Download e-book for iPad: Algebra Through Practice: A Collection of Problems in by T. S. Blyth, E. F. Robertson

By T. S. Blyth, E. F. Robertson

ISBN-10: 0521253004

ISBN-13: 9780521253000

Problem-solving is an artwork primary to figuring out and skill in arithmetic. With this sequence of books, the authors have supplied a variety of labored examples, issues of entire ideas and attempt papers designed for use with or rather than general textbooks on algebra. For the ease of the reader, a key explaining how the current books can be utilized at the side of a few of the significant textbooks is incorporated. every one quantity is split into sections that commence with a few notes on notation and conditions. nearly all of the fabric is geared toward the scholars of regular skill yet a few sections include tougher difficulties. by way of operating in the course of the books, the scholar will achieve a deeper figuring out of the elemental ideas concerned, and perform within the formula, and so answer, of different difficulties. Books later within the sequence disguise fabric at a extra complex point than the sooner titles, even though every one is, inside its personal limits, self-contained.

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Extra resources for Algebra Through Practice: A Collection of Problems in Algebra with Solutions

Example text

Für allgemeines n wird die Rechnung mit den Koordinaten zu kompliziert, hier hilft ein Trick. Im Fall w = o sind beide Seiten gleich Null; es genügt also, den Fall w o zu behandeln, wir setzen λ := w, w > 0 μ := − v, w . und Dann ist 0 ≤ λv + μw, λv + μw = λ2 v, v + 2λμ v, w + μ2 w, w = λ( v, v · w, w − 2 v, w 2 + v, w 2 ) = λ( v, v · w, w − v, w 2 ). Da λ > 0 folgt daraus v, v · w, w − v, w 2 ≥ 0 und wegen der Monotonie der Quadratwurzel erhält man die behauptete Ungleichung. Mit der Frage, wann die Ungleichung von C AUCHY-S CHWARZ eine Gleichung ist, beschäftigen wir uns im nächsten Abschnitt.

Der Leser möge das zur Übung ausführen. 3 Winkel im R n 27 problematisch ist, weil sie die Kenntnis des Winkels im Bogenmaß erfordert. Für die Theorie ist es geschickter, die algebraischen Definitionen von Skalarprodukt und Norm voranzustellen, und dann den Winkel so zu erklären, dass die Gleichung (∗) erfüllt ist. Bei diesem Kniff gibt es eine Hürde, die überwunden wird durch die Ungleichung von C AUCHY-S CHWARZ Für v, w ∈ R n gilt | v, w | ≤ v · w . Als unmittelbare Folgerung erhält man die Ungleichungen −1 ≤ falls v v, w ≤ 1, v · w 0.

M ) (0, . . , 0). Die Handhabung dieser Definition erfordert etwas Übung. 7 eine Charakterisierung mit Hilfe des Vektorproduktes geben. 4 aus den Komponenten von v1 , v2 , v3 eine Zahl berechnen, die über die lineare Unabhängigkeit entscheidet. 6 sind zwei Vektoren v = ( x1 , x2 ) und w = (y1 , y2 ) im R2 genau dann linear unabhängig, wenn det x1 y1 x2 y2 : = x 1 y 2 − x2 y1 0. Es ist eine naheliegende Frage, ob es ein ähnliches Kriterium im R3 gibt. Sind zwei Vektoren v = ( x1 , x2 , x3 ) und w = (y1 , y2 , y3 ) im R3 gegeben, so kann man sie in Form einer „Matrix“ A= x1 y1 x2 y2 x3 y3 untereinander schreiben.

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Algebra Through Practice: A Collection of Problems in Algebra with Solutions by T. S. Blyth, E. F. Robertson

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