Algèbre linéaire by Joseph Grifone

By Joseph Grifone

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1. Montrer que les solutions forment un espace vectoriel S sur R. 2. A l’aide du th´ eor` eme d’existence et unicit´ e, montrer que dimR S = 2. II. Soit l’´ equation (dite ´ equation caract´ eristique) r 2 + ar + b = 0 1. Montrer que si elle admet deux racines r´ eelles r1 , r2 , alors la solution g´ en´ erale est : y = A e r1 x + B e r2 x 2. Montrer que si l’´ equation caract´ eristique admet une racine double r, la solution g´ en´ erale est : y = (A + B x) er x 3. 4 n’est pas satisfait (ce qui montre qu’il ne peut ˆ etre d´ eduit des autres axiomes).

0, a1 , b1 , . . . , l1 ) ( 0, 1, . . , 0, a2 , b2 , . . . , l2 ) ····················· ( 0, 0, . . , 1, an−k , bn−k , . . , ln−k ) (obtenues en donnant au (n − k)-uplet (λ1 , . . , λn−k ) successivement les valeurs (1, 0, . . , 0), (0, 1, . . , 0) . . (0, . . , 0, 1) ). Elles forment un syst`eme de g´en´erateurs, car la solution g´en´erale est combinaison lin´eaire de ces solutions. Il est facile de voir qu’il s’agit d’une famille libre et donc d’une base de l’espace des solutions2 .

2 On note R+ \{0}, l’ensemble des nombres r´ eels strictement positifs. Montrer que les lois : x⊕ y := xy conf` erent ` a * R+ \{0} λ . x : = xλ (x, y ∈ R+ \{0} , λ ∈ R) une structure d’espace vectoriel sur R. 2 (commutativit´ e de la somme) peut ˆ etre d´ eduit des autres axiomes. 4 Dans R3 muni des lois habituelles, lequel vectoriel ?  F = (x, y, z) ∈ R3 de ces sous-ensembles est-il un sous-espace ˛  ff ˛ x−y+z =0 ˛ ˛ 2x −y = 0 G = {(x, y, z) ∈ R3 | (x − y)2 = 2x + y} 5 Dans M2 (R) muni les lois de l’exemple 3, page 5, lequel de ces sous-ensembles est-il un sous-espace vectoriel ?

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