# TP3_Sol.pdf - Solutions des exercices supplémentaires du...

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Solutions des exercices supplémentaires du TP3 MAT 1600 24 janvier 2018 Chapitre 1 Solution 16.a. Supposons le contraire. Si { 3 v 1 , 2 v 1 - v 2 , v 1 + v 3 } un ensemble linéairement dépendant, alors il existe a, b, c des nombres réels non tous nuls tel que : a (3 v 1 ) + b (2 v 1 - v 2 ) + c ( v 1 + v 3 ) = 0 on a alors que (3 a + 2 b + c ) v 1 + ( - b ) v 2 + c 3 = 0 Puisque { v 1 , v 2 , v 3 } est un ensemble linéairement indépendant, il faut que : (3 a + 2 b + c ) = 0 - b = 0 c = 0 La seule solution de ce système et que a, b, c sont tous égales à 0. Ceci est une contradic- tion. Solution 16.b. Supposons le contraire. Si { v 1 , v 1 + v 2 , ..., v 1 + v 2 + ... + v n } un ensemble linéairement dépendant, alors il existe a 1 , a 2 , ..., a n des nombres réels non tous nuls tel que : a 1 ( v 1 ) + a 2 ( v 1 + v 2 ) + ... + a n ( v 1 + v 2 + ... + v n ) = 0 on a alors que ( a 1 + a 2 + ... + a n ) v 1 + ( a 2 + ... + a n ) v 2 + ... + ( a n ) v n = 0 Puisque { v 1 , v 2 , .., v n } est un ensemble linéairement indépendant, il faut que : a 1 + a 2 + ... + a n = 0 a 2 + ... + a n = 0 .

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