Download e-book for iPad: An Introduction to Linear Algebra by L. Mirsky

By L. Mirsky

ISBN-10: 0198531168

ISBN-13: 9780198531166

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As Abel instructed: "Read without delay from the Masters". this is often THE ebook!

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Extra info for An Introduction to Linear Algebra

Example text

Im Ring Z der ganzen Zahlen ist jedes Ideal ein Hauptideal, also von der Form (m) = mZ = {mq | q ∈ Z} f¨ ur ein m ∈ Z. Beweis. a): Sei I ⊆ Z ein Ideal. Ist I = {0}, so ist I das von 0 erzeugte Hauptideal. Ist I = {0}, so gibt es positive Zahlen in I, da zu jedem a ∈ I auch −a ∈ I gilt. Sei m die kleinste positive Zahl in I und n ∈ I beliebig. 2 (Division mit Rest) k¨ onnen wir n = mq + r mit q, r ∈ Z, 0 ≤ r < m schreiben. Da r = n − mq ∈ I wegen n ∈ I, m ∈ I gilt und m nach Definition die kleinste positive Zahl in I ist, folgt aus 0 ≤ r < m, dass r = 0 gilt.

C) Zeigen Sie, dass 2 kein √ Primelement √ in Z[ −5] ist (Hinweis: Betrachten Sie das Produkt (1 + −5)(1 − −5)). 6. 11 definierte p-adische Bewertung (fortgesetzt durch νp (0) = −∞). Zeigen Sie: a) νp (n + m) ≥ min{νp (n), νp (m)} b) νp (n + m) = min{νp (n), νp (m)}, falls νp (n) = νp (m) gilt. c) νp (nm) = νp (n) + νp (m) d) Der durch p−νp (n) n = 0 |n|p := 0 n=0 definierte p-adische Betrag hat (wie der gew¨ohnliche Absolutbetrag) die Eigenschaften • |n|p ≥ 0 und |n|p = 0 genau dann, wenn n = 0 ist.

Ein Element a ∈ R, a = 0 heißt ein Nullteiler, wenn es b = 0 in R mit ab = 0 gibt; gibt es in R keine Nullteiler, so heißt R nullteilerfrei oder ein Integrit¨ atsbereich. 2. Sei R ein Integrit¨atsbereich. Dann gilt: a) b) c) d) e) f) (K¨ urzungsregel) Sind a, b, c ∈ R, c = 0 mit ac = bc, so ist a = b. F¨ ur alle a ∈ R gilt a|0 und a|a. F¨ ur alle a ∈ R gilt 1|a Sind a, b ∈ R mit a|b und ǫ ∈ R× eine Einheit, so gilt a|ǫb, ǫa|b. Sind a ∈ R, ǫ ∈ R× mit a|ǫ, so ist a ∈ R× . Sind a, b, c ∈ R mit c = 0, so gilt genau dann ca|cb, wenn a|b gilt.

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An Introduction to Linear Algebra by L. Mirsky


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