Calculus and Linear Algebra. Volume 2: Vector Spaces, by Wilfred Kaplan, Donald J. Lewis

By Wilfred Kaplan, Donald J. Lewis

Within the moment quantity of Calculus and Linear Algebra, the concept that of linear algebra is additional built and utilized to geometry, many-variable calculus, and differential equations.

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Extra resources for Calculus and Linear Algebra. Volume 2: Vector Spaces, Many-Variable Calculus, and Differential Equations

Example text

Insbesondere gilt V isomorph W als algebraische Mengen ⇐⇒ K[V ] isomorph K[W ] . 9 Eine Aquivalenz von Kategorien 35 Beweis. Injektivit¨ at von FV,W : F¨ ur α, β ∈ Mor(V, W ) sei α∗ = β ∗ . Zu zeigen: α = β . 8 eines Morphismus gibt es f1 , . . , fm und g1 , . . , gm ∈ K[X1 , . . , Xn ] mit α(v) = (f1 (v), . . , fm (v)) und β(v) = (g1 (v), . . , gm (v)) f¨ ur alle v ∈ V . F¨ ur alle ϕ ∈ K[W ] = K[Y1 , . . , Ym ]/I(W ) = Mor(W, K ) gilt ϕ◦α = Def von ∗ α∗ (ϕ) = β ∗ (ϕ) = Def von ∗ ϕ◦β. Es folgt yi ◦ α = yi ◦ β f¨ ur alle i = 1, .

Hierdurch erh¨ alt man injektive Gruppenhomomorphismen λ : G → GL(K[G]), g → λg ρ : G → GL(K[G]), g → ρg , wobei K[G] als K-Vektorraum aufgefasst wird und GL(K[G]) die Gruppe der K-linearen bijektiven Abbildungen K[G] → K[G] bez¨ uglich der Hintereinanderausf¨ uhrung von Abbildungen bezeichnet. 9 ist auch f¨ ur Rechtstranslationen anwendbar. 10 Linearisierung affiner Gruppen 61 Satz Sei G eine lineare algebraische Gruppe. Dann gibt es ein n ∈ N und eine abgeschlossene Untergruppe H von GLn (K) so, dass G H ist.

Gm ist. Es folgt Gj ⊂ Gi f¨ ur alle j = 1, . . , m , da Gj ⊂ G1 · . . · Gm gilt. Hieraus folgt Gj = Gi f¨ ur alle j = 1, . . 1 ist. Es ist also m = 1 . (2) Als irreduzible Komponente ist G0 abgeschlossen in G . (i) G0 ist eine Untergruppe von G , denn: Als Bild der Multiplikation G0 ×G0 → G0 G0 ist G0 G0 irreduzibel. Da G0 ⊂ G0 G0 gilt und G0 maximal ist, folgt G0 = G0 G0 . Da i : G → G−1 , x → x−1 , ein Hom¨oomorphismus ist, ist (G0 )−1 eine irreduzible Komponente von G , die e enth¨alt, und also gilt G0 = (G0 )−1 nach (1).

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