9.16. Hesse-Darstellungen


In diesem Abschnitt betrachten wir ausschließlich affine Unterräume des m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ . Mit Hilfe der Orthonormalsysteme wird es uns gelingen, das Lot auf einen affinen Unterraum zu fällen, und so einen geometrisch ausgezeichneten Aufpunkt zur Verfügung zu stellen.

Zum zweiten soll in diesem Abschnitt das Zusammenspiel zwischen affinen Unterräumen und linearen Gleichungssystemen noch einmal beleuchtet werden. Dabei spielen Gleichungssysteme, die auf eine bestimmte Weise strukturiert sind, eine besondere Rolle bei der Berechnung der Lotvektoren.


 

Bemerkung und Definition:  Ist M=a+< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaadggacqGHRaWkcqGH8aapcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4226@ ein affiner Unterraum des m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ , so gibt es genau einen Vektor lM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabgIGiolaad2eaaaa@3930@ , der in < w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaaaa@4064@ liegt.

l heißt Lotvektor von M, seine Länge | l | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGSbaacaGLhWUaayjcSdaaaa@39FC@ nennen wir den Nullabstand von M.

Beweis:

Zur Existenz: Da m =< w 1 ,, w k >< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaOGaeyypa0JaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+iabgwPiflabgYda8iaadEhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadEhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpdaahaaWcbeqaaiabgwQiEbaaaaa@4EA8@ , gibt es eine Zerlegung für den Aufpunkt: a=a'+a" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadggacaGGNaGaey4kaSIaamyyaiaackcaaaa@3BD4@ . Wir setzen nun
 

l=a"< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9iaadggacaGGIaGaeyicI4SaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaaaa@456B@

und zeigen: lM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabgIGiolaad2eaaaa@3930@ :  Da  la=a"(a'+a")=a'< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabgkHiTiaadggacqGH9aqpcaWGHbGaaiOiaiabgkHiTiaacIcacaWGHbGaai4jaiabgUcaRiaadggacaGGIaGaaiykaiabg2da9iabgkHiTiaadggacaGGNaGaeyicI4SaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@4F29@ , hat man: la+< w 1 ,, w k >M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabgIGiolaadggacqGHRaWkcqGH8aapcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4JaeyicI4Saamytaaaa@4519@ . Man darf jetzt also den Aufpunkt von M austauschen:
 
M=l+< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaadYgacqGHRaWkcqGH8aapcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4231@ .

Zur Eindeutigkeit: Ist l' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacEcaaaa@3785@ ein weiterer Vektor der genannten Art, so erhält man die folgenden Informationen:

l'l< w 1 ,, w k >,   da   l'M=l+< w 1 ,, w k >, l'l< w 1 ,, w k > ,   da   l   und   l'   zum Untervektorraum   < w 1 ,, w k >    gehören. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9F40@

Also ist l'l< w 1 ,, w k >< w 1 ,, w k > ={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaacEcacqGHsislcaWGSbGaeyicI4SaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+iabgMIihlabgYda8iaadEhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadEhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpdaahaaWcbeqaaiabgwQiEbaakiabg2da9iaacUhacaaIWaGaaiyFaaaa@5367@ , d.h.: l=l' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9iaadYgacaGGNaaaaa@397C@ .
 

Beachte:

Die nachfolgenden Bemerkungen bereiten zunächst diese Schritte vor. Für den Rest dieses Abschnitts setzen wir stets 0<k<m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaadUgacqGH8aapcaWGTbaaaa@3A8D@ voraus. Ferner sei die Darstellung a+< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgUcaRiabgYda8iaadEhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadEhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@404E@ eines affinen Unterraums M stets so gewählt, dass die Erzeugersequenz w 1 ,, w k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaaaaa@3C70@ linear unabhängig ist.

 

Bemerkung:  Zu jeder linear unabhängigen Sequenz w 1 ,, w k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaaaaa@3C70@ des m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ gibt es eine ON-Sequenz n k+1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaaaaa@3E32@ , so dass
 
< w 1 ,, w k > =< n k+1 ,, n m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaGccqGH+aGpaaa@4BD3@

Die Sequenz n k+1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaaaaa@3E32@ nennen wir eine orthonormale Fortsetzung von w 1 ,, w k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaaaaa@3C70@ .

Beweis:

Über den Basisergänzungssatz und das Orthonormalisierungsverfahren gewinnen wir eine ON-Basis n 1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@3C60@ , mit < w 1 ,, w k >=< n 1 ,, n k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4817@ . Also hat man nach Ergebnissen aus 9.15:
 

< w 1 ,, w k > =< n 1 ,, n k > =< n k+1 ,, n m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaGccqGH+aGpaaa@574C@
 

Beachte:

 

Aufgabe:  Ermittle zu den folgenden Sequenzen jeweils eine orthonormale Fortsetzung.
 
  • ( 4 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGinaaqaaiabgkHiTiaaiodaaaaacaGLOaGaayzkaaaaaa@39E7@    ?
     

    <( 4 3 ) > =Ker( 4 3 )=<( 3 4 )>=< ( 3 4 ) ° > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaeWaaeaafaqabeGabaaabaGaaGinaaqaaiabgkHiTiaaiodaaaaacaGLOaGaayzkaaGaeyOpa4ZaaWbaaSqabeaacqGHLkIxaaGccqGH9aqpcaWGlbGaamyzaiaadkhacaaMc8+aaeWaaeaafaqabeGabaaabaGaaGinaaqaaiabgkHiTiaaiodaaaaacaGLOaGaayzkaaGaeyypa0JaeyipaWZaaeWaaeaafaqabeGabaaabaGaaG4maaqaaiaaisdaaaaacaGLOaGaayzkaaGaeyOpa4Jaeyypa0JaeyipaWZaaeWaaeaafaqabeGabaaabaGaaG4maaqaaiaaisdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGHWcaSaaGccqGH+aGpaaa@5584@ .

    Also ist 1 5 ( 3 4 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGynaaaadaqadaqaauaabeqaceaaaeaacaaIZaaabaGaaGinaaaaaiaawIcacaGLPaaaaaa@3A84@ eine orthonormale Fortsetzung von ( 4 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGinaaqaaiabgkHiTiaaiodaaaaacaGLOaGaayzkaaaaaa@39E7@ .

  • ( 1 2 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaaaa@39B3@    ?
     

    Zunächst errechnet man den Senkrechtraum

    <( 1 2 1 ) > =Ker( 1 2 1 )=><( 1 0 1 ),( 0 1 2 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0Jaam4saiaadwgacaWGYbGaaGPaVpaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIYaaabaGaaGymaaaaaiaawIcacaGLPaaacqGH9aqpcqGH8aapdaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaGimaaqaaiabgkHiTiaaigdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiaaicdaaeaacaaIXaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGH+aGpaaa@53E0@ .
     
    Durch Orthonormalisieren der Sequenz ( 1 0 1 ),( 0 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaicdaaeaacqGHsislcaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqadeaaaeaacaaIWaaabaGaaGymaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaaaaa@4004@ erhält man nun mit
     
    1 2 ( 1 0 1 ), 1 3 ( 1 1 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaaaleqaaaaakmaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaacaGGSaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIZaaaleqaaaaakmaabmaabaqbaeqabmqaaaqaaiabgkHiTiaaigdaaeaacaaIXaaabaGaeyOeI0IaaGymaaaaaiaawIcacaGLPaaaaaa@444A@ eine orthonormale Fortsetzung von ( 1 2 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaaaa@39B3@ .

  • ( 0 2 2 1 ),( 1 1 1 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeabbaaaaeaacaaIWaaabaGaaGOmaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaeeaaaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaaaaa@3FA6@    ?
     

    Über die Berechnung

    <( 0 2 2 1 ),( 1 1 1 1 ) > =Ker( 0 1 2 1 2 1 1 1 )=<( 1 0 1 2 ),( 0 1 1 0 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5EFA@
     
    und die anschließende Orthonormalisierung der Erzeugersequenz erhält man
    1 6 ( 1 0 1 2 ), 1 66 ( 1 6 5 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaI2aaaleqaaaaakmaabmaabaqbaeqabqqaaaaabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacaGGSaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaI2aGaaGOnaaWcbeaaaaGcdaqadaqaauaabeqaeeaaaaqaaiaaigdaaeaacaaI2aaabaGaeyOeI0IaaGynaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaaaaa@4696@ als eine orthonormale Fortsetzung von ( 0 2 2 1 ),( 1 1 1 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeabbaaaaeaacaaIWaaabaGaaGOmaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaeeaaaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaaaaa@3FA6@ .

Die Errechnung einer orthonormalen Fortsetzung ist, die Beispiele zeigen dies deutlich, ist i.a. recht aufwendig. In zwei Sonderfällen jedoch gibt es eine schnelle Methode, eine orthonormale Fortsetzung zu finden:
 

Bemerkung: 
  1. Ist w eine linear unabhängige Sequenz der Länge 1 in 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ , so ist ( w 2 w 1 ) ° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaam4DamaaBaaaleaacaaIYaaabeaaaOqaaiabgkHiTiaadEhadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHWcaSaaaaaa@3E60@ eine orthonormale Fortsetzung von w.
     
  2. Ist v,w eine linear unabhängige Sequenz der Länge 2 in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ , so ist (v×w)° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAhacqGHxdaTcaWG3bGaaiykaiabgclaWcaa@3D3C@ eine orthonormale Fortsetzung von v,w.
    Dabei ist das Vektorprodukt (oder auch Kreuzprodukt) der Vektoren v und w gegeben durch
     
    v×w=( v 2 w 3 v 3 w 2 v 1 w 3 + v 3 w 1 v 1 w 2 v 2 w 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5760@ .

Beweis:

Zu 1.:  Mit w ist auch ( w 2 w 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaam4DamaaBaaaleaacaaIYaaabeaaaOqaaiabgkHiTiaadEhadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaaaaa@3C47@ von 0 verschieden, so dass ( w 2 w 1 ) ° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaam4DamaaBaaaleaacaaIYaaabeaaaOqaaiabgkHiTiaadEhadaWgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHWcaSaaaaaa@3E60@ wohldefiniert ist. Die Rechnung w 1 w 2 + w 2 ( w 1 )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaadEhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWG3bWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiabgkHiTiaadEhadaWgaaWcbaGaaGymaaqabaGccaGGPaGaeyypa0JaaGimaaaa@4287@ bestätigt w ( w 2 w 1 ) ° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEnaabmaabaqbaeqabiqaaaqaaiaadEhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHsislcaWG3bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiSaalaaaaa@410D@ und damit
 

<w>=< ( w 2 w 1 ) ° > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaam4Daiabg6da+iabg2da9iabgYda8maabmaabaqbaeqabiqaaaqaaiaadEhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHsislcaWG3bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiSaalaaOGaeyOpa4daaa@4484@ .

Zu 2.:  Der Beweis ergibt sich aus den folgenden Eigenschaften des Vektorprodukts: 

v,wlinear unabhängigv×w0 v×wvv×ww. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaaqaaiaadAhacaGGSaGaam4DaiaaykW7caqGSbGaaeyAaiaab6gacaqGLbGaaeyyaiaabkhacaqGGaGaaeyDaiaab6gacaqGHbGaaeOyaiaabIgacaqGKdGaaeOBaiaabEgacaqGPbGaae4zaiaaywW7cqGHshI3caaMf8UaamODaiabgEna0kaadEhacqGHGjsUcaaIWaaabaGaamODaiabgEna0kaadEhacqGHLkIxcaWG2bGaaGzbVlabgEIizlaaywW7caWG2bGaey41aqRaam4DaiabgwQiEjaadEhacaGGUaaaaaaa@6906@

Diese und weitere Eigenschaften sind auf einer eigenen Seite notiert und nachgewiesen.
 

 

Bemerkung und Bezeichnung:  Ist M=a+< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaadggacqGHRaWkcqGH8aapcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4226@ ein affiner Unterraum des m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ , so gilt für jede orthonormale Fortsetzung n k+1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaaaaa@3E32@ von w 1 ,, w k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaam4DamaaBaaaleaacaWGRbaabeaaaaa@3C70@ :
 
xM( n k+1 n m )(xa)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaad2eacaaMf8Uaeyi1HSTaaGzbVpaabmaabaqbaeqabmqaaaqaaiaad6gadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaaGcbaGaeSO7I0eabaGaamOBamaaBaaaleaacaWGTbaabeaaaaaakiaawIcacaGLPaaacaGGOaGaamiEaiabgkHiTiaadggacaGGPaGaeyypa0JaaGimaaaa@4DF4@ .

Die Vektoren n k+1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaaaaa@3E32@ bezeichnen wir in diesem Zusammenhang als Normalenvektoren von M und nennen das Gleichungssystem eine Hesse-Darstellung von M.

Beweis:

xM xa< w 1 ,, w k >=< n k+1 ,, n m > =Ker( n k+1 n m ) ( n k+1 n m )(xa)=0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7936@
 

Beachte:

Mit Hilfe einer Hesse-Darstellung, bzw. den damit gegebenen Normalenvektoren, ist es nun leicht, den Lotvektor und den Nullabstand zu berechnen:
 

Bemerkung:  Sind n k+1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaaaaa@3E32@ Normalenvektoren des affinen Unterraums M=a+< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaadggacqGHRaWkcqGH8aapcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG3bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaa@4226@ , so gilt für den Lotvektor l:
 
  1. l=( n k+1 ·a) n k+1 ++( n m ·a) n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiabl+y6NjaadggacaGGPaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaGGOaGaamOBamaaBaaaleaacaWGTbaabeaakiabl+y6NjaadggacaGGPaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@4FBC@ .
     
  2. | l |=| ( n k+1 n m )a |= ( n k+1 ·a) 2 ++ ( n m ·a) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGSbaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaadaqadaqaauaabeqadeaaaeaacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaaaOqaaiabl6Uinbqaaiaad6gadaWgaaWcbaGaamyBaaqabaaaaaGccaGLOaGaayzkaaGaamyyaaGaay5bSlaawIa7aiabg2da9iaacIcacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiabl+y6NjaadggacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeSOjGSKaey4kaSIaaiikaiaad6gadaWgaaWcbaGaamyBaaqabaGccqWIpM+zcaWGHbGaaiykamaaCaaaleqabaGaaGOmaaaaaaa@5D58@ .

Beweis:

Zu 1.:  Da der Lotvektor l eindeutig bestimmt ist, reicht es zu zeigen:
 

( n k+1 ·a) n k+1 ++( n m ·a) n m M< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaOGaeS4JPFMaamyyaiaacMcacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiabgUcaRiablAciljabgUcaRiaacIcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaeS4JPFMaamyyaiaacMcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaeyicI4SaamytaiabgMIihlabgYda8iaadEhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadEhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpdaahaaWcbeqaaiabgwQiEbaaaaa@5C3E@ .

 

  • Nach Konstruktion ist ( n k+1 ·a) n k+1 ++( n m ·a) n m < n k+1 ,, n m >=< w 1 ,, w k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6533@ .
     
  • Die Zeilenvektoren der Matrix ( n k+1 n m ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaaakeaacqWIUlstaeaacaWGUbWaaSbaaSqaaiaad2gaaeqaaaaaaOGaayjkaiaawMcaaaaa@3F40@ bilden ein ON-System. Wendet man eine solche Matrix auf einen Vektor x< n k+1 ,, n m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaad6gadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaeyOpa4daaa@42C9@ an, so erhält man nach einem Ergebnis in 9.15 genau den Koordinatenvektor von x:
     
    ( n k+1 n m )(( n k+1 ·a) n k+1 ++( n m ·a) n m )=( n k+1 ·a n m ·a )=( n k+1 n m )a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@74CB@ .

    Damit aber ist ( n k+1 ·a) n k+1 ++( n m ·a) n m M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6gadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaOGaeS4JPFMaamyyaiaacMcacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaakiabgUcaRiablAciljabgUcaRiaacIcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaeS4JPFMaamyyaiaacMcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaeyicI4Saamytaaaa@5025@ .

    Zu 2.:  Der Lotvektor l gehört zu M, erfüllt also die Hesse-Darstellung von M. Die Matrix ( n k+1 n m ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaaakeaacqWIUlstaeaacaWGUbWaaSbaaSqaaiaad2gaaeqaaaaaaOGaayjkaiaawMcaaaaa@3F40@ operiert auf dem Erzeugnis ihrer Zeilenvektoren längentreu. Also hat man insgesamt:
     

    | l |=| ( n k+1 n m )l |=| ( n k+1 n m )a | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGSbaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaadaqadaqaauaabeqadeaaaeaacaWGUbWaaSbaaSqaaiaadUgacqGHRaWkcaaIXaaabeaaaOqaaiabl6Uinbqaaiaad6gadaWgaaWcbaGaamyBaaqabaaaaaGccaGLOaGaayzkaaGaamiBaaGaay5bSlaawIa7aiabg2da9maaemaabaWaaeWaaeaafaqabeWabaaabaGaamOBamaaBaaaleaacaWGRbGaey4kaSIaaGymaaqabaaakeaacqWIUlstaeaacaWGUbWaaSbaaSqaaiaad2gaaeqaaaaaaOGaayjkaiaawMcaaiaadggaaiaawEa7caGLiWoaaaa@56D1@ .
     

Beachte:

 

Wir betrachten nun einige Beispiele. Für den 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ und den 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ nutzen wir dabei die oben vorgestellten konstruktiven Möglichkeiten zur Ermittlung einer orthonormalen Fortsetzung!

 

Beispiel 1 (Geraden in 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ ):  Ist g=a+<w> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaadggacqGHRaWkcqGH8aapcaWG3bGaeyOpa4daaa@3CAB@ eine Gerade in 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ , so ist n= ( w 2 w 1 ) ° = 1 |w| ( w 2 w 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9maabmaabaqbaeqabiqaaaqaaiaadEhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHsislcaWG3bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiSaalaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaiiFaiaadEhacaGG8baaamaabmaabaqbaeqabiqaaaqaaiaadEhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHsislcaWG3bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaaa@4B8E@ ein Normalenvektor von g und die Gleichungen (die Matrix ( n ) ist hier einzeilig!)
 
( w 2 |w| w 1 |w| )(xa)=0 ( w 2 |w| w 1 |w| )x=( w 2 |w| w 1 |w| )a w 2 |w| x 1 w 1 |w| x 2 = w 2 a 1 w 1 a 2 |w| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7F72@

sind die verschiedenen Formen einer Hesse Darstellung von g. Ferner ist
  
l=(n·a)n= w 2 a 1 w 1 a 2 |w| 1 |w| ( w 2 w 1 )= 1 |w | 2 ( w 2 2 a 1 w 1 w 2 a 2 w 1 w 2 a 1 + w 1 2 a 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@75E8@ der Lotvektor und
  |l|=|(n·a)n|= | w 2 a 1 w 1 a 2 | |w| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadYgacaGG8bGaeyypa0JaaiiFaiaacIcacaWGUbGaeS4JPFMaamyyaiaacMcacaWGUbGaaiiFaiabg2da9maalaaabaGaaiiFaiaadEhadaWgaaWcbaGaaGOmaaqabaGccaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaam4DamaaBaaaleaacaaIXaaabeaakiaadggadaWgaaWcbaGaaGOmaaqabaGccaGG8baabaGaaiiFaiaadEhacaGG8baaaaaa@50FE@ der Nullabstand von g.

 

So errechnet man etwa für g=( 0 3 )+<( 1 2 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9maabmaabaqbaeqabiqaaaqaaiaaicdaaeaacaaIZaaaaaGaayjkaiaawMcaaiabgUcaRiabgYda8maabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaiabg6da+aaa@40E3@ :
 
n= ( 2 1 ) ° = 1 5 ( 2 1 ), ( n )a=( 2 5 1 5 )( 0 3 )= 3 5 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5736@
 

Also sind die Gleichungen
 

( 2 5 1 5 )(x( 0 3 ))=0 ( 2 5 1 5 )x= 3 5 2 5 x 1 1 5 x 2 = 3 5 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5A6C@

Hesse-Darstellungen von g. l= 3 5 1 5 ( 2 1 )= 3 5 ( 2 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9iabgkHiTmaalaaabaGaaG4maaqaamaakaaabaGaaGynaaWcbeaaaaGcdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaiwdaaSqabaaaaOWaaeWaaeaafaqabeGabaaabaGaaGOmaaqaaiabgkHiTiaaigdaaaaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIZaaabaGaaGynaaaadaqadaqaauaabeqaceaaaeaacqGHsislcaaIYaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@46B3@ ist der Lotvektor und |l|= 3 5 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadYgacaGG8bGaeyypa0ZaaSaaaeaacaaIZaaabaWaaOaaaeaacaaI1aaaleqaaaaaaaa@3B87@ ist der Nullabstand von g.  

 

Beispiel 2 (Ebenen in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ ):  Ist E=a+<v,w> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2da9iaadggacqGHRaWkcqGH8aapcaWG2bGaaiilaiaadEhacqGH+aGpaaa@3E34@ eine Ebene in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ , so ist n=(v×w)° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaSaaa@3F35@ ein Normalenvektor von E und (da hier die Matrix ( n ) einzeilig ist) sind
 
(v×w)°(xa)=0 (v×w)°x=(v×w)°a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaaqaaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaScaGGOaGaamiEaiabgkHiTiaadggacaGGPaGaeyypa0JaaGimaaqaaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaScaWG4bGaeyypa0JaaiikaiaadAhacqGHxdaTcaWG3bGaaiykaiabgclaWkaadggaaaaaaa@54C1@

die verschiedenen Formen einer Hesse Darstellung von E. Ferner ist
 
l=(n·a)n=((v×w)°·a)(v×w)°= (v×w)°·a |v×w | 2 (v×w) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9iaacIcacaWGUbGaeS4JPFMaamyyaiaacMcacaWGUbGaeyypa0JaaiikaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaScqWIpM+zcaWGHbGaaiykaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaScqGH9aqpdaWcaaqaaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqGHWcaScqWIpM+zcaWGHbaabaGaaiiFaiaadAhacqGHxdaTcaWG3bGaaiiFamaaCaaaleqabaGaaGOmaaaaaaGccaGGOaGaamODaiabgEna0kaadEhacaGGPaaaaa@6AF7@   der Lotvektor und
 
|l|=|n·a|=| (v×w)·a |v×w| | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadYgacaGG8bGaeyypa0JaaiiFaiaad6gacqWIpM+zcaWGHbGaaiiFaiabg2da9iaacYhadaWcaaqaaiaacIcacaWG2bGaey41aqRaam4DaiaacMcacqWIpM+zcaWGHbaabaGaaiiFaiaadAhacqGHxdaTcaWG3bGaaiiFaaaacaGG8baaaa@520A@   der Nullabstand von E.

So hat man etwa für E=( 2 1 2 )+<( 0 2 3 ),( 1 0 3 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2da9maabmaabaqbaeqabmqaaaqaaiaaikdaaeaacqGHsislcaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkcqGH8aapdaqadaqaauaabeqadeaaaeaacaaIWaaabaGaeyOeI0IaaGOmaaqaaiaaiodaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaG4maaaaaiaawIcacaGLPaaacqGH+aGpaaa@497E@ :
 

n=(( 0 2 3 )×( 1 0 3 ))°= 1 7 ( 6 3 2 ), ( n )a=( 6 7 3 7 2 7 )( 2 1 2 )= 13 7 . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6305@

Also erhält man
 

( 6 7 3 7 2 7 )(x( 2 1 2 ))=0 ( 6 7 3 7 2 7 )x= 13 7 6 7 x 1 + 3 7 x 2 + 2 7 x 3 = 13 7 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@60A3@

als Hesse-Darstellungen, l= 13 7 1 7 ( 6 3 2 )= 13 49 ( 6 3 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9maalaaabaGaaGymaiaaiodaaeaacaaI3aaaamaalaaabaGaaGymaaqaaiaaiEdaaaWaaeWaaeaafaqabeWabaaabaGaaGOnaaqaaiaaiodaaeaacaaIYaaaaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaiaaiodaaeaacaaI0aGaaGyoaaaadaqadaqaauaabeqadeaaaeaacaaI2aaabaGaaG4maaqaaiaaikdaaaaacaGLOaGaayzkaaaaaa@4766@ als Lotvektor und |l|= 13 7 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadYgacaGG8bGaeyypa0ZaaSaaaeaacaaIXaGaaG4maaqaaiaaiEdaaaaaaa@3C29@ als Nullabstand.
 

 

 

Beispiel 3 (Geraden in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ ):  g=( 2 1 3 )+<( 2 2 1 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9maabmaabaqbaeqabmqaaaqaaiaaikdaaeaacaaIXaaabaGaaG4maaaaaiaawIcacaGLPaaacqGHRaWkcqGH8aapdaqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGOmaaqaaiaaigdaaaaacaGLOaGaayzkaaGaeyOpa4daaa@4260@ ist eine Gerade in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ . Zunächst ermitteln wir (über den Gauß-Algorithmus)
 
<( 2 2 1 ) > =Ker( 2 2 1 )=<( 1 0 2 ),( 0 1 2 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaeWaaeaafaqabeWabaaabaGaaGOmaaqaaiaaikdaaeaacaaIXaaaaaGaayjkaiaawMcaaiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0Jaam4saiaadwgacaWGYbGaaGPaVpaabmaabaqbaeqabeWaaaqaaiaaikdaaeaacaaIYaaabaGaaGymaaaaaiaawIcacaGLPaaacqGH9aqpcqGH8aapdaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaGimaaqaaiabgkHiTiaaikdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiaaicdaaeaacaaIXaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGH+aGpaaa@53E3@ ,

und  gewinnen durch Orthonormalisieren
 
1 5 ( 1 0 2 )und 1 3 5 ( 4 5 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaI1aaaleqaaaaakmaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacaaMf8UaaeyDaiaab6gacaqGKbGaaGzbVpaalaaabaGaaGymaaqaaiaaiodadaGcaaqaaiaaiwdaaSqabaaaaOWaaeWaaeaafaqabeWabaaabaGaeyOeI0IaaGinaaqaaiaaiwdaaeaacqGHsislcaaIYaaaaaGaayjkaiaawMcaaaaa@4A51@

als Normalenvektoren. Also sind die Gleichungen
 
( 1 5 0 2 5 4 3 5 5 3 5 2 3 5 )(x( 2 1 3 ))=0 ( 1 5 0 2 5 4 3 5 5 3 5 2 3 5 )x=( 4 5 9 3 5 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@68A2@

Hesse-Darstellungen,
 

l= 1 5 (( 1 0 2 )( 2 1 3 ))( 1 0 2 )+ 1 45 (( 4 5 2 )·( 2 1 3 ))( 4 5 2 )= 4 5 ( 1 0 2 ) 9 45 ( 4 5 2 )=( 0 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2da9maalaaabaGaaGymaaqaaiaaiwdaaaGaaiikamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGHflY1daqadaqaauaabeqadeaaaeaacaaIYaaabaGaaGymaaqaaiaaiodaaaaacaGLOaGaayzkaaGaaiykamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI0aGaaGynaaaacaGGOaWaaeWaaeaafaqabeWabaaabaGaeyOeI0IaaGinaaqaaiaaiwdaaeaacqGHsislcaaIYaaaaaGaayjkaiaawMcaaiabl+y6NnaabmaabaqbaeqabmqaaaqaaiaaikdaaeaacaaIXaaabaGaaG4maaaaaiaawIcacaGLPaaacaGGPaWaaeWaaeaafaqabeWabaaabaGaeyOeI0IaaGinaaqaaiaaiwdaaeaacqGHsislcaaIYaaaaaGaayjkaiaawMcaaiabg2da9iabgkHiTmaalaaabaGaaGinaaqaaiaaiwdaaaWaaeWaaeaafaqabeWabaaabaGaaGymaaqaaiaaicdaaeaacqGHsislcaaIYaaaaaGaayjkaiaawMcaaiabgkHiTmaalaaabaGaaGyoaaqaaiaaisdacaaI1aaaamaabmaabaqbaeqabmqaaaqaaiabgkHiTiaaisdaaeaacaaI1aaabaGaeyOeI0IaaGOmaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacaaIWaaabaGaeyOeI0IaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaaaaa@7734@

der Lotvektor und
 

|l|= 5 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadYgacaGG8bGaeyypa0ZaaOaaaeaacaaI1aaaleqaaaaa@3ABA@

der Nullabstand von g.
 

 


 9.15
9.17.