9.3. Lineare Unabhängigkeit


Im letzten Abschnitt haben wir präzise angeben, wann eine Erzeugersequenz verlängert werden kann, ohne das Erzeugnis zu ändern. Wir greifen diesen Gedanken noch einmal auf, betrachten diesmal jedoch die umgekehrte Richtung: Unter welchen Umständen läßt sich eine Erzeugersequenz "verlustfrei" verkürzen? Oder: Welche Sequenzen enthalten "überflüssige" Erzeuger, welche nicht?

Die folgenden Begriffe beschreiben beide Situationen:
 
Definition:Es sei V ein Vektorraum.

Eine Sequenz v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ von Vektoren aus V heißt

  • linear abhängig, falls es ein v i { v 1 ,, v k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaaiyFaaaa@421B@ gibt, so dass
    < v 1 ,, v k >=< v 1 ,, v i1 , v i+1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiilaiaadAhadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@52DA@ . *)
     
  • linear unabhängig, falls sie nicht linear abhängig ist.

_______
*) Gelegentlich kürzen wir < v 1 ,, v i1 , v i+1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaGGSaGaamODamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4939@ durch < v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@4325@ ab.


Beachte:

Da bei einem Erzeugnis < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ die Reihenfolge der Erzeuger unwesentlich ist, spielt auch bei der linearen Abhängigkeit, bzw. Unabhängigkeit die Reihenfolge der Vektoren keine Rolle.

 
Bei Sequenzen der Länge 1 bzw. 2 ist die lineare Anhängigkeit leicht zu überblicken:
 
Bemerkung:
  1. v  linear abhängigv=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaabYgacaqGPbGaaeOBaiaabwgacaqGHbGaaeOCaiaabccacaqGHbGaaeOyaiaabIgacaqGKdGaaeOBaiaabEgacaqGPbGaae4zaiaaywW7cqGHuhY2caaMf8UaamODaiabg2da9iaaicdaaaa@4D13@ .
  2. v,w  linear abhängigv=αw      w=αv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaaeiBaiaabMgacaqGUbGaaeyzaiaabggacaqGYbGaaeiiaiaabggacaqGIbGaaeiAaiaabsoacaqGUbGaae4zaiaabMgacaqGNbGaaGzbVlabgsDiBlaaywW7caWG2bGaeyypa0JaeqySdeMaam4DaiaaysW7cqGHOiI2caaMe8Uaam4Daiabg2da9iabeg7aHjaadAhaaaa@5A06@ .

Beweis:

Zu 1.: v  linear abhängig<v>=<v>=   <>={0}v=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaabYgacaqGPbGaaeOBaiaabwgacaqGHbGaaeOCaiaabccacaqGHbGaaeOyaiaabIgacaqGKdGaaeOBaiaabEgacaqGPbGaae4zaiaaywW7cqGHuhY2caaMf8UaeyipaWJaamODaiabg6da+iabg2da9iabgYda8iaadAhacqGH+aGpcqGH9aqpcaaMe8UaeyipaWJaeyybIySaeyOpa4Jaeyypa0Jaai4EaiaaicdacaGG9bGaaGzbVlabgsDiBlaaywW7caWG2bGaeyypa0JaaGimaaaa@6377@ .

Zu 2.: 
" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":  Ist etwa v<v,w>=<w> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolabgYda8iaadAhacaGGSaGaam4Daiabg6da+iabg2da9iabgYda8iaadEhacqGH+aGpaaa@4129@ , so gibt es ein geeignetes α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3788@ , so dass v=αw MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iabeg7aHjaadEhaaaa@3A85@ .
" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":  Ist etwa v=αw MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iabeg7aHjaadEhaaaa@3A85@ , so hat man:  v<w>=<v,w> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolabgYda8iaadEhacqGH+aGpcqGH9aqpcqGH8aapcaWG2bGaaiilaiaadEhacqGH+aGpaaa@4129@ , also ist die Sequenz v,w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3baaaa@3890@ linear abhängig.


Bei längeren Sequenzen ist meist ein deutlich größerer Aufwand erforderlich, um die lineare Abhängigkeit zu testen. Es lohnt sich daher, nach zusätzlichen Kriterien zu suchen.

3. in der folgenden Bemerkung ist das Standardkriterium für lineare Abhängigkeit.
  
Bemerkung: Es sei V ein Vektorraum, v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3FA3@ , dann sind die folgenden Aussagen äquivalent:
  1. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3D3A@ linear abhängig
     
  2. Es gibt ein v i { v 1 ,, v k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiabl+UimjaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaaiyFaaaa@42E7@ , so dass v i < v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4794@
     
  3. Es gibt α 1 ,, α k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqydaWgaaWcbaGaam4Aaaqabaaaaa@3DB6@ , mit mindestens einem α i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaaGimaaaa@3B2D@ , so dass α 1 v 1 ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F1@
     

Eine Darstellung des Nullvektors gemäß 3. nennt man eine nicht-triviale Darstellung der Null (in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ ). Die Äquivalenz 1. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@3848@ 3. läßt sich daher auch so formulieren:

v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3D3A@ ist genau dann linear abhängig, wenn sich der Nullvektor nicht-trivial aus v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3D3A@ linear kombinieren lässt.

Beweis:

1. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ 2.:  Nach Definition gibt es also ein v i { v 1 ,, v k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolaacUhacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiabl+UimjaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaaiyFaaaa@42E7@ , so dass
 

< v 1 ,, v k >=< v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4CC6@ .

Mit v i < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@42F3@ , hat man daher aber auch: v i < v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4794@ .

2. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ 3.:   v i < v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4794@ bedeutet: es gibt α 1 ,, α k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqydaWgaaWcbaGaam4Aaaqabaaaaa@3DB6@ so dass

v i = α 1 v 1 ++ α i1 v i1 + α i+1 v i+1 ++ α k v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaamODamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaOGaamODamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@5962@ .

Folgt:   0= α 1 v 1 ++ α i1 v i1 +(1) v i + α i+1 v i+1 ++ α k v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaamODamaaBaaaleaacaWGPbGaeyOeI0IaaGymaaqabaGccqGHRaWkcaGGOaGaeyOeI0IaaGymaiaacMcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaadMgacqGHRaWkcaaIXaaabeaakiaadAhadaWgaaWcbaGaamyAaiabgUcaRiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaeqySde2aaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaaaaa@5DFF@ . Dabei ist der Koeffizient von v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaaaaa@37FE@ ungleich Null; es liegt also eine nicht-triviale Darstellung der Null vor.

3. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ 1.:  Ist nun α 1 v 1 + α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@430F@ eine nicht-triviale Darstellung der Null, etwa α i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaaGimaaaa@3B2D@ , so lässt sich diese Gleichung nach v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaaaaa@37FE@ umstellen:

v i = α 1 α i v 1 α i1 α i v i1 α i+1 α i v i+1 α k α i v k < v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@749C@ .
Das bedeutet aber: < v 1 ,, v k >=< v 1 ,, v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4CC6@ .
 

Das folgende Beispiel benutzt neben dem Standardkriterium (in a.) auch die Alternative 2. (in b. und in c.):
 
Beispiel:
  1. In 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3846@ ist die Sequenz ( 1 2 1 ),( 2 2 0 ),( 4 1 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqadeaaaeaacaaIYaaabaGaeyOeI0IaaGOmaaqaaiaaicdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiabgkHiTiaaisdaaeaacaaIXaaabaGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaaa@4573@ linear abhängig, denn: 2( 1 2 1 )+3( 2 2 0 )+2( 4 1 1 )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIYaaabaGaaGymaaaaaiaawIcacaGLPaaacqGHRaWkcaaIZaWaaeWaaeaafaqabeWabaaabaGaaGOmaaqaaiabgkHiTiaaikdaaeaacaaIWaaaaaGaayjkaiaawMcaaiabgUcaRiaaikdadaqadaqaauaabeqadeaaaeaacqGHsislcaaI0aaabaGaaGymaaqaaiabgkHiTiaaigdaaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@49CC@ ist eine nicht-triviale Darstellung der Null.
     
  2. In MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecuaaa@409E@ ist X 2 +X, X 2 X,X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadIfacaGGSaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIfacaGGSaGaamiwaaaa@3F52@ , denn: X 2 X= X 2 +X2X< X 2 +X,X> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIfacqGH9aqpcaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiwaiabgkHiTiaaikdacaWGybGaeyicI4SaeyipaWJaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadIfacaGGSaGaamiwaiabg6da+aaa@494D@ .
     
  3. In F() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@4498@ ist sin 2 , cos 2 ,1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaaiilaiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiaacYcacaaIXaaaaa@3F98@ linear abhängig, denn: 1= sin 2 + cos 2 < sin 2 , cos 2 > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9iGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabgIGiolabgYda8iGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiaacYcaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqGH+aGpaaa@4BF1@ .

  4.  
 

In einer ersten Anwendung notieren wir drei allgemeine Beispiele:

Beispiel:  In jedem Vektorraum V sind Sequenzen, die den Nullvektor 0 enthalten, oder bei denen Vektoren doppelt vorkommen sofort linear abhängig:
  1. v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@3E25@ ist linear abhängig für alle x< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4105@ .
     
  2. v 1 ,, v k , 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaaIWaaaaa@3DE2@ ist linear abhängig.
     
  3. v 1 ,, v k , v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@3F3F@ ist linear abhängig.
     

Beweis:

Zum ersten Beispiel: x< v 1 ,, v k ,x> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+aaa@42B2@ ! Die restlichen sind Spezialfälle des gerade bewiesenen.

 

In der folgenden Bemerkung stehen technische Aspekte im Vordergrund; beim Beweis nutzen wir das Standardkriterium:

Bemerkung:  Es sei V ein Vektorraum, v 1 ,, v k ,xV,   α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4bGaeyicI4SaamOvaiaacYcacaaMe8UaeqySdeMaeyiyIKRaaGimaaaa@46E1@ , dann gilt:
  1. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear abhängig v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaaiilaiaadIhaaaa@439E@ linear abhängig.
     
  2. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear abhängig v 1 ,,α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqycaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4826@ linear abhängig.
     
  3. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear abhängig v 1 ,, v i , v j + v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaadAhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4D7A@ linear abhängig.
     
  4. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear abhängig v 1 ,, v i , v j +α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaadAhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcqaHXoqycaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4F19@ linear abhängig.
     

Beweis:

Zu 1.:  Ist α 1 v 1 ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F1@ eine nicht-triviale Darstellung der Null in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ , so ist offensichtlich 

α 1 v 1 ++ α k v k +0x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaaIWaGaamiEaiabg2da9iaaicdaaaa@468A@

eine nicht-triviale Darstellung der Null in < v 1 ,, v k ,x> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4bGaeyOpa4daaa@4031@ .

Zu 2.:  Für die Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ " sei wieder eine nicht-triviale Darstellung der Null in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ gegeben:

α 1 v 1 ++ α k v k +0x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaaIWaGaamiEaiabg2da9iaaicdaaaa@468A@ .

Da nun α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyiyIKRaaGimaaaa@3A09@ , lässt sich diese Gleichung auch so notieren:

α 1 v 1 ++ α i α α v i ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRmaalaaabaGaeqySde2aaSbaaSqaaiaadMgaaeqaaaGcbaGaeqySdegaaiabeg7aHjaadAhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyypa0JaaGimaaaa@4F07@ .
Das ist aber eine nicht-triviale Darstellung der Null in < v 1 ,,α v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@44C4@ .

Die Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ " ergibt sich direkt aus dem gerade Bewiesenen:   Mit v 1 ,,α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@42AE@ ist dann nämlich auch v 1 ,, 1 α α v i ,, v k = v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaWaaSaaaeaacaaIXaaabaGaeqySdegaaiabeg7aHjaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4CAD@ linear abhängig.

Zu 3.: Für i=j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaadQgaaaa@38CC@ liegt 2. (mit α=2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyypa0JaaGOmaaaa@394A@ ) vor; sei also o.E. ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgcMi5kaadQgaaaa@398D@ .
  " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":  Sei noch einmal 

v 1 ++ α i v i ++ α j v j ++ v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOAaaqabaGccaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaamODamaaBaaaleaacaWGRbaabeaakiabg2da9iaaicdaaaa@4E2E@

eine nicht-triviale Darstellung der Null in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ . Dann ist offensichtlich  

v 1 ++( α i α j ) v i ++ α j ( v j + v i )++ v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiaacIcacqaHXoqydaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHXoqydaWgaaWcbaGaamOAaaqabaGccaGGPaGaamODamaaBaaaleaacaWGPbaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGQbaabeaakiaacIcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaamODamaaBaaaleaacaWGPbaabeaakiaacMcacqGHRaWkcqWIMaYscqGHRaWkcaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyypa0JaaGimaaaa@5792@

eine Darstellung der Null in < v 1 ,, v i ,, v j + v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGQbaabeaakiabgUcaRiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4AC8@ , und zwar eine nicht-triviale, denn: Falls α j 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadQgaaeqaaOGaeyiyIKRaaGimaaaa@3B2E@ , ist nichts zu zeigen und falls α j =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaaGimaaaa@3A6D@ , liegt die Originaldarstellung vor!

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":  Wir argumentieren der Reihe nach (mit der Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ " aus 2., 3. und wieder 2.):

v 1 ,, v i ,, v j + v i ,, v k linear abhängig v 1 ,, v i ,, v j + v i ,, v k linear abhängig v 1 ,, v i ,, v j + v i v i ,, v k linear abhängig v 1 ,, v i ,, v j ,, v k linear abhängig. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@CD18@

4. ergibt sich in ähnlicher Weise aus 2. und 3.

 

Diese Bemerkung eröffnet eine neue Möglichkeit, die lineare Abhängigkeit einer Sequenz zu begründen: Man wende die Äquivalenzen 2. bis 4. solange an, bis eine Sequenz entsteht, deren Abhängigkeit direkt zu erkennen ist, z.B. weil in ihr der Nullvektor auftritt oder Vektoren doppelt vorkommen.
 
Beispiel:

  ( 3 1 ),( 1 1 ),( 1 1 )  linear abhängig Addiere den 1. Vektor zum 2. ( 3 1 ),( 2 2 ),( 1 1 )  linear abhängig Multipliziere den 3. Vektor mit 2 ( 3 1 ),( 2 2 ),( 2 2 )  linear abhängig MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaaqaamaabmaabaqbaeqabiqaaaqaaiaaiodaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaceaaaeaacqGHsislcaaIXaaabaGaaGymaaaaaiaawIcacaGLPaaacaGGSaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaaeiBaiaabMgacaqGUbGaaeyzaiaabggacaqGYbGaaeiiaiaabggacaqGIbGaaeiAaiaabsoacaqGUbGaae4zaiaabMgacaqGNbaabaWaaCbeaeaacqGHuhY2aSabaeqabaGaaeyqaiaabsgacaqGKbGaaeyAaiaabwgacaqGYbGaaeyzaiaabccacaqGKbGaaeyzaiaab6gaaeaacaqGXaGaaeOlaiaabccacaqGwbGaaeyzaiaabUgacaqG0bGaae4BaiaabkhacaqGGaGaaeOEaiaabwhacaqGTbGaaeiiaiaabkdacaqGUaaaaeqaaaGcbaWaaeWaaeaafaqabeGabaaabaGaaG4maaqaaiaaigdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiqaaaqaaiaaikdaaeaacaaIYaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaceaaaeaacaaIXaaabaGaaGymaaaaaiaawIcacaGLPaaacaqGSbGaaeyAaiaab6gacaqGLbGaaeyyaiaabkhacaqGGaGaaeyyaiaabkgacaqGObGaaei5aiaab6gacaqGNbGaaeyAaiaabEgaaeaadaWfqaqaaiabgsDiBdWceaqabeaacaqGnbGaaeyDaiaabYgacaqG0bGaaeyAaiaabchacaqGSbGaaeyAaiaabQhacaqGPbGaaeyzaiaabkhacaqGLbGaaeiiaiaabsgacaqGLbGaaeOBaaqaaiaabodacaqGUaGaaeiiaiaabAfacaqGLbGaae4AaiaabshacaqGVbGaaeOCaiaabccacaqGTbGaaeyAaiaabshacaqGGaGaaeOmaaaabeaaaOqaamaabmaabaqbaeqabiqaaaqaaiaaiodaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaceaaaeaacaaIYaaabaGaaGOmaaaaaiaawIcacaGLPaaacaGGSaWaaeWaaeaafaqabeGabaaabaGaaGOmaaqaaiaaikdaaaaacaGLOaGaayzkaaGaaeiBaiaabMgacaqGUbGaaeyzaiaabggacaqGYbGaaeiiaiaabggacaqGIbGaaeiAaiaabsoacaqGUbGaae4zaiaabMgacaqGNbaaaaaa@B86D@
 

Die folgende Bemerkung gibt Auskunft über einen bestimmten strukturellen Aspekt des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ . In der Natur seiner Elemente, also der n-Tupel, liegt es begründet, dass nicht beliebig lange, linear unabhängige Sequenzen gebildet werden können.
 
Bemerkung:  Für; x 1 ,, x k n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@4090@ gilt:
  1. k>n x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg6da+iaad6gacaaMf8UaeyO0H4TaaGzbVlaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadIhadaWgaaWcbaGaam4Aaaqabaaaaa@44D6@   linear abhängig.
  2. x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C72@   linear unabhängig kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caWGRbGaeyizImQaamOBaaaa@3EFA@ .

Beweis: Es reicht offensichtlich, nur den Fall k=n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaad6gacqGHRaWkcaaIXaaaaa@3A6F@ zu betrachten (Siehe 1. in der letzten Bemerkung). Wir führen den Beweis per Induktion über n. Im Induktionsschluss muß dabei zwischen n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3A16@ und n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ vermittelt werden; wir benutzen dabei die folgenden Zusammenhänge:
 

  • x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AFA@ , so ist ( x 0 ) n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaamiEaaqaaiaaicdaaaaacaGLOaGaayzkaaGaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3EE7@ .
  • Ist x n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaaaaa@3C97@ und setzt man x =( x 1 x n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafaGaeyypa0ZaaeWaaeaafaqabeWabaaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiabl6UinbqaaiaadIhadaWgaaWcbaGaamOBaaqabaaaaaGccaGLOaGaayzkaaaaaa@3F92@  , so ist x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafaGaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3B06@ und x=( x x n+1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9maabmaabaqbaeqabiqaaaqaaiqadIhagaqbaaqaaiaadIhadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaaa@3E4E@ .

Es gilt nun zunächst der folgende Hilfssatz:

Ist x 1 ,, x k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGRbaabeaaaaa@3C72@ linear abhängig in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ , so ist ( x 1 0 ),,( x k 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiaaicdaaaaacaGLOaGaayzkaaGaaiilaiablAciljaacYcadaqadaqaauaabeqaceaaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaaGimaaaaaiaawIcacaGLPaaaaaa@411C@ linear abhängig in n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3A16@ .(+)

Denn ist α 1 x 1 ++ α k x k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F5@ eine nicht-triviale Darstellung der Null in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ , so ist offensichtlich

α 1 ( x 1 0 )++ α k ( x k 0 )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaafaqabeGabaaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiaaicdaaaaacaGLOaGaayzkaaGaey4kaSIaeSOjGSKaey4kaSIaeqySde2aaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaafaqabeGabaaabaGaamiEamaaBaaaleaacaWGRbaabeaaaOqaaiaaicdaaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4895@
eine nicht-triviale Darstellung der Null in n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3A16@ .

Nun zum eigentlichen Induktionsbeweis:

n=1: ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaacaWGUbGaeyypa0JaaGymaaaaaaa@38AD@   Die Vektoren des 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIXaaaaaaa@3841@ sind gewöhnliche reelle Zahlen. Die Sequenz x 1 , x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaaaa@3A6C@ in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ ist sicherlich linear abhängig, falls x 1 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabg2da9iaaicdaaaa@3997@ ist; sei daher x 1 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiabgcMi5kaaicdaaaa@3A58@ . Dann ist aber

x 2 = x 2 x 1 x 1 < x 1 > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaamiEamaaBaaaleaacaaIXaaabeaakiabgIGiolabgYda8iaadIhadaWgaaWcbaGaaGymaaqabaGccqGH+aGpaaa@4437@ ,
d.h. x 1 , x 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaaaa@3A6C@ ist linear abhängig.

n      n+1: ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaacaWGUbGaaGjbVlabgkDiElaaysW7caWGUbGaey4kaSIaaGymaiaacQdaaaaaaa@3FB1@    Seien nun n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaikdaaaa@387A@ Vektoren des n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3A16@ gegeben:  x 1 ,, x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaaaaa@3E13@ . Dann sind x 1 ,, x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaafaWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcaceWG4bGbauaadaWgaaWcbaGaamOBaiabgUcaRiaaikdaaeqaaaaa@3E2B@   n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaikdaaaa@387A@ Vektoren des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ , nach Induktionsvoraussetzung ist diese Sequenz also linear abhängig. Ist nun die Sequenz  x 1 ,, x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaaaaa@3E13@ von der Form ( x 1 0 ),,( x n+2 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGabmiEayaafaWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaaaaiaawIcacaGLPaaacaGGSaGaeSOjGSKaaiilamaabmaabaqbaeqabiqaaaqaaiqadIhagaqbamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaaakeaacaaIWaaaaaGaayjkaiaawMcaaaaa@42D5@ , so ist sie nach (+) ebenfalls linear abhängig. Wir nehmen daher an, einer der Vektoren x i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3800@ ist in der letzten Koordinate nicht mit 0 besetzt; dies sei - die Reihenfolge ist ja unerheblich - etwa der letzte: x n+2,n+1 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbGaey4kaSIaaGOmaiaacYcacaWGUbGaey4kaSIaaGymaaqabaGccqGHGjsUcaaIWaaaaa@3F6E@ .
Wir setzen nun für i{1,,n+1} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgIGiolaacUhacaaIXaGaaiilaiablAciljaacYcacaWGUbGaey4kaSIaaGymaiaac2haaaa@4028@ :

y i = x i x i,n+1 x n+2,n+1 x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsisldaWcaaqaaiaadIhadaWgaaWcbaGaamyAaiaacYcacaWGUbGaey4kaSIaaGymaaqabaaakeaacaWG4bWaaSbaaSqaaiaad6gacqGHRaWkcaaIYaGaaiilaiaad6gacqGHRaWkcaaIXaaabeaaaaGccaWG4bWaaSbaaSqaaiaad6gacqGHRaWkcaaIYaaabeaaaaa@4C4E@ .

Nach Induktionsvoraussetzung ist  y 1 ,, y n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaafaWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcaceWG5bGbauaadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3E2C@ eine linear abhängige Sequenz in n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ . Da gemäß Konstruktion bei den Vektoren  y 1 ,, y n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyEamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@3E14@ die letzte Koordinate jeweils mit 0 belegt ist, ist nach (+) diese Sequenz, und damit auch  y 1 ,, y n+1 , x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyEamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaaaaa@4288@ linear abhängig in n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaaa@3A16@ . Dies ist aber nach 4. in der Bemerkung zuvor äquivalent zur linearen Abhängigkeit von  x 1 ,, x n+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaaaaa@3E13@ .
 

 

In einigen Fällen ist die lineare Abhängigkeit also durch einfaches Zählen festzustellen. So ist z.B. die Sequenz ( 1 2 ),( 2 3 ),( 3 4 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiqaaaqaaiaaikdaaeaacaaIZaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaceaaaeaacaaIZaaabaGaaGinaaaaaiaawIcacaGLPaaaaaa@4076@ allein auf Grund ihrer Länge linear abhängig.

 


Linear unabhängig sein bedeutet nach Definition: nicht linear abhängig sein. Daher ergibt sich aus nahezu jedem der bisherigen Ergebnisse über die lineare Abhängigkeit durch bloßes Verneinen eine entsprechende (und sogar schon bewiesene) Aussage über die lineare Unabhängigkeit!

 
Bemerkung:
  1. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyybIymaaa@3762@ ist linear unabhängig.
  2. v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaaaa@36E4@ linear unabhängig v0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bGaeyiyIKRaaGimaaaa@3EDD@
  3. v,w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3baaaa@3890@ linear unabhängig vαw      wαv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bGaeyiyIKRaeqySdeMaam4DaiaaysW7cqGHNis2caaMe8Uaam4DaiabgcMi5kabeg7aHjaadAhaaaa@4AE3@

Beweis:

2. und 3. ergeben sich direkt aus der entsprechenden Bemerkung für die lineare Abhängigkeit. 

Zu 1. argumentiert man folgendermaßen: MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyybIymaaa@3762@ kann nicht linear abhängig sein, denn sonst gäbe es in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyybIymaaa@3762@ einen überflüssigen Vektor; die leere Sequenz enthält aber überhaupt keine Vektoren.

 

 
Bemerkung: Es sei V ein Vektorraum, v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3ED7@ , dann sind die folgenden Aussagen äquivalent:
  1. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear unabhängig
  2. Es gibt keine Darstellung der Null  α 1 v 1 ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F1@ , so dass α i 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyiyIKRaaGimaaaa@3B2D@ für ein i
    d.h.: ist  α 1 v 1 ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F1@ , so folgt:  α 1 == α k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeSOjGSKaeyypa0JaeqySde2aaSbaaSqaaiaadUgaaeqaaOGaeyypa0JaaGimaaaa@402C@ .

     Es gilt also das folgende Standardkriterium für die lineare Unabhängigkeit:

v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ ist genau dann linear unabhängig, wenn sich der Nullvektor nur trivial aus v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear kombinieren lässt.

Wir üben das Standardkriterium an einigen Beispielen und betrachten zunächst zwei Sequenzen, auf die wir als Referenzbeispiele oft zurückgreifen werden:
 
Beispiel:
  1. In n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ gilt: e 1 ,, e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaaaaa@3C4F@ ist linear unabhängig.
  2. In n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaaaaa@41BB@ gilt: 1,X,, X n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacYcacaWGybGaaiilaiablAciljaacYcacaWGybWaaWbaaSqabeaacaWGUbaaaaaa@3CB0@ ist linear unabhängig.

Beweis:

Zu a.: Ist ( α 1 α n )= α 1 e 1 ++ α n e n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeSO7I0eabaGaeqySde2aaSbaaSqaaiaad6gaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWGLbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaaGimaaaa@4DB9@ eine Darstellung des Nullvektors, so hat man offensichtlich: α 1 == α n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeSOjGSKaeyypa0JaeqySde2aaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaaGimaaaa@402F@ . D.h.: Aus den Vektoren e 1 ,, e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaaaaa@3C4F@ lässt sich der Nullvektor nur trivial kombinieren.

Zu b.: Eine Darstellung des Nullvektors α 0 1+ α 1 X++ α n X n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaicdaaeqaaOGaaGymaiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiaadIfacqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaamOBaaqabaGccaWGybWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaaGimaaaa@46F7@ ist hier eine Darstellung des Nullpolynoms; gemäß Nullpolynomtest ist dies aber nur möglich, wenn alle Koeffizienten mit 0 besetzt sind: α 1 == α n =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeSOjGSKaeyypa0JaeqySde2aaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaaGimaaaa@402F@ .

 
Beispiel:
  1. ( 1 3 1 ),( 2 2 0 ),( 1 0 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaiodaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGOmaaqaaiaaicdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@42A6@ ist linear unabhängig,
    denn: Aus einer beliebigen Darstellung des Nullvektors α( 1 3 1 )+β( 2 2 0 )+γ( 1 0 1 )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaiodaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgUcaRiabek7aInaabmaabaqbaeqabmqaaaqaaiaaikdaaeaacaaIYaaabaGaaGimaaaaaiaawIcacaGLPaaacqGHRaWkcqaHZoWzdaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaGimaaqaaiaaigdaaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@49B1@ ergibt sich das folgende lineare Gleichungssystem:
     
    α+2β+γ=0 2β=0 β=0 3α+2β=0 3α+2β=0 α=0 α+γ=0 α+γ=0 γ=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@72EC@

    Also lässt sich der Nullvektor aus den gegebenen Vektoren nur trivial kombinieren.

  2. sin , cos  ist linear unabhängig: 
    Sei αsin+βcos=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaci4CaiaacMgacaGGUbGaey4kaSIaeqOSdiMaci4yaiaac+gacaGGZbGaeyypa0JaaGimaaaa@4176@ . Da dies eine Funktionengleichung ist, kann man durch Einsetzen gezielt gewählter Werte beliebig viele Zahlengleichungen ableiten; so ergibt sich z.B. für 0 die Gleichung 
     
    αsin0+βcos0=0β=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaci4CaiaacMgacaGGUbGaaGimaiabgUcaRiabek7aIjGacogacaGGVbGaai4CaiaaicdacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7cqaHYoGycqGH9aqpcaaIWaaaaa@4BC3@ ,

    und für π 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@3872@ die Gleichung 
    αsin π 2 +βcos π 2 =0α=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaci4CaiaacMgacaGGUbWaaSaaaeaacqaHapaCaeaacaaIYaaaaiabgUcaRiabek7aIjGacogacaGGVbGaai4CamaalaaabaGaeqiWdahabaGaaGOmaaaacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7cqaHXoqycqGH9aqpcaaIWaaaaa@4F5F@ .
     
  3. e aX , e bX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaamyyaiaadIfaaaGccaGGSaGaamyzamaaCaaaleqabaGaamOyaiaadIfaaaaaaa@3C58@   linear unabhängig ab MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWGHbGaeyiyIKRaamOyaaaa@3EF5@
    Beweis:
    " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ": Wäre a=b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadkgaaaa@38BC@ , so wäre e aX , e bX = e aX , e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaamyyaiaadIfaaaGccaGGSaGaamyzamaaCaaaleqabaGaamOyaiaadIfaaaGccqGH9aqpcaWGLbWaaWbaaSqabeaacaWGHbGaamiwaaaakiaacYcacaWGLbWaaWbaaSqabeaacaWGHbGaamiwaaaaaaa@43D6@ als Sequenz mit einer Verdopplung linear abhängig.
    " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ": Mit ab MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgcMi5kaadkgaaaa@397D@ ist auch < e a e b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaamyyaaaakiabgcMi5kaadwgadaahaaWcbeqaaiaadkgaaaaaaa@3BB5@ . Sei nun
    α e aX +β e bX =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaamyzamaaCaaaleqabaGaamyyaiaadIfaaaGccqGHRaWkcqaHYoGycaWGLbWaaWbaaSqabeaacaWGIbGaamiwaaaakiabg2da9iaaicdaaaa@4194@ .
    Setzt man in diese Gleichung zunächst 0 und anschließend 1 ein, erhält man das folgende Gleichungssystem:
     
    α+β=0 α e a +β e b =0 β=α α( e a e b )=0 α=0 β=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeGabiqaaaqaaiabeg7aHjabgUcaRiabek7aIjabg2da9iaaicdaaeaacqaHXoqycaWGLbWaaWbaaSqabeaacaWGHbaaaOGaey4kaSIaeqOSdiMaamyzamaaCaaaleqabaGaamOyaaaakiabg2da9iaaicdaaaGaaGzbVlabgkDiElaaywW7faqaceGabaaabaGaeqOSdiMaeyypa0JaeyOeI0IaeqySdegabaGaeqySdeMaaiikaiaadwgadaahaaWcbeqaaiaadggaaaGccqGHsislcaWGLbWaaWbaaSqabeaacaWGIbaaaOGaaiykaiabg2da9iaaicdaaaGaaGzbVlabgkDiElaaywW7faqaceGabaaabaGaeqySdeMaeyypa0JaaGimaaqaaiabek7aIjabg2da9iaaicdaaaaaaa@667F@

    Also ist eine Kombination des Nullvektors aus e aX , e bX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaamyyaiaadIfaaaGccaGGSaGaamyzamaaCaaaleqabaGaamOyaiaadIfaaaaaaa@3C58@ nur trivial möglich.

 

Bemerkung:  Es sei V ein Vektorraum, v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3ED7@ , α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyiyIKRaaGimaaaa@3A09@ , dann gilt:
  1. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear unabhängig v 1 ,,α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqycaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@4826@ linear unabhängig.
  2. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear unabhängig v 1 ,, v i ,, v j + v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@4E2A@ linear unabhängig.
  3. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ linear unabhängig v 1 ,, v i ,, v j +α v i ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@4FC9@ linear unabhängig.

Wie schon bei der entsprechenden Bemerkung zur linearen Abhängigkeit ergibt sich auch hier die Möglichkeit einer zweiten Testmethode auf lineare Unabhängigkeit: Man forme eine gegebene Sequenz solange äquivalent um, bis eine bekannte, linear unabhängige Referenzsequenz entsteht.
 
Beispiel:
 

  •   ( 1 2 1 ),( 0 1 0 ),( 0 2 1 )  linear unabhängig Subtrahiere den 3. Vektor vom 1. ( 1 0 0 ),( 0 1 0 ),( 0 2 1 )  linear unabhängig Addiere das 2-fache des 2. Vektors zum 3. e 1 , e 2 , e 3   linear unabhängig MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C9D2@
     

  •   X 3 , X 2 +X, X 2 X,X+1  linear unabhängig Addiere den 3. Vektor zum 2. X 3 ,2 X 2 , X 2 X,X+1  linear unabhängig Multipliziere den 2. Vektor mit ½ X 3 , X 2 , X 2 X,X+1  linear unabhängig Subtrahiere den 2. Vektor vom 3. X 3 , X 2 ,X,X+1  linear unabhängig Addiere den 3. Vektor zum 4. X 3 , X 2 ,X,1  linear unabhängig Multipliziere den 3. Vektor mit 1 X 3 , X 2 ,X,1  linear unabhängig MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@64A2@
 

In der folgenden Bemerkung untersuchen wir, wie sich die lineare Unabhängigkeit bei Änderungen der Sequenzlänge verhält. Dabei dürfte das Verkürzen von Sequenzen keine Probleme bereiten, liegt es doch "im Trend" der Unabhängigkeit. Interessanterweise ist unter genau kalkulierbaren Umständen auch das Verlängern von Sequenzen unschädlich.
 
Bemerkung:  Es sei V ein Vektorraum, v 1 ,, v k ,xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4bGaeyicI4SaamOvaaaa@4087@ , dann gilt:
  1. v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@3E28@ linear unabhängig v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgkDiElaaywW7caWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@41EA@ linear unabhängig.
  2. v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ linear unabhängig       x< v 1 ,, v k > v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlabgEIizlaaysW7caWG4bGaeyycI8SaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iaaywW7cqGHuhY2caaMf8UaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@5386@ linear unabhängig.

Beweis:

Zu 1.:  Wäre v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ linear abhängig, so wäre auch die verlängerte Sequenz v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@3E28@ linear abhängig.

Zu 2.:
" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3849@ ": Um zu zeigen, dass v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@3E28@ linear unabhängig ist, setzen wir eine Darstellung der Null an:

α 1 v 1 ++ α k v k +αx=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcqaHXoqycaWG4bGaeyypa0JaaGimaaaa@4772@ (*)

Zunächst gilt nun: α=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyypa0JaaGimaaaa@394B@ , denn sonst wäre x= α 1 α v 1 α k α v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabgkHiTmaalaaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqySdegaaiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHsislcqWIMaYscqGHsisldaWcaaqaaiabeg7aHnaaBaaaleaacaWGRbaabeaaaOqaaiabeg7aHbaacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyicI4SaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@52B7@ , ein Widerspruch also.

Damit aber reduziert sich Darstellung (*) auf

α 1 v 1 ++ α k v k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43F4@ ,

so dass aus der linearen Unabhängigkeit von v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ folgt:  α 1 == α k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeSOjGSKaeyypa0JaeqySde2aaSbaaSqaaiaadUgaaeqaaOGaeyypa0JaaGimaaaa@402F@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3845@ ": Aus 1. erhält man sofort: v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C71@ ist linear unabhängig. Wäre nun x< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4108@ , so wäre einem der allgemeinen Beispiele im oberen Teil die Sequenz v 1 ,, v k ,x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4baaaa@3E28@ linear abhängig.

 


 9.2
9.4.