Aigebres Connexes et Homologie des Espaces de Lacets by Jean Michel Lemaire PDF

By Jean Michel Lemaire

ISBN-10: 0387069682

ISBN-13: 9780387069685

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U ................................... V ................................... W ... ....................................................................................................... α◦β Fig 1: Verkettung (α ◦ β)(u) = α(β(u)). Die Verkettung ergibt wieder eine lineare Abbildung, denn (α ◦ β)(u1 + u2 ) = = (α ◦ β)(ru) = = α(β(u1 + u2 )) = α(β(u1 )) + α(β(u2 )) (α ◦ β)(u1 ) + (α ◦ β)(u2 ) f¨ ur u1 , u2 ∈ U, α(β(ru)) = α(rβ(u)) rα(β(u)) = r(α ◦ β)(u) f¨ ur u ∈ U und r ∈ K. 2 Verkettung linearer Abbildungen 41 Sind die Vektorr¨aume U, V, W endlich-dimensional und ist dim U = k, dim V = m, dim W = n, dann kann man bez¨ uglich gegebener Basen von U, V, W obige Abbildungen durch Matrizen aus K n,m bzw.

Dm = 0 m¨oglich ist. Also ist dim V /U = r. b) (U1 + U2 )/U2 ist isomorph zu U1 /(U1 ∩ U2 ), denn die lineare Abbildung α : (a + b) + U2 → a + (U1 ∩ U2 ) (a ∈ U1 , b ∈ U2 ) ist bijektiv. Genau dann ist n¨amlich ur a, a ∗ ∈ U1 , a + (U1 ∩ U2 ) = a ∗ + (U1 ∩ U2 ) f¨ wenn a − a ∗ ∈ U2 , wenn also (a + b) + U2 = (a ∗ + b) + U2 . Daher gilt dim(U1 + U2 ) − dim U2 = dim((U1 + U2 )/U2 ) = dim(U1 /(U1 ∩ U2 )) = dim U1 − dim(U1 ∩ U2 ). ✷ Deﬁnition 3: Mit Hom(V, W ) bezeichnen wir die Menge aller Homomorphismen (lineare Abbildungen) des K-Vektorraums V in den K-Vektorraum W .

R................ P (p1 , p2 ) . . . . . . . . ✲ . α .. x gegeben (Fig. 3). Verkettet man diese noch mit einer Streckung am Zentrum O mit dem Faktor r, dann hat diese Drehstreckung die Matrix a −b b a cos ϕ − sin ϕ sin ϕ cos ϕ =r x1 p1 = = = p2 = = = . Wir betrachten nun die Menge a −b b a C= ✻ r cos(α + ϕ) r cos α cos ϕ − r sin α sin ϕ p1 cos ϕ − p2 sin ϕ r sin(α + ϕ) r sin α cos ϕ + r cos α sin ϕ p1 sin ϕ + p2 cos ϕ Fig. 3: Drehung | a, b ∈ IR mit den u ¨blichen Matrizenoperationen (Addition, Multiplikation).