α 1 X++ α n X n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamiwaiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGUbaabeaakiaadIfadaahaaWcbeqaaiaad6gaaaaaaa@4475@ .

  3. Die Standardbasis des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ , e 1 ,, e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaaaaa@3C52@ , spielt eine Sonderrolle: Jedes x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AFD@ ist sein eigener Koordinatenvektor, denn:
     
    x= x 1 e 1 ++ x n e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGccaWGLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaamiEamaaBaaaleaacaWGUbaabeaakiaadwgadaWgaaWcbaGaamOBaaqabaaaaa@42CD@ .

Wir untersuchen nun das Zusammenspiel zwischen den Vektoren xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3948@ und ihren Koordinatenvektoren α n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3B9F@ genauer: Schreibt man T v 1 ,, v n (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bGaaiykaaaa@3FE7@ statt α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@378B@ , so ist 

T v 1 ,, v n :V n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacQdacaWGwbGaeyOKH4QaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@43A7@

offenbar diejenige Funktion, die jedem x den ihm zukommenden Koordinatenvektor zuweist. Sie heißt die zur Basis v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamODamaaBaaameaacaaIXaaabeaaliaacYcacqWIMaYscaGGSaGaamODamaaBaaameaacaWGUbaabeaaaaa@3C78@ gehörige Koordinatentransformation.

Die Koordinatentransformationen endlicher Vektorräume stellen eine so enge Vebindung zu den Räumen n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ her, dass man jeden endlichen Vektorraum letztendlich als Kopie eines passenden n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ auffassen kann. Insbesondere wird man alle Eigenschaften des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ in entsprechender Form wiederfinden. Die folgende Bemerkung führt diese Überlegungen detailiert aus.
 
Bemerkung:  Es sei V ein endlicher Vektorraum mit v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamODamaaBaaameaacaaIXaaabeaaliaacYcacqWIMaYscaGGSaGaamODamaaBaaameaacaWGUbaabeaaaaa@3C78@ als ausgewählter Basis und T v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaaaaa@3D87@ als zugehöriger Koordinatentransformation.

Dann gilt:

  1. T v 1 ,, v n (0)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaaIWaGaaiykaiabg2da9iaaicdaaaa@4164@ .
  2. T v 1 ,, v n ( v i )= e i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabg2da9iaadwgadaWgaaWcbaGaamyAaaqabaaaaa@4413@ .
  3. T v 1 ,, v n (x+y)= T v 1 ,, v n (x)+ T v 1 ,, v n (y) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGH9aqpcaWGubWaaSbaaSqaaiaadAhadaWgaaadbaGaaGymaaqabaWccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaadbaGaamOBaaqabaaaleqaaOGaaiikaiaadIhacaGGPaGaey4kaSIaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG5bGaaiykaaaa@57A6@ .
  4. T v 1 ,, v n (τx)=τ T v 1 ,, v n (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacqaHepaDcaWG4bGaaiykaiabg2da9iabes8a0jaadsfadaWgaaWcbaGaamODamaaBaaameaacaaIXaaabeaaliaacYcacqWIMaYscaGGSaGaamODamaaBaaameaacaWGUbaabeaaaSqabaGccaGGOaGaamiEaiaacMcaaaa@4E72@ .
  5. T v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaaaaa@3D87@ ist bijektiv.
  6. v< x 1 ,, x k > T v 1 ,, v n (v)< T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6C9A@ .
  7. x 1 ,, x k   linear unabhängig T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k )  linear unabhängig MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7A50@ .
  8. x 1 ,, x k   maximal in  V T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k )  maximal in  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FED@ .
  9. x 1 ,, x k   Basis von  V T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k )  Basis von  n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DF1@ .

Wir führen den recht umfangreichen Beweis auf einer eigenen Seite.

Bei endlichen Vektorräumen ist Anzahl der Basiselemente eine Konstante; sie charaktersiert die Struktur des Raumes:
 
Bemerkung und Bezeichnung:  Es sei V ein endlicher Vektorraum mit v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamODamaaBaaameaacaaIXaaabeaaliaacYcacqWIMaYscaGGSaGaamODamaaBaaameaacaWGUbaabeaaaaa@3C78@ als ausgewählter Basis.

Dann hat jede andere Basis von V ebenfalls n Elemente. Die Zahl n heißt Dimension von V (in Zeichen: n=dimV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iGacsgacaGGPbGaaiyBaiaadAfaaaa@3B88@ ).

Wir setzen zusätzlich: dim{0}=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacMgacaGGTbGaai4EaiaaicdacaGG9bGaeyypa0JaaGimaaaa@3D2E@ .

Beweis:

Ist x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ eine weitere Basis von V, so ist T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaacYcacqWIMaYscaGGSaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa@4E7B@ eine Basis von n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ . Folgt: k=n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaad6gaaaa@38D5@ .

Um also die Dimension eines endlichen Vektorraums zu ermitteln, reicht es eine Basis zu finden und deren Elemente zu zählen!
 
Beispiel:
  1. dim n =n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacMgacaGGTbGaeSyhHe6aaWbaaSqabeaacaWGUbaaaOGaeyypa0JaamOBaaaa@3D47@ .
  2. dim n =n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacMgacaGGTbWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaGccqGH9aqpcaWGUbGaey4kaSIaaGymaaaa@4826@ .
  3. dim< v 1 ,, v k >k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacMgacaGGTbGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabgsMiJkaadUgaaaa@43F4@ .
  4. dim< v 1 ,, v k >=k v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacMgacaGGTbGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iaadUgacaaMf8Uaeyi1HSTaaGzbVlaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4Aaaqabaaaaa@4F42@ linear unabhängig.

Mit Hilfe der Koordinatentransformationen lassen sich nun alle spezifischen Ergebnisse des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@387C@ auf einen beliebigen endlichen Vektorraum übertragen:
 
Bemerkung:  Es sei V ein endlicher Vektorraum der Dimension n.

Dann gilt:

  1. x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ linear unabhängig kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caWGRbGaeyizImQaamOBaaaa@3EFD@ .
  2. x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ maximal kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caWGRbGaeyyzImRaamOBaaaa@3F0E@ .
  3. x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3C78@ Basis x 1 ,, x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG4bWaaSbaaSqaaiaad6gaaeqaaaaa@41F0@ linear unabhängig.

Beweis: Sei v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamODamaaBaaameaacaaIXaaabeaaliaacYcacqWIMaYscaGGSaGaamODamaaBaaameaacaWGUbaabeaaaaa@3C78@ eine Basis und T v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaaaaa@3D87@ die zugehörige Transformation. Dann hat man:

Zu 1.:  T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaacYcacqWIMaYscaGGSaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa@4E7B@ linear unabhängig in n kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaOGaaGzbVlabgkDiElaaywW7caWGRbGaeyizImQaamOBaaaa@4197@ .

Zu 2.:  T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaacYcacqWIMaYscaGGSaGaamivamaaBaaaleaacaWG2bWaaSbaaWqaaiaaigdaaeqaaSGaaiilaiablAciljaacYcacaWG2bWaaSbaaWqaaiaad6gaaeqaaaWcbeaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa@4E7B@ maximal in n kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaOGaaGzbVlabgkDiElaaywW7caWGRbGaeyyzImRaamOBaaaa@41A8@ .

Zu 3.:   x 1 ,, x n   Basis von  V T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x n )  Basis von  n T v 1 ,, v n ( x 1 ),, T v 1 ,, v n ( x n )  linear unabhängig x 1 ,, x n   linear unabhängig. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B7F5@

 


Für theoretische Überlegungen ist es oft notwendig, Basen eines bestimmten Zuschnitts zu haben. So erlaubt es z.B. der folgende Satz, eine beliebige Sequenz linear unabhängiger Vektoren zu einer Basis aufzustocken, wobei sogar eine Vorauswahl der hinzuzufügenden Vektoren möglich ist.
 
Satz (Basisergänzungssatz):  Es sei V ein endlicher Vektorraum und v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ eine vorgewählte Basis von V.

Ist x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ eine linear unabhängige Sequenz von Vektoren aus V, so gibt es Basisvektoren v 1 ,, v nk { v 1 ,, v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmODayaafaWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcaceWG2bGbauaadaWgaaWcbaGaamOBaiabgkHiTiaadUgaaeqaaOGaeyicI4Saai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOBaaqabaGccaGG9baaaa@4889@ derart, dass x 1 ,, x k , v 1 ,, v nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcaceWG2bGbauaadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiqadAhagaqbamaaBaaaleaacaWGUbGaeyOeI0Iaam4Aaaqabaaaaa@45AC@ eine Basis von V ist.

Beweis:
Ist k=n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaad6gaaaa@38D5@ , so ist nichts zu zeigen, denn dann ist x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ bereits eine Basis, die durch Hinzunahme von 0 Vektoren aus { v 1 ,, v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOBaaqabaGccaGG9baaaa@3E7E@ entstanden ist. Sei also k<n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgYda8iaad6gaaaa@38D3@

Wir zeigen zunächst: Es gibt einen Vektor v i { v 1 ,, v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaaiyFaaaa@4221@ , so dass x 1 ,, x k , v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaaaa@3F44@ linear unabhängig ist.

Da k<n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgYda8iaad6gaaaa@38D3@ , ist x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ nicht maximal, es gibt also ein xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3948@ , so dass x< x 1 ,, x k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgMGiplabgYda8iaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadIhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@410E@ ist. Nun ist aber v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ eine Basis; für geeignete α i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaaa@38A5@ hat man also:

x= α 1 v 1 ++ α n v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWG2bWaaSbaaSqaaiaad6gaaeqaaaaa@4433@ .
Wären nun alle Basisvektoren v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ Elemente von < x 1 ,, x k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E8B@ , so hätte man auch:
x= α 1 v 1 ++ α n v n < x 1 ,, x k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaeyicI4SaeyipaWJaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@4E60@ .

Also muss es ein v i { v 1 ,, v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaaiyFaaaa@4221@ geben, so dass v i < x 1 ,, x k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgMGiplabgYda8iaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadIhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4230@ . Damit aber, so haben wir in Teil 3 gezeigt, ist x 1 ,, x k , v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaaaa@3F44@ linear unabhängig.

Nun zum eigentlichen Beweis. Wir betrachten das System aller linear unabhängigen Erweiterungen von x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C75@ mit Elementen aus { v 1 ,, v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOBaaqabaGccaGG9baaaa@3E7E@ :
 

B={ x 1 ,, x k , v 1 ,, v j | v 1 ,, v j { v 1 ,, v n }       x 1 ,, x k , v 1 ,, v j   ist linear unabhängig} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7F70@

B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabgcMi5kabgwGigdaa@39F3@ , denn z.B. ist x 1 ,, x k B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiabgIGiolaadkeaaaa@3ECA@ . Da B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36B3@ endlich ist, können wir in B MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36B3@ eine Sequenz x 1 ,, x k , v 1 ,, v j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcaceWG2bGbauaadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiqadAhagaqbamaaBaaaleaacaWGQbaabeaaaaa@43CB@ mit maximaler Länge auswählen. Diese Sequenz ist linear unabhängig, also muss k+jn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgUcaRiaadQgacqGHKjYOcaWGUbaaaa@3B55@ sein. Wäre nun k+j<n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgUcaRiaadQgacqGH8aapcaWGUbaaaa@3AA4@ , so gäbe es nach dem zuvor Bewiesenen ein v i { v 1 , , v n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaaiyFaaaa@4221@ , so dass x 1 ,, x k , v 1 ,, v j , v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcaceWG2bGbauaadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiqadAhagaqbamaaBaaaleaacaWGQbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaaaa@469A@ linear unabhängig ist, im Widerspruch zur maximalen Länge von x 1 ,, x k , v 1 ,, v j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcaceWG2bGbauaadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiqadAhagaqbamaaBaaaleaacaWGQbaabeaaaaa@43CB@ . Also ist k+j=n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgUcaRiaadQgacqGH9aqpcaWGUbaaaa@3AA6@ , d.h. j=nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2da9iaad6gacqGHsislcaWGRbaaaa@3AB1@ , so dass x 1 ,, x k , v 1 ,, v nk MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcaceWG2bGbauaadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiqadAhagaqbamaaBaaaleaacaWGUbGaeyOeI0Iaam4Aaaqabaaaaa@45AC@ eine Basis von V ist.
 


Der folgende Satz stellte i.w. nur eine andere Formulierung des Basisergänzungssatzes dar:
 
Satz (Steinitzscher Austauschsatz):  Es sei V ein endlicher Vektorraum. Sind v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ und w 1 ,, w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGUbaabeaaaaa@3C76@ zwei Basen von V, so lassen sich beliebige Vektoren aus v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ durch geeignete Vektoren aus w 1 ,, w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGUbaabeaaaaa@3C76@ ersetzen, ohne dass die Basiseigenschaft verloren geht.

Beweis: Streicht man in v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C74@ k ausgewählte Vektoren, so ist die verbleibende Sequenz linear unabhängig und damit nach dem Basisergänzungssatz durch k Vektoren aus w 1 ,, w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGUbaabeaaaaa@3C76@ zu einer Basis zu ergänzen.

 


 9.4
9.6.