Einführung in die lineare Algebra by Rolf Walter (auth.)

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By Rolf Walter (auth.)

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Diese Regeln kann man unmittelbar aus der Definition (1) nachrechnen; (4) und (5) beruhen wiederum darauf, daß die Komponenten von u und v linear in die Definition CI) eingehen. Die Durchführung sei dem Leser überlassen. 0 Die Beziehung (6) ist die Antisymmetrie oder Alternierungseigenscha[t des Vektorproduktes. Mit ihrer Hilfe übertragen sich die Eigenschaften (4), (5) sofort auf den ersten Faktor: (4') (5') (u + u') X v = u X v + u' X V (Au) X v = A· u X v, insbesondere folgt für A = 0 bzw. A = - 1 (7) u X 0 = 0 = 0 X v, (- u) X v = - u X v = u X (- v).

Eine Teilmenge E von Rn heißt Ebene, wenn ihre Elemente u mit Hilfe eines festen Elementes ß E Rn und zweier fester, linear unabhängiger Elemente v, v' E Rn in der Fonn (2) erhalten werden können, wobei X, X' unabhängig voneinander alle reellen Zahlen durchlaufen. Die Formel (2) heißt wieder Parameterdarstellung von E mit dem Anfangspunkt ß und den Richtungsvektoren v, v' sowie den Parametern X, X'. Satz und Defmition C. Zu drei Punkten ß, w, ~ E Rn, die nicht auf einer Gerade liegen, existiert genau eine Ebene E C Rn, die ß, ~ enthält.

U Bild IS Zur Dreiecksungleichung Beweis von D. l) bzw. 4) bzw. 3): (l3) (14) lu + V 12 =

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