Algèbre linéaire by Joseph Grifone

By Joseph Grifone

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Relation de d´ ependance : 2 v1 − v2 + v3 = 0. La famille n’est pas g´ en´ eratrice, car le vecteur v = (a, b, c) ne peut s’´ ecrire sous la forme v = x1 v1 + x2 v2 + x3 v3 que si : 4 a − 5 b − c = 0. 8 2. Il s’agit d’une base. 9 „ 1 0 0 0 « „ , 1 0 0 −1 « , „ 0 2 1 0 « est g´ en´ eratrice et libre. 10 0 B B B E11 = B B @ 1. 1 0 0 . 0 0 0 0 B Enn = @ ... 0 ... 0 ... 1 ... 0 . C 0 .. A ... 1 0 .. . 0 1 0 0 1 C B C B C B 0 , E = B C 12 B .. C A @ . 0 0 ... 0 ... 0 ... 0 .. . 0 1 C C C C ...

Soient F1 , F2 , F3 trois sous-espaces vectoriels d’un espace vectoriel E. Montrer que : F1 ∩ (F2 + F3 ) ⊃ F1 ∩ F2 + F1 ∩ F3 2. A-t-on l’inclusion contraire ? * 17 Soient : F =  A ∈ M2 (R) | A = „ a −b 2a + b −a «ff G =  A ∈ M2 (R) | A = „ a −b 3a +b −2 a + b «ff Montrer que M2 (R) = F ⊕ G. * 18 Soit A ∈ Mn (K) et tA la matrice de Mn (K) dont les lignes sont les colonnes de A. 0 1 0 1 1 2 −1 1 0 8 Par exemple, si A = @ 0 1 3 A alors t A = @ 2 1 1 A. 8 1 −1 −1 3 −1 t A est dite transpos´ ee de A.

Ep }, - une base de E2 du type {a1 , . . , ar , εr+1 , . . , εq }. Or tout vecteur de E1 + E2 s’´ecrit comme somme d’un vecteur de E1 , et d’un vecteur de E2 et donc il est de la forme : x= λ1 a1 + · · · + λr ar + λr+1 er+1 + · · · + λp ep +µ1 a1 + · · · + µr ar + µr+1 εr+1 + · · · + µq εq c’est-`a-dire, en posant νi = λi + µi , pour i = 1, . . , r : x = ν1 a1 + · · · + νr ar + λr+1 er+1 + · · · + λp ep + µr+1 εr+1 + · · · + µq εq Par cons´equent, la famille {a1 , . . , ar , er+1 , . .

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