9.2. Untervektorräume und Erzeugnisse


In einem Vektorraum (V, + , · ) ist der Menge V eine hochkomplexe algebraische Struktur unterlegt. Es ist daher nicht unbedingt zu erwarten, dass beliebige Teilmengen von V automatisch an dieser Struktur teilhaben. In diesem Abschnitt werden diejenigen Teilmengen von V charakterisiert und untersucht, die die Vektorraumstruktur von V übernehmen.

   
Definition:  Es sei (V, + , · ) ein Vektorraum. Eine Teilmenge UV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabgkOimlaadAfaaaa@399A@ ; heißt ein Untervektorraum von V  falls für alle v,wU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaeyicI4Saamyvaaaa@3AEE@ und α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ gilt:
  1. 0U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadwfaaaa@3901@ .
  2. v,wUv+wU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaeyicI4SaamyvaiaaywW7cqGHshI3caaMf8UaamODaiabgUcaRiaadEhacqGHiiIZcaWGvbaaaa@459E@ .
  3. vUαvU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaadwfacaaMf8UaeyO0H4TaaGzbVlabeg7aHjabgwSixlaadAhacqGHiiIZcaWGvbaaaa@45FD@ .

Beachte:

Die Wortwahl Untervektorraum erklärt sich aus der folgenden Bemerkung:

Bemerkung:  Es sei (V, + , · ) ein Vektorraum und U eine Teilmenge von V. Dann gilt:

U  ist Untervektorraum von V    MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSnaaa@3845@    (U, + , · ) ist ein Vektorraum.

Beweis:

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":
Nach 2. und 3. in der Definition des Untervektorraums sind + und · auch Rechenoperationen auf U:
 

+:U×UU :×UU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabgUcaRiaacQdacaWGvbGaey41aqRaamyvaiabgkziUkaadwfaaeaacqGHflY1caGG6aGaeSyhHeQaey41aqRaamyvaiabgkziUkaadwfaaaaaaa@4857@

Nun müssen die Axiome (V1) bis (V8) nachgeprüft werden. Allerdings sind alle Rechenregeln, die + und · in V erfüllen, erst recht in U gültig, brauchen also nicht mehr bewiesen zu werden. Bis auf (V3) und (V4) sind alle Axiome durch dieses Argument abgedeckt!

(V3):  Der Nullvektor 0V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadAfaaaa@3902@ verhält sich zu allen Vektoren aus V neutral, also erst recht zu allen aus U. Gemäß 1. ist aber 0U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadwfaaaa@3901@ , daher besitzt auch U ein neutrales Element, nämlich den Nullvektor aus V.

(V4):  Ist vU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaadwfaaaa@3942@ , so ist nach 3. auch  v=(1)vU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamODaiabg2da9iaacIcacqGHsislcaaIXaGaaiykaiabgwSixlaadAhacqGHiiIZcaWGvbaaaa@417B@ . v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaamODaaaa@37D1@ erfüllt die Inversengleichung in V, also auch in U. Jedes Element aus U besitzt daher ein inverses Element in U (und zwar dasselbe inverse Element, das es in V besitzt!).

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":
Zu 1.:  Ist (U, + , · ) ein Vektorraum, so ist per Definition
 

+:U×UU :×UU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabgUcaRiaacQdacaWGvbGaey41aqRaamyvaiabgkziUkaadwfaaeaacqGHflY1caGG6aGaeSyhHeQaey41aqRaamyvaiabgkziUkaadwfaaaaaaa@4857@

Dies sind aber genau die Bedingungen 2. und 3. Ferner besitzt U als Vektorraum ein neutrales Element 0'U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaacEcacqGHiiIZcaWGvbaaaa@39AC@ , und da die Vektorsubtraktion auf U abgeschlossen ist, folgt daraus: 0=0'0'U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaaicdacaGGNaGaeyOeI0IaaGimaiaacEcacqGHiiIZcaWGvbaaaa@3DBE@ .
 

 
Nach diesem Kriterium sind der Nullraum und der Vektorraum V selbst automatisch Untervektorräume von V:

Beispiel:  Es sei (V, + , · ) ein Vektorraum, dann sind
  1. {0}V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaaicdacaGG9bGaeyOGIWSaamOvaaaa@3B7A@
  2. VV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiabgkOimlaadAfaaaa@399B@

Untervektorräume von V.
 

 
Beispiel:
a.   U{( x 3x )|x} 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg2da9iaacUhadaqadaqaauaabeqaceaaaeaacaWG4baabaGaaG4maiaadIhaaaaacaGLOaGaayzkaaGaaiiFaiaadIhacqGHiiIZcqWIDesOcaGG9bGaeyOGIWSaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@475C@ ist ein Untervektorraum, denn:
  • 0=( 0 30 )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9maabmaabaqbaeqabiqaaaqaaiaaicdaaeaacaaIZaGaeyyXICTaaGimaaaaaiaawIcacaGLPaaacqGHiiIZcaWGvbaaaa@4018@ .
     
  • ( x 3x ),( y 3y )U( x 3x )+( y 3y )=( x+y 3(x+y) )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaamiEaaqaaiaaiodacaWG4baaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaceaaaeaacaWG5baabaGaaG4maiaadMhaaaaacaGLOaGaayzkaaGaeyicI4SaamyvaiaaywW7cqGHshI3caaMf8+aaeWaaeaafaqabeGabaaabaGaamiEaaqaaiaaiodacaWG4baaaaGaayjkaiaawMcaaiabgUcaRmaabmaabaqbaeqabiqaaaqaaiaadMhaaeaacaaIZaGaamyEaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqaceaaaeaacaWG4bGaey4kaSIaamyEaaqaaiaaiodacaGGOaGaamiEaiabgUcaRiaadMhacaGGPaaaaaGaayjkaiaawMcaaiabgIGiolaadwfaaaa@5D54@ .
     
  • ( x 3x )Uα( x 3x )=( αx 3(αx) )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaamiEaaqaaiaaiodacaWG4baaaaGaayjkaiaawMcaaiabgIGiolaadwfacaaMf8UaeyO0H4TaaGzbVlabeg7aHnaabmaabaqbaeqabiqaaaqaaiaadIhaaeaacaaIZaGaamiEaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqaceaaaeaacqaHXoqycaWG4baabaGaaG4maiaacIcacqaHXoqycaWG4bGaaiykaaaaaiaawIcacaGLPaaacqGHiiIZcaWGvbaaaa@5441@ .
     
b.   U{( x+1 3x )|x} 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg2da9iaacUhadaqadaqaauaabeqaceaaaeaacaWG4bGaey4kaSIaaGymaaqaaiaaiodacaWG4baaaaGaayjkaiaawMcaaiaacYhacaWG4bGaeyicI4SaeSyhHeQaaiyFaiabgkOimlabl2riHoaaCaaaleqabaGaaGOmaaaaaaa@48F9@ ist kein Untervektorraum, da der Nullvektor fehlt:
     Gäbe es nämlich ein x, so dass ( x+1 3x )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaamiEaiabgUcaRiaaigdaaeaacaaIZaGaamiEaaaaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3D93@ , also x+1=03x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgUcaRiaaigdacqGH9aqpcaaIWaGaaGzbVlabgEIizlaaywW7caaIZaGaamiEaiabg2da9iaaicdaaaa@4287@ , so wäre x=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabgkHiTiaaigdaaaa@3994@ und x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaaicdaaaa@38A6@ . Widerspruch.
 
  1. U{f𝔽()|f(0)=0}𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg2da9iaacUhacaWGMbGaeyicI48efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaiaacYhacaWGMbGaaiikaiaaicdacaGGPaGaeyypa0JaaGimaiaac2hacqGHckcZcqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@57A7@ ist ein Untervektorraum, denn:
    • 0U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadwfaaaa@3901@ , da 0(0) = 0.
    • f,gUf(0)=0g(0)=0f+g(0)=0f+gU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacYcacaWGNbGaeyicI4SaamyvaiaaywW7cqGHshI3caaMf8UaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaaicdacaaMf8Uaey4jIKTaaGzbVlaadEgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIWaGaaGzbVlabgkDiElaaywW7caWGMbGaey4kaSIaam4zaiaacIcacaaIWaGaaiykaiabg2da9iaaicdacaaMf8UaeyO0H4TaaGzbVlaadAgacqGHRaWkcaWGNbGaeyicI4Saamyvaaaa@6523@ .
    • fUf(0)=0αf(0)=0αfU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadwfacaaMf8UaeyO0H4TaaGzbVlaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIWaGaaGzbVlabgkDiElaaywW7cqaHXoqycaWGMbGaaiikaiaaicdacaGGPaGaeyypa0JaaGimaiaaywW7cqGHshI3caaMf8UaeqySdeMaamOzaiabgIGiolaadwfaaaa@59A0@ .
       
  2. U{f𝔽()|f(0)0}𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg2da9iaacUhacaWGMbGaeyicI48efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaiaacYhacaWGMbGaaiikaiaaicdacaGGPaGaeyyzImRaaGimaiaac2hacqGHckcZcqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@5867@ ist kein Untervektorraum, denn die Eigenschaft 3 ist verletzt: So ist etwa cosU MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaeyicI4Saamyvaaaa@3B1A@ , aber (1)cos=cos MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgkHiTiaaigdacaGGPaGaeyyXICTaci4yaiaac+gacaGGZbGaeyypa0JaeyOeI0Iaci4yaiaac+gacaGGZbaaaa@42CD@ nicht.

 

 
Fast alle klassischen Funktionenmengen der Analysis treten in der linearen Algebra als Untervektorräume auf! Die Eigenschaften 1 bis 3 sind dabei durch die jeweiligen Rechenregeln meistens schon erfüllt. Für ein Zusammenspiel zwischen linearer Algebra und Analysis sind also die folgenden Beispiele ein erstes Indiz.

Beispiel:  Es sei A MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgkOimlabl2riHcaa@3A1B@ . Die folgenden Mengen sind Untervektorräume des jeweils angegebenen Vektorraums:
 
   a. C 0 (A)𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaakiaacIcacaWGbbGaaiykaiabgkOimprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xHWBKaaiikaiabl2riHkaacMcaaaa@4A69@ Die Menge der stetigen Funktionen auf A.
   b. C n (A)𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaamOBaaaakiaacIcacaWGbbGaaiykaiabgkOimprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xHWBKaaiikaiabl2riHkaacMcaaaa@4AA2@   Die Menge der n-mal stetig differenzierbaren Funktionen auf A.
   c. C (A)𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaeyOhIukaaOGaaiikaiaadgeacaGGPaGaeyOGIW8efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@4B20@ Die Menge der beliebig oft differenzierbaren Funktionen auf A.
   d. I(A)𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaacIcacaWGbbGaaiykaiabgkOimprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xHWBKaaiikaiabl2riHkaacMcaaaa@497E@ Die Menge der integrierbaren  Funktionen auf A.
   e. B(A)𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaacIcacaWGbbGaaiykaiabgkOimprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xHWBKaaiikaiabl2riHkaacMcaaaa@4977@ Die Menge der beschränkten Funktionen auf A.
   f. 𝔽 konv ()𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrdaWgaaWcbaGaam4Aaiaad+gacaWGUbGaamODaaqabaGccaGGOaGaeSyfHuQaaiykaiabgkOimlab=vi8gjaacIcacqWIvesPcaGGPaaaaa@4F9A@ Die Menge der konvergenten Folgen.
   g. 𝔽 0konv ()𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrdaWgaaWcbaGaaGimaiabgkHiTiaadUgacaWGVbGaamOBaiaadAhaaeqaaOGaaiikaiablwriLkaacMcacqGHckcZcqWFfcVrcaGGOaGaeSyfHuQaaiykaaaa@5141@ Die Menge der Nullfolgen.
   h. 𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabgkOimprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xHWBKaaiikaiabl2riHkaacMcaaaa@4766@ Die Menge aller Polynome über MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ .
   i. n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiaad6gaaaGccqGHckcZcqWFzecuaaa@44D0@ Die Menge aller Polynome vom Grad n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImQaamOBaaaa@3891@ , zuzüglich des Nullpolynoms 0.
 

Beweis:  Wir zeigen nur e. (alle anderen Fälle sind durch die entsprechenden Rechenregeln bereits erledigt):

  • Die konstante Funktion 0 auf A ist beschränkt, also: 0B(A) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadkeacaGGOaGaamyqaiaacMcaaaa@3B0D@ .
     
  • Sind f 1 , f 2 B(A) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiaacYcacaWGMbWaaSbaaSqaaiaaikdaaeqaaOGaeyicI4SaamOqaiaacIcacaWGbbGaaiykaaaa@3EBC@ , so gibt Zahlen c1,c2, so dass | f i (x)| c i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEaiaacMcacaGG8bGaeyizImQaam4yamaaBaaaleaacaWGPbaabeaaaaa@4005@ für alle xA MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadgeaaaa@3930@ . Mit der Dreiecksungleichung erhält man daher für alle xA MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadgeaaaa@3930@ :
     
    | f 1 (x)+ f 2 (x)|| f 1 (x)|+| f 2 (x)| c 1 + c 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiEaiaacMcacqGHRaWkcaWGMbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadIhacaGGPaGaaiiFaiabgsMiJkaacYhacaWGMbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadIhacaGGPaGaaiiFaiabgUcaRiaacYhacaWGMbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadIhacaGGPaGaaiiFaiabgsMiJkaadogadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaikdaaeqaaaaa@566C@ .

    Also ist auch  f1f2  beschränkt:   f 1 + f 2 B(A) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqabaGccqGHiiIZcaWGcbGaaiikaiaadgeacaGGPaaaaa@3EEE@ .
     
  • Ist fB(A) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadkeacaGGOaGaamyqaiaacMcaaaa@3B3E@ , etwa: |f(x)|c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadAgacaGGOaGaamiEaiaacMcacaGG8bGaeyizImQaam4yaaaa@3DC7@ für alle xA MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadgeaaaa@3930@ , so gilt für diese x ebenfalls: 
     
    |αf(x)|=|α||f(x)||α|c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiabeg7aHjaadAgacaGGOaGaamiEaiaacMcacaGG8bGaeyypa0JaaiiFaiabeg7aHjaacYhacaGG8bGaamOzaiaacIcacaWG4bGaaiykaiaacYhacqGHKjYOcaGG8bGaeqySdeMaaiiFaiaadogaaaa@4CEB@ ,

    also auch wieder: αfB(A) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaamOzaiabgIGiolaadkeacaGGOaGaamyqaiaacMcaaaa@3CDD@ .

Beachte:

 

Der Unterraumbegriff verträgt sich nur schlecht mit den Mengenoperationen. Lediglich die Schnittbildung führt nicht aus der Klasse der Untervektorräume von V hinaus. Für spätere Anwendungen ist dies allerdings eine wichtige Eigenschaft.
 
Bemerkung:  Sind UV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabgkOimlaadAfaaaa@399A@ und WV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabgkOimlaadAfaaaa@399C@ Untervektorräume von V, so ist auch UW MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabgMIihlaadEfaaaa@393D@ ein Untervektorraum von V.

Beweis:

  1. Man hat: 0U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadwfaaaa@3901@ und 0W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadEfaaaa@3903@ , also ist auch 0UW MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadwfacqGHPiYXcaWGxbaaaa@3B7B@ .
  2. v,wUWv,wU      v,wWv+wU      v+wWv+wUW MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaeyicI4SaamyvaiabgMIihlaadEfacaaMf8UaeyO0H4TaaGzbVlaadAhacaGGSaGaam4DaiabgIGiolaadwfacaaMe8Uaey4jIKTaaGjbVlaadAhacaGGSaGaam4DaiabgIGiolaadEfacaaMf8UaeyO0H4TaaGzbVlaadAhacqGHRaWkcaWG3bGaeyicI4SaamyvaiaaysW7cqGHNis2caaMe8UaamODaiabgUcaRiaadEhacqGHiiIZcaWGxbGaaGzbVlabgkDiElaaywW7caWG2bGaey4kaSIaam4DaiabgIGiolaadwfacqGHPiYXcaWGxbaaaa@7390@ .
  3. vUWvU      vWαvU      αvWαvUW MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaadwfacqGHPiYXcaWGxbGaaGzbVlabgkDiElaaywW7caWG2bGaeyicI4SaamyvaiaaysW7cqGHNis2caaMe8UaamODaiabgIGiolaadEfacaaMf8UaeyO0H4TaaGzbVlabeg7aHjaadAhacqGHiiIZcaWGvbGaaGjbVlabgEIizlaaysW7cqaHXoqycaWG2bGaeyicI4Saam4vaiaaywW7cqGHshI3caaMf8UaeqySdeMaamODaiabgIGiolaadwfacqGHPiYXcaWGxbaaaa@6DCF@ .

 


 

Reelle Vektorräume sind, abgesehen vom Nullraum, stets unendliche Mengen. Für endlich viele Vektoren v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ aus V ist daher die Menge { v 1 ,, v k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGG9baaaa@3E78@ i.d.R. kein Untervektorraum von V. Ziel ist es nun, { v 1 ,, v k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGG9baaaa@3E78@ durch möglichst wenige Vektoren zu einem Untervektorraum aufzustocken.
 
Definition:  Es sei (V, + , · ) ein Vektorraum und v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3ED7@ , dann heißt die Menge

< v 1 ,, v k >{ α 1 v 1 ++ α k v k | α 1 ,, α k } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iaacUhacqaHXoqydaWgaaWcbaGaaGymaaqabaGccaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaeqySde2aaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaakiaacYhacqaHXoqydaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiabeg7aHnaaBaaaleaacaWGRbaabeaakiabgIGiolabl2riHkaac2haaaa@599D@

das Erzeugnis der Vektoren v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ . Die Vektoren selbst nennen wir Erzeuger von < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ . Ein Element von < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ , also ein Vektor der Form

α 1 v 1 ++ α k v k = i=1 k α i v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpdaaeWbqaaiabeg7aHnaaBaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGRbaaniabggHiLdaaaa@4DE5@

heißt eine Linearkombinationen von v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ .

Um auch den Fall k = 0 zur Verfügung zu haben, setzen wir zusätzlich:

<><>{0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaeyOpa4Jaeyypa0JaeyipaWJaeyybIySaeyOpa4Jaeyypa0Jaai4EaiaaicdacaGG9baaaa@4040@ .

 

 
Beachte:

 
 
Beispiel: In 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ hat man für U<( 2 1 1 ),( 1 3 1 )>={ α 1 ( 2 1 1 )+ α 2 ( 1 3 1 )| α 1 , α 2 }={( 2 α 1 + α 2 α 1 +3 α 2 α 1 + α 2 )| α 1 , α 2 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7E62@
etwa: ( 0 10 2 )=2( 2 1 1 )4( 1 3 1 )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGimaaqaaiabgkHiTiaaigdacaaIWaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGH9aqpcaaIYaWaaeWaaeaafaqabeWabaaabaGaaGOmaaqaaiaaigdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgkHiTiaaisdadaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaG4maaqaaiaaigdaaaaacaGLOaGaayzkaaGaeyicI4Saamyvaaaa@49A6@ , aber auch 0=0( 2 1 1 )+0( 1 3 1 )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaaicdadaqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaey4kaSIaaGimamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIZaaabaGaaGymaaaaaiaawIcacaGLPaaacqGHiiIZcaWGvbaaaa@43F2@ .

Die folgenden Beispiele zeigen, wie man überprüfen kann, ob ein Vektor in einem Erzeugnis liegt oder nicht:

a.( 8 6 2 )U es gibt   α 1 , α 2 ,so dass   α 1 ( 2 1 1 )+ α 2 ( 1 3 1 )=( 8 6 2 ) es gibt   α 1 , α 2 ,so dass   2 α 1 + α 2 =8 α 1 +3 α 2 =6 α 1 + α 2 =2 es gibt   α 1 , α 2 ,so dass   α 1 =6 α 1 +3 α 2 =6 α 2 =2 α 1 =4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaiaabggacaqGUaGaaGzbVpaabmaabaqbaeqabmqaaaqaaiabgkHiTiaaiIdaaeaacaaI2aaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGHiiIZcaWGvbaabaGaaGzbVlabgsDiBlaaywW7caqGLbGaae4CaiaabccacaqGNbGaaeyAaiaabkgacaqG0bGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiabeg7aHnaaBaaaleaacaaIYaaabeaakiabgIGiolabl2riHkaacYcacaqGZbGaae4BaiaabccacaqGKbGaaeyyaiaabohacaqGZbGaeqySde2aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaafaqabeWabaaabaGaaGOmaaqaaiaaigdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgUcaRiabeg7aHnaaBaaaleaacaaIYaaabeaakmaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIZaaabaGaaGymaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacqGHsislcaaI4aaabaGaaGOnaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaaabaaabaGaaGzbVlabgsDiBlaaywW7caqGLbGaae4CaiaabccacaqGNbGaaeyAaiaabkgacaqG0bGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiabeg7aHnaaBaaaleaacaaIYaaabeaakiabgIGiolabl2riHkaacYcacaqGZbGaae4BaiaabccacaqGKbGaaeyyaiaabohacaqGZbqbaeWabmqaaaqaaiaabkdacqaHXoqydaWgaaWcbaGaaeymaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsislcaaI4aaabaGaeqySde2aaSbaaSqaaiaabgdaaeqaaOGaey4kaSIaaG4maiabeg7aHnaaBaaaleaacaaIYaaabeaakiabg2da9iaaiAdaaeaacqaHXoqydaWgaaWcbaGaaeymaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsislcaaIYaaaaaqaaaqaaiaaywW7cqGHuhY2caaMf8UaaeyzaiaabohacaqGGaGaae4zaiaabMgacaqGIbGaaeiDaiabeg7aHnaaBaaaleaacaaIXaaabeaakiaacYcacqaHXoqydaWgaaWcbaGaaGOmaaqabaGccqGHiiIZcqWIDesOcaGGSaGaae4Caiaab+gacaqGGaGaaeizaiaabggacaqGZbGaae4CauaabmqadeaaaeaacqaHXoqydaWgaaWcbaGaaeymaaqabaGccqGH9aqpcqGHsislcaaI2aaabaGaeqySde2aaSbaaSqaaiaabgdaaeqaaOGaey4kaSIaaG4maiabeg7aHnaaBaaaleaacaaIYaaabeaakiabg2da9iaaiAdaaeaacqaHXoqydaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsislcaaIYaGaeyOeI0IaeqySde2aaSbaaSqaaiaabgdaaeqaaOGaeyypa0JaaGinaaaaaaaaaa@DA87@

Da die mittlere Gleichung ebenfalls von α 1 =6       α 2 =4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGOnaiaaysW7cqGHNis2caaMe8UaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGinaaaa@4449@ gelöst wird, ist das Gleichungssystem (eindeutig) lösbar; daher ist  ( 8 6 2 )=6( 2 1 1 )+4( 1 3 1 )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaeyOeI0IaaGioaaqaaiaaiAdaaeaacqGHsislcaaIYaaaaaGaayjkaiaawMcaaiabg2da9iabgkHiTiaaiAdadaqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaey4kaSIaaGinamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIZaaabaGaaGymaaaaaiaawIcacaGLPaaacqGHiiIZcaWGvbaaaa@49DF@ .

b.( 2 0 1 )U es gibt   α 1 , α 2 ,so dass   α 1 ( 2 1 1 )+ α 2 ( 1 3 1 )=( 2 0 1 ) es gibt   α 1 , α 2 ,so dass   2 α 1 + α 2 =2 α 1 +3 α 2 =0 α 1 + α 2 =1 es gibt   α 1 , α 2 ,so dass   α 1 =1 α 1 +3 α 2 =0 α 2 =1 α 1 =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D2E9@

Hier nun zeigt die mittlere Gleichung, dass dieses Gleichungssystem keine Lösung besitzt, daher ist ( 2 0 1 )U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGOmaaqaaiaaicdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabgMGiplaadwfaaaa@3C12@ .

 

Beispiel:

In    ist  U< X 2 ,X+1,X1> ={ α 1 X 2 + α 2 (X+1)+ α 3 (X1)| α 1 , α 2 , α 3 } ={ α 1 X 2 +( α 2 + α 3 )X+( α 2 α 3 )1| α 1 , α 2 , α 3 } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9C36@

Also hat man etwa: 4 X 2 +7X+3=4 X 2 +(5+2)X+(52)1U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI3aGaamiwaiabgUcaRiaaiodacqGH9aqpcaaI0aGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaacIcacaaI1aGaey4kaSIaaGOmaiaacMcacaWGybGaey4kaSIaaiikaiaaiwdacqGHsislcaaIYaGaaiykaiaaigdacqGHiiIZcaWGvbaaaa@4D5B@ .

Die Gleichheit von Polynomen über MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ ist die Gleichheit ihrer Koeffizienten (Identätssatz für Polynome). Man kann daher wie im vorangehenden Beispiel argumentieren:

XU es gibt   α 1 , α 2 , α 3 ,so dass   α 1 X 2 +( α 2 + α 3 )X+( α 2 α 3 )1=0 X 2 +1X+01 es gibt   α 1 , α 2 , α 3 ,so dass   α 1 =0 α 2 + α 3 =1 α 2 α 3 =0 es gibt   α 1 , α 2 , α 3 ,so dass   α 1 =0 2 α 2 =1 α 3 = α 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@DE46@

Das liefert die (eindeutige) Darstellung X=0 X 2 +( 1 2 + 1 2 )X+( 1 2 1 2 )1U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabg2da9iaaicdacqGHflY1caWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiikamaalaaabaGaaGymaaqaaiaaikdaaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGPaGaamiwaiabgUcaRiaacIcadaWcaaqaaiaaigdaaeaacaaIYaaaaiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaaiykaiaaigdacqGHiiIZcaWGvbaaaa@4CF7@ .

 

 

Wir benötigen gelegentlich die folgenden Standardbeispiele:

Bemerkung:
  1. In jedem Vektorraum V gilt:  <  0 >  = {0}.
     
  2. In MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecuaaa@409B@ gilt: <1,X,, X n1 , X n >= n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaaGymaiaacYcacaWGybGaaiilaiablAciljaacYcacaWGybWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGymaaaakiaacYcacaWGybWaaWbaaSqabeaacaWGUbaaaOGaeyOpa4Jaeyypa0JaamiuamaaCaaaleqabaGaamOBaaaaaaa@4620@ .
     
  3. In n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ gilt: < e 1 ,, e n >= n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaakiabg6da+iabg2da9iabl2riHoaaCaaaleqabaGaamOBaaaaaaa@41FB@ .

Beweis:

Zu 1.:  <0>={α0|α}={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaaGimaiabg6da+iabg2da9iaacUhacqaHXoqycaaIWaGaaiiFaiabeg7aHjabgIGiolabl2riHkaac2hacqGH9aqpcaGG7bGaaGimaiaac2haaaa@4761@ .

Zu 2.:  <1,X,, X n1 , X n >={ α 0 1+ α 1 X++ α n X n | α 0 ,, α n }= n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6441@ .

Zu 3.:   < e 1 ,, e n >={ α 1 e 1 ++ α n e n | α 1 ,, α n }={( α 1 α n )| α 1 ,, α n }= n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@74AD@ .

 

Ziel bei der Definition des Erzeugnisbegriffs war es, {v1,...,vk} durch möglichst wenige Vektoren zu einem Untervektorraum aufzustocken. Die folgende Bemerkung stellt sicher, dass diese zentrale Eigenschaft tatsächlich vorliegt .
 
Bemerkung:  Es sei (V, + , · ) ein Vektorraum und v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3ED7@ , dann ist < v1,...,vk > der kleinste Untervektorraum, der die Vektoren v1,...,vk enthält.

Beweis:

Die Aussage ist offensichtlich richtig für k = 0, so dass wir k > 0 annehmen dürfen. Wir zeigen nun:

  1. < v1,...,vk > ist ein Untervektorraum von V.
  2. v 1 ,, v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4697@ .
  3. Für jeden (anderen) Untervektorraum UV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabgkOimlaadAfaaaa@399A@ , der v1,...,vk enthält, gilt: < v 1 ,, v k >U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabgkOimlaadwfaaaa@415A@ .

Zu 1.: 
Es sind die drei definierenden Eigenschaften eines Untervektorraums nachzuweisen: 
  • 0=0 v 1 ++0 v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaaicdacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaaGimaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4A2F@ .
  • Für zwei Linearkombinationen  α 1 v 1 ++ α k v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4C50@ und β 1 v 1 ++ β k v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabek7aInaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4C54@ hat man nach Umsortieren und Zusammenfassen:
    α 1 v 1 ++ α k v k + β 1 v 1 ++ β k v k =( α 1 + β 1 ) v 1 ++( α 1 + β k ) v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7066@ .
  • Ist  α 1 v 1 ++ α k v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4C50@ , so gilt für jedes α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ :
    α( α 1 v 1 ++ α k v k )=(α α 1 ) v 1 ++(α α k ) v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6286@ .

Zu 2.: 
Für ein beliebiges i schreibt man unter Verwendung des Kronecker-Symbols δ i,j { 1,  falls  i=j 0,  falls  ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaadMgacaGGSaGaamOAaaqabaGccqGH9aqpdaGabaqaauaabaqaceaaaeaacaaIXaGaaiilaiaabAgacaqGHbGaaeiBaiaabYgacaqGZbGaamyAaiabg2da9iaadQgaaeaacaaIWaGaaiilaiaabAgacaqGHbGaaeiBaiaabYgacaqGZbGaamyAaiabgcMi5kaadQgaaaaacaGL7baaaaa@4F1B@
 

v i = δ i,1 v 1 ++ δ i,i v i ++ δ i,k v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGPbaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaaiilaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabes7aKnaaBaaaleaacaWGPbGaaiilaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaakiabgUcaRiablAciljabgUcaRiabes7aKnaaBaaaleaacaWGPbGaaiilaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@5C29@ .

Zu 3.: 
Wir müssen zeigen: Jede Linearkombination der Vektoren v1,...,vk liegt in U. Seien dazu α 1 ,, α k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqydaWgaaWcbaGaam4AaaqabaGccqGHiiIZcqWIDesOaaa@40B4@ vorgegeben. Als Untervektorraum ist U bzgl. der skalaren Multiplikation und bzgl. der Vektoraddition abgeschlossen, man kann also der Reihe nach argumentieren:

v 1 ,, v k U α 1 v 1 ,, α k v k U α 1 v 1 ++ α k v k U MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@63A6@
 

Beachte:

 
Bemerkung:  Es sei (V, + , · ) ein Vektorraum und v 1 ,, v k ,xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiaacYcacaWG4bGaeyicI4SaamOvaaaa@4084@ , dann gilt:
  1. < v 1 ,, v k >< v 1 ,, v k ,x> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabgkOimlabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+aaa@4AC8@ .
  2. Im Allgemeinen ist < v 1 ,, v k >< v 1 ,, v k ,x> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabgcMi5kabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+aaa@4A93@ .
  3. < < v 1 ,, v k >=< v 1 ,, v k ,x>x< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+iaaywW7cqGHuhY2caaMf8UaamiEaiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@5A66@ .

Beweis:

Zu 1.:
Wir müssen zeigen: Jede Linearkombination von v1,...,vk ist auch eine Linearkombination von v1,...,vk, x. Dies ist aber über die folgende Gleichheit leicht einzusehen:
 

α 1 v 1 ++ α k v k = α 1 v 1 ++ α k v k +0x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcqaHXoqydaWgaaWcbaGaaGymaaqabaGccaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaeqySde2aaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaakiabgUcaRiaaicdacaWG4baaaa@5218@ .

Zu 2.:
Hier reicht ein Gegenbeispiel. So gilt etwa in 𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@4495@ : <X><X,1> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamiwaiabg6da+iabgcMi5kabgYda8iaadIfacaGGSaGaaGymaiabg6da+aaa@3EED@ , denn 1<X> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgMGiplabgYda8iaadIfacqGH+aGpaaa@3B13@ .

Zu 3.:
" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":  Da jeder Erzeuger zu seinem Erzeugnis gehört, hat man hier: x< v 1 ,, v k ,x>=< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4C53@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":  Nach 1. ist bereits eine Teilmengenbeziehung erfüllt; bleibt zu zeigen < v 1 ,, v k >< v 1 ,, v k ,x> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabgoOijlabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiEaiabg6da+aaa@4AC6@ .
Nach Voraussetzung liegt x in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ , es gibt daher eine Darstellung von x in < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ , etwa:
 

x= β 1 v 1 ++ β k v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabek7aInaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHYoGydaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaaaa@442E@ .

Sei nun y= α 1 v 1 ++ α k v k + α k+1 x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiaadIhaaaa@4A76@ ein beliebiger Vektor aus < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E84@ . Folgt:

y = α 1 v 1 ++ α k v k + α k+1 ( β 1 v 1 ++ β k v k ) =( α 1 + α k+1 β 1 ) v 1 ++( α k + α k+1 β k ) v k < v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8130@

 

 
Beachte:

 

Neben dem (geeigneten) Verlängern oder Verkürzen von Erzeugersequenzen gibt es weitere Manipulationen, die das Erzeugnis unverändert lassen.
  
Definition:  Es sei (V, + , · ) ein Vektorraum und v 1 ,, v k V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabgIGiolaadAfaaaa@3ED7@ und α 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHe6aaWbaaSqabeaacqGHGjsUcaaIWaaaaaaa@3D2A@ , dann gilt:
  1. < v 1 ,, v i ,, v k >=< v 1 ,,α v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiabeg7aHjaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@5306@ .
  2. < v 1 ,, v i ,, v j ,, v k >=< v 1 ,, v i ,, v j + v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGQbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@5DAC@ .
  3. < v 1 ,, v i ,, v j ,, v k >=< v 1 ,, v i ,, v j +α v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F4B@ .
  4. < v 1 ,, v i ,, v j ,, v k >=< v 1 ,, v i ,, v j α v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F56@ .

Beweis:

Zu 1.:  Wir benutzen 3. aus der vorherigen Bemerkung zweimal (beachte im zweiten Schritt: α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyiyIKRaaGimaaaa@3A09@ ).

< v 1 ,, v i ,, v k > =   < v 1 ,, v i ,α v i ,, v k > (Verlängern möglich, da  α v i < v 1 ,, v i ,, v k >) =   < v 1 ,,α v i ,, v k > (Verkürzen möglich, da   v i = 1 α α v i < v 1 ,,α v i ,, v k >) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C2F7@

Zu 2.:   Der Fall i = j ist bereits in 1. bewiesen (mit a = 2). Für ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgcMi5kaadQgaaaa@398D@ gehen wir wie gerade vor:
 

< v 1 ,, v i ,, v j ,, v k > =   < v 1 ,, v i ,, v j , v j + v i ,, v k > (Verlängern möglich, da   v j + v i < v 1 ,, v k >) =   < v 1 ,, v i ,, v j + v i ,, v k > (Verkürzen möglich, da   v j =( v j + v i ) v i < v 1 ,, v i ,, v j + v i ,, v k >) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D9C7@

Zu 3.:  Durch Kombination von 1. und 2. erhält man:
 

< v 1 ,, v i ,, v j ,, v k > =   < v 1 ,,α v i ,, v j ,, v k > =   < v 1 ,,α v i ,, v j +α v i ,, v k > =   < v 1 ,, v i ,, v j +α v i ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaiabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOAaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaeaacqGH9aqpcaaMe8UaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGQbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aqaaaqaaiabg2da9iaaysW7cqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHXoqycaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aqaaaqaaiabg2da9iaaysW7cqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeqySdeMaamODamaaBaaaleaacaWGPbaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+aaaaaa@91AA@

 

4. ergibt sich als Spezialfall von 3. wenn man α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3788@ durch α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeqySdegaaa@3875@ ersetzt.
 

 
Beachte:
 

Erzeugnisse bleiben also unverändert, wenn man

  • einen Erzeuger mit einem Faktor α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyiyIKRaaGimaaaa@3A09@ multipliziert oder durch ihn dividiert.
     
  • zu einem Erzeuger einen anderen addiert oder von ihm subtrahiert.
     
  • zu einem Erzeuger ein Vielfaches eines anderen addiert oder von ihm subtrahiert.

 
Die folgenden Beispiele zeigen, wie man dieses Prinzip einsetzen kann. Oft lassen sich dadurch Erzeugnisse übersichtlicher und kürzer darstellen.

Beispiel:  
  1. In MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecuaaa@409B@ ist

    =<2 X 2 ,X X 2 ,X1,X+1> (Dividiere den ersten Vektor durch 2.) =< X 2 ,X X 2 ,X1,X+1> (Addiere den ersten Vektor zum zweiten.) =< X 2 ,X,X1,X+1> (Addiere den dritten Vektor zum vierten.) =< X 2 ,X,X1,2X> (2X< X 2 ,X,X1>,kann also weggelassen werden.) =< X 2 ,X,X1> (Subtrahieren vom dritten Vektor den zweiten.) =< X 2 ,X,1> (Multiplizieren den dritten Vektor mit −1.) =< X 2 ,X,1> = 2 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DD9@
     

  2. In 𝔽() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFfcVrcaGGOaGaeSyhHeQaaiykaaaa@4495@ ist

    = < sin 2 , cos 2 ,1> (Addiere den ersten Vektor zum zweiten.) =< sin 2 , sin 2 + cos 2 ,1> ( sin 2 + cos 2 =1< sin 2 ,1>.) =< sin 2 ,1>. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@966E@
     

  3. In 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ ist

    =<( 2 0 2 ),( 1 1 2 ),( 1 1 0 )> (Subtrahiere den ersten Vektor vom zweiten.) =<( 2 0 2 ),( 1 1 0 ),( 1 1 0 )> (Verkürze die Sequenz.) =<( 2 0 2 ),( 1 1 0 )> (Teile den ersten Vektor durch 2.) =<( 1 0 1 ),( 1 1 0 )> (Addiere den ersten Vektor zum zweiten.) =<( 1 0 1 ),( 0 1 1 )>. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@FC95@

 

 


 9.1
9.3.