9.8. Matrizen


Matrizen sind ein sehr praktisches Hilfsmittel zur Lösung linearer Gleichungssysteme. Im nächsten Abschnitt wird dies ausführlich dargestellt. Gleichzeitig aber sind Matrizen auch als Funktionen eines bestimmten Typs einsetzbar. Diese Doppelgesichtigkeit spiegelt die innere Verwandtschaft dieser beiden Themen.

In diesem Teil wird zunächst der Funktionenaspekt betont.

Matrizen sind (hier) an den n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ gebunden. Über das Basiskonzept ist eine Ausweitung auf endliche (!) Vektorräume möglich. Wir gehen darauf nicht ein.
 
Definition:  Ein rechteckiges Zahlenschema der Form

( a 11 a 1m a n1 a nm ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWadaaabaGaamyyamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacqWIVlctaeaacaWGHbWaaSbaaSqaaiaaigdacaWGTbaabeaaaOqaaiabl6UinbqaaiablgVipbqaaiabl6UinbqaaiaadggadaWgaaWcbaGaamOBaiaaigdaaeqaaaGcbaGaeS47IWeabaGaamyyamaaBaaaleaacaWGUbGaamyBaaqabaaaaaGccaGLOaGaayzkaaaaaa@4C5A@

nennen wir eine n×m   - Matrix ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaacaWGUbGaey41aqRaamyBaiaaysW7caqGTaGaaeiiaiaab2eacaqGHbGaaeiDaiaabkhacaqGPbGaaeiEaaaaaaa@425C@ (über MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ ). Ist n=m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaad2gaaaa@38D4@ , sprechen wir von einer quadratischen Matrix.

Beachte:

 

 
Beispiel:
  • ( 2 0 3 2 1 2 0 0 2 2 4 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWaeaaaaeaacaaIYaaabaGaaGimaaqaaiabgkHiTiaaiodaaeaacaaIYaaabaGaaGymaaqaaiaaikdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiaaikdaaeaacaaIYaaabaGaaGinaaqaaiaaigdaaaaacaGLOaGaayzkaaaaaa@4232@ ist eine 3 × 4 - Matrix.
     
  • ( 2 13 4 0 3.5 0.2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeqagaaaaeaacaaIYaaabaGaaGymaiaaiodaaeaacqGHsislcaaI0aaabaGaaGimaaqaaiaaiodacaGGUaGaaGynaaqaaiaaicdacaGGUaGaaGOmaaaaaiaawIcacaGLPaaaaaa@4076@ ist eine 1 × 6 - Matrix.
     
  • ( 3 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaG4maaqaaiaaiodaaaaacaGLOaGaayzkaaaaaa@38F9@ ist eine 2 × 1 - Matrix.

 

Zwei wichtige Beispiele quadratischer Matrizen zeichnen wir durch einen Namen aus:
Beispiel:
  • Die Matrix E= ( δ ij ) 1i,jn =( 1 0 0 0 1 0 0 0 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiabg2da9iaacIcacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykamaaBaaaleaacaaIXaGaeyizImQaamyAaiaacYcacaWGQbGaeyizImQaamOBaaqabaGccqGH9aqpdaqadaqaauaabeqaeqaaaaaabaGaaGymaaqaaiaaicdaaeaacqWIVlctaeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacqWIVlctaeaacaaIWaaabaGaeSO7I0eabaGaeSO7I0eabaGaeSy8I8eabaGaeSO7I0eabaGaaGimaaqaaiabl+UimbqaaiaaicdaaeaacaaIXaaaaaGaayjkaiaawMcaaaaa@5B68@ ist die Einheitsmatrix (des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ ).
    Sowohl die Spalten- wie auch Zeilenvektoren bilden hier jeweils die Standardbasis des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ :
    ( δ ij )=( e 1 e n )=( e 1 e n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaeyypa0JaaiikaiaadwgadaWgaaWcbaGaaGymaaqabaGccqWIMaYscaWGLbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabg2da9maabmaabaqbaeqabmqaaaqaaiaadwgadaWgaaWcbaGaaGymaaqabaaakeaacqWIUlstaeaacaWGLbWaaSbaaSqaaiaad6gaaeqaaaaaaOGaayjkaiaawMcaaaaa@4AE3@ .
  • Die Matrix 0= (0) 1i,jn =( 0 0 0 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaacIcacaaIWaGaaiykamaaBaaaleaacaaIXaGaeyizImQaamyAaiaacYcacaWGQbGaeyizImQaamOBaaqabaGccqGH9aqpdaqadaqaauaabeqadmaaaeaacaaIWaaabaGaeS47IWeabaGaaGimaaqaaiabl6UinbqaaiablgVipbqaaiabl6UinbqaaiaaicdaaeaacqWIVlctaeaacaaIWaaaaaGaayjkaiaawMcaaaaa@50D0@ ist die Nullmatrix (des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ ).
    Jeder Zeilen- und Spaltenvektoren ist hier der Nullvektor:
    (0)=(00)=( 0 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGPaGaeyypa0JaaiikaiaaicdacqWIVlctcaaIWaGaaiykaiabg2da9maabmaabaqbaeqabmqaaaqaaiaaicdaaeaacqWIUlstaeaacaaIWaaaaaGaayjkaiaawMcaaaaa@43BD@ .

Bei der folgenden Konstruktion treten die Spaltenvektoren einer Matrix in den Vordergrund.
Definition:  ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ sei eine n×m   - Matrix MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhaaaa@424C@ . Für einen beliebigen Vektor x m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamyBaaaaaaa@3AF9@ setzen wir:

( a ij )x= j=1 m x j a j = x 1 a 1 ++ x m a m n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaadIhacqGH9aqpdaaeWbqaaiaadIhadaWgaaWcbaGaamOAaaqabaGccaWGHbWaaSbaaSqaaiabgkci3kaadQgaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aOGaeyypa0JaamiEamaaBaaaleaacaaIXaaabeaakiaadggadaWgaaWcbaGaeyOiGCRaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaWG4bWaaSbaaSqaaiaad2gaaeqaaOGaamyyamaaBaaaleaacqGHIaYTcaWGTbaabeaakiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@5ACB@

Der Vektor ( a ij )x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaadIhaaaa@3B38@ (gelesen: " ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ mal x") ist also eine Linearkombination der Spaltenvektoren von ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ , wobei die Koordinaten des Vektors x den Koeffizientensatz liefern.

Die gerade eingeführte Methode ordnet jedem Vektor x m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamyBaaaaaaa@3AF9@ genau einen Ergebnisvektor ( a ij )x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaadIhacqGHiiIZcqWIDesOdaahaaWcbeqaaiaad6gaaaaaaa@3F4C@ zu! Wir können daher eine n×m   - Matrix MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhaaaa@424C@ auch auffassen als eine Funktion von m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ nach n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ :

( a ij ): m n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaacQdacqWIDesOdaahaaWcbeqaaiaad2gaaaGccqGHsgIRcqWIDesOdaahaaWcbeqaaiaad6gaaaaaaa@420F@ .

Unter diesem Gesichtspunkt, nennen wir den Vektor ( a ij )x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaadIhaaaa@3B38@ auch einen Bildvektor (dann gelesen: " ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ von x"), der durch Anwenden der Matrix ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ einstanden sei.

 

 
Beispiel: 
  • ( 2 0 3 2 1 2 0 0 2 2 4 1 )( 0 2 1 3 )=0( 2 1 2 )+2( 0 2 2 )( 3 0 4 )+3( 2 0 1 )=( 9 4 3 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@699E@ .
     
  • ( 3 1 2 0 0 4 )( 6 3 )=6( 3 2 0 )+3( 1 0 4 )=( 21 12 12 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWacaaabaGaaG4maaqaaiaaigdaaeaacaaIYaaabaGaaGimaaqaaiaaicdaaeaacaaI0aaaaaGaayjkaiaawMcaaiaaykW7daqadaqaauaabeqaceaaaeaacaaI2aaabaGaaG4maaaaaiaawIcacaGLPaaacqGH9aqpcaaI2aGaaGPaVpaabmaabaqbaeqabmqaaaqaaiaaiodaaeaacaaIYaaabaGaaGimaaaaaiaawIcacaGLPaaacqGHRaWkcaaIZaGaaGPaVpaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIWaaabaGaaGinaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacaaIYaGaaGymaaqaaiaaigdacaaIYaaabaGaaGymaiaaikdaaaaacaGLOaGaayzkaaaaaa@559D@ .
     
  • ( 0 1 1 0 )( x 1 x 2 )= x 1 ( 0 1 )+ x 2 ( 1 0 )=( x 2 x 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@53BE@ .

 
Bemerkung:  ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ sei eine beliebige n×m   - Matrix MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhaaaa@424C@ , (0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGPaaaaa@37FC@ die Nullmatrix und ( δ ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaaaa@3AFA@ die Einheitsmatrix des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ . Dann gilt:
  1. ( a ij )0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaaicdacqGH9aqpcaaIWaaaaa@3CB5@ .
  2. (0)x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGPaGaamiEaiabg2da9iaaicdaaaa@3AB9@ .
  3. ( δ ij )x=x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiabg2da9iaadIhaaaa@3DFA@ .

Beweis:

  ( a ij )0=0 a 1 ++0 a m =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaaicdacqGH9aqpcaaIWaGaeyyXICTaamyyamaaBaaaleaacqGHIaYTcaaIXaaabeaakiabgUcaRiablAciljabgUcaRiaaicdacqGHflY1caWGHbWaaSbaaSqaaiabgkci3kaad2gaaeqaaOGaeyypa0JaaGimaaaa@4D98@ .
  (0)x= x 1 0++ x n 0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaicdacaGGPaGaamiEaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGccqGHflY1caaIWaGaey4kaSIaeSOjGSKaey4kaSIaamiEamaaBaaaleaacaWGUbaabeaakiabgwSixlaaicdacqGH9aqpcaaIWaaaaa@48C1@ .
  ( δ ij )x= x 1 e 1 ++ x n e n =x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWGLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeSOjGSKaey4kaSIaamiEamaaBaaaleaacaWGUbaabeaakiabgwSixlaadwgadaWgaaWcbaGaamOBaaqabaGccqGH9aqpcaWG4baaaa@4E7C@ .

Die beiden letzten Beispiele sind unter dem Funktionengesichtspunkt interessant:
Offensichtlich stellt die Nullmatrix die Nullfunktion des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ und die Einheitsmatrix die Identität des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ dar!

Die Matrixanwendung ist mit den Rechenregeln des m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ verträglich:
Bemerkung:  ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ sei eine n×m   - Matrix MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhaaaa@424C@ , x,y m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3CA7@ und α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ . Dann gilt:
  1. ( a ij )(αx)=α(( a ij )x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaacIcacqaHXoqycaWG4bGaaiykaiabg2da9iabeg7aHjaacIcacaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiaacMcaaaa@477D@
  2. ( a ij )(x+y)=( a ij )x+( a ij )y MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGH9aqpcaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiabgUcaRiaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacaWG5baaaa@4AF8@

Beweis:

Zu 1.: ( a ij )(αx)=(α x 1 ) a 1 ++(α x m ) a m =α( x 1 a 1 ++ x m a m )=α(( a ij )x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7769@ .

Zu 2.:  ( a ij )(x+y) =( x 1 + y 1 ) a 1 ++( x m + y m ) a m = x 1 a 1 ++ x m a m + y 1 a 1 ++ y m a m =( a ij )x+( a ij )y. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8DF9@

 


Die folgende Begriffsbildung betont den Funktionencharakter von Matrizen.
Definition:  ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ sei eine n×m   - Matrix MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhaaaa@424C@ , dann heißt die Menge
  1. Ker( a ij )={x m |( a ij )x=0} m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacqGH9aqpcaGG7bGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamyBaaaakiaacYhacaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiabg2da9iaaicdacaGG9bGaeyOGIWSaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@5331@ der Kern von ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ .
  2. Im( a ij )={( a ij )x|x m } n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2gacaaMc8UaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiabg2da9iaacUhacaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiaacYhacaWG4bGaeyicI4SaeSyhHe6aaWbaaSqabeaacaWGTbaaaOGaaiyFaiabgkOimlabl2riHoaaCaaaleqabaGaamOBaaaaaaa@5081@ das Bild von ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ .

Beachte:

 
Bemerkung:  Ker( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcaaaa@3E77@ ist ein Untervektorraum von n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ .

Beweis:

  1. Da ( a ij )0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaaicdacqGH9aqpcaaIWaaaaa@3CB5@ , ist 0Ker( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaaaa@40B5@ , also Ker( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacqGHGjsUcqGHfiIXaaa@41B7@ .
     
  2. Sei xKer( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaaaa@40F8@ und α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ , dann ist:
    ( a ij )(αx)=α(( a ij )x)=α0=0αxKer( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaacIcacqaHXoqycaWG4bGaaiykaiabg2da9iabeg7aHjaacIcacaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiaacMcacqGH9aqpcqaHXoqycaaIWaGaeyypa0JaaGimaiaaywW7cqGHshI3caaMf8UaeqySdeMaamiEaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaaaa@5EC3@ .
     
  3. Für x,yKer( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI4Saam4saiaadwgacaWGYbGaaGPaVlaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcaaaa@42A6@ hat man:
    ( a ij )(x+y)=( a ij )x+( a ij )y=0+0=0x+yKer( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGH9aqpcaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaamiEaiabgUcaRiaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacaWG5bGaeyypa0JaaGimaiabgUcaRiaaicdacqGH9aqpcaaIWaGaaGzbVlabgkDiElaaywW7caWG4bGaey4kaSIaamyEaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaaaa@627C@ .

 
Bemerkung:   Im( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2gacaaMc8UaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3D86@ ist ein Untervektorraum von m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaaaa@3878@ . Es gilt sogar:

Im( a ij )=< a 1 ,, a m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2gacaaMc8UaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiabg2da9iabgYda8iaadggadaWgaaWcbaGaeyOiGCRaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadggadaWgaaWcbaGaeyOiGCRaamyBaaqabaGccqGH+aGpaaa@4A09@ ,

d.h. Im( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2gacaaMc8UaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3D86@ ist das Erzeugnis der Spaltenvektoren der Matrix ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@

Beweis:

yIm( a ij ) es gibt ein   x m    mit   y=( a ij )x es gibt ein   x m    mit   y= x 1 a 1 ++ x m a m y< a 1 ,, a m >. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A396@

Mit Hilfe der Begriffe Bild und Kern lassen sich neue Kriterien zur Maximalität bzw. zur linearen Unabhängigkeit formulieren:
Bemerkung:  Für Vektoren v 1 ,, v m n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGTbaabeaakiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@408E@ gilt:
  1. v 1 ,, v m    maximalIm( v 1 v m )= n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGTbaabeaakiaaysW7caqGTbGaaeyyaiaabIhacaqGPbGaaeyBaiaabggacaqGSbGaaGzbVlabgsDiBlaaywW7ciGGjbGaaiyBaiaaykW7caGGOaGaamODamaaBaaaleaacaaIXaaabeaakiablAciljaadAhadaWgaaWcbaGaamyBaaqabaGccaGGPaGaeyypa0JaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@5768@ .
  2. v 1 ,, v m    linear unabhängigKer( v 1 v m )={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGTbaabeaakiaaysW7caqGSbGaaeyAaiaab6gacaqGLbGaaeyyaiaabkhacaqGGaGaaeyDaiaab6gacaqGHbGaaeOyaiaabIgacaqGKdGaaeOBaiaabEgacaqGPbGaae4zaiaaywW7cqGHuhY2caaMf8Uaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeSOjGSKaamODamaaBaaaleaacaWGTbaabeaakiaacMcacqGH9aqpcaGG7bGaaGimaiaac2haaaa@61EA@ .

Beweis:

Zu 1.: Wegen Im( v 1 v m )=< v 1 ,, v m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2gacaaMc8UaaiikaiaadAhadaWgaaWcbaGaaGymaaqabaGccqWIMaYscaWG2bWaaSbaaSqaaiaad2gaaeqaaOGaaiykaiabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamyBaaqabaGccqGH+aGpaaa@4961@ , ist hier nichts zu zeigen.

Zu 2.: 

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ " Da ( v 1 v m )0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAhadaWgaaWcbaGaaGymaaqabaGccqWIMaYscaWG2bWaaSbaaSqaaiaad2gaaeqaaOGaaiykaiaaicdacqGH9aqpcaaIWaaaaa@3EED@ , hat man bereits:

Ker( v 1 v m ){0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeSOjGSKaamODamaaBaaaleaacaWGTbaabeaakiaacMcacqGHdksYcaGG7bGaaGimaiaac2haaaa@4563@ .

Sei nun xKer( v 1 v m ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamODamaaBaaaleaacaaIXaaabeaakiablAciljaadAhadaWgaaWcbaGaamyBaaqabaGccaGGPaaaaa@4330@ , also: x 1 v 1 ++ x m v m =( v 1 v m )x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaWG4bWaaSbaaSqaaiaad2gaaeqaaOGaamODamaaBaaaleaacaWGTbaabeaakiabg2da9iaacIcacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeSOjGSKaamODamaaBaaaleaacaWGTbaabeaakiaacMcacaWG4bGaeyypa0JaaGimaaaa@4B3E@ .
Da v 1 ,, v m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGTbaabeaaaaa@3C70@ linear unabhängig, folgt: x i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaaicdaaaa@39CA@ für alle i, d.h. x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaaicdaaaa@38A6@ . Also gilt auch:

Ker( v 1 v m ){0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGPaVlaacIcacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeSOjGSKaamODamaaBaaaleaacaWGTbaabeaakiaacMcacqGHckcZcaGG7bGaaGimaiaac2haaaa@4565@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ " Sei jetzt x 1 v 1 ++ x m v m =( v 1 v m )x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaWG4bWaaSbaaSqaaiaad2gaaeqaaOGaamODamaaBaaaleaacaWGTbaabeaakiabg2da9iaacIcacaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeSOjGSKaamODamaaBaaaleaacaWGTbaabeaakiaacMcacaWG4bGaeyypa0JaaGimaaaa@4B3E@ , d.h. xKer( v 1 v m )={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadUeacaWGLbGaamOCaiaaykW7caGGOaGaamODamaaBaaaleaacaaIXaaabeaakiablAciljaadAhadaWgaaWcbaGaamyBaaqabaGccaGGPaGaeyypa0Jaai4EaiaaicdacaGG9baaaa@46F0@ .
Folgt: x=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaaicdaaaa@38A6@ , also: x i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaaicdaaaa@39CA@ für alle i. v 1 ,, v m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGTbaabeaaaaa@3C70@ ist somit linear unabhängig.

Da Matrizen als Elemente des nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaeyyXICTaamyBaaaaaaa@3BB5@ aufgefasst werden können, trägt die Menge aller n×m   - Matrizen MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabQhacaqGLbGaaeOBaaaa@4427@ in natürlicher Weise eine Vektorraumstruktur, nämlich die des nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbGaeyyXICTaamyBaaaaaaa@3BB5@ . Die folgende Bemerkung stellt diesen Sachverhalt noch einmal in der matrixtypischen Notation dar.

Bemerkung: Sind ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3B@ und ( b ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkgadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaaaa@3A3C@ zwei n×m   - Matrizen MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabQhacaqGLbGaaeOBaaaa@4427@ , α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyicI4SaeSyhHekaaa@3A7C@ , so setzen wir:

( a ij )+( b ij )=( a ij + b ij ) α( a ij )=(α a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacqGHRaWkcaGGOaGaamOyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaeyypa0JaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaabaGaeqySdeMaaiikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiabg2da9iaacIcacqaHXoqycaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcaaaaaaa@5598@

Die Menge (General Linear Group)

GL(m,n)={( a ij )|( a ij )   ist eine   n×m   - Matrix} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadYeacaGGOaGaamyBaiaacYcacaWGUbGaaiykaiabg2da9iaacUhacaGGOaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGaaiiFaiaacIcacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacaaMe8UaaeyAaiaabohacaqG0bGaaeiiaiaabwgacaqGPbGaaeOBaiaabwgacaaMe8UaamOBaiabgEna0kaad2gacaaMe8UaaeylaiaabccacaqGnbGaaeyyaiaabshacaqGYbGaaeyAaiaabIhacaGG9baaaa@5EC4@

ist mit den eingeführten Operationen ein Vektorraum.

In der Funktionensichtweise entspricht die Matrizenaddition exakt der Funktionenaddition:

( a ij )+( b ij )x =( a ij + b ij )x = x 1 ( a 1 + b 1 )++ x m ( a m + b m ) = x 1 a 1 ++ x m a m + x 1 b 1 ++ x m b m =( a ij )x+( b ij )x. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8F49@

und die skalare Multiplikation genau der Vervielfachung von Funktionen:

α( a ij )x =(α a ij )x = x 1 (α a 1 )++ x m (α a m ) =α( x 1 a 1 ++ x m a m ) =α(( a ij )x). MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@74F1@


 9.7
9.9.