# heure_7 - Combinaison lin´ eaire et syst` eme g´ en´...

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Unformatted text preview: Combinaison lin´ eaire et syst` eme g´ en´ erateur D´ efinition Soit un espace vectoriel V . Soient k vecteurs v 1 , v 2 ,..., v k de V et k scalaires l 1 , l 2 ,..., l k ∈ R . I On appelle combinaison lin´ eaire des vecteurs v 1 , v 2 ,..., v k tout vecteur v = l 1 v 1 + l 2 v 2 + ... + l k v k I On dit que v est lin´ eairement d´ ependant des vecteurs v 1 , v 2 ,..., v k . Combinaison lin´ eaire et syst` eme g´ en´ erateur Exemple Soient u = i + 2 j , v 1 = 3 i + j et v 2 = i + 3 j . Montrez qu’on peut ´ ecrire u comme comb. lin´ eaire de v 1 et v 2 . Soit l 1 , l 2 ∈ R tel que u = l 1 v 1 + l 2 v 2 , i.e. , i + 2 j = l 1 (3 i + j ) + l 2 ( i + 3 j ) = (3 l 1 + l 2 ) i + ( l 1 + 3 l 2 ) j 3 l 1 + l 2 = 1 l 1 + 3 l 2 =2 = ⇒ 3 l 1 + l 2 = 1- 8 l 2 =- 5 = ⇒ 3 l 1 = 3 8 l 2 = 5 8 = ⇒ l 1 = 1 8 l 2 = 5 8 u = 1 8 v 1 + 5 8 v 2 Combinaison lin´ eaire et syst` eme g´ en´ erateur Exemple Soient u = i + 5 j + 6 k , v 1 = i- j + 3 k et v 2 = 2 i + 4 j ....
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## This note was uploaded on 11/07/2010 for the course CIV 3930 taught by Professor Montes during the Spring '10 term at École Polytechnique de Montréal.

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heure_7 - Combinaison lin´ eaire et syst` eme g´ en´...

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