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Montrer que f est un projecteur si et seulement si f est diagonalisable et ses valeurs propres ∈ {0, 1}. ´ ´ CHAPITRE 2. ALGEBRE LINEAIRE 50 Exercice 47 Soit f ∈ L(E). Montrer que f est bijectif si et seulement si 0 n’est pas valeur propre. Montrer qu’alors, si λ est valeur propre de f , alors λ−1 est valeur propre de f −1 . Montrer qu’en g´en´eral, si λ est valeur propre de f , alors λk est valeur propre de f k . Exercice 48 Si f ∈ L(E), on dit que P := det(XIdE − f ) est le polynme caract´eristique de f .

D´eterminer une base de SC0 puis une base de AC0 . En d´eduire une base de C0 puis une base de C. 4. Trouver un carr´e magique de trace 27 dont toutes les entr´ees sont distinctes. Exercice 30 Soit E l’espace des polynmes de degr´e au plus 3 sur R. Soient f1 , f2 , f3 et f4 les formes lin´eaires qui envoient P sur P (0), P (1), P (0) et P (1) respectiveˇ D´eterminer la base de E dont c’est la base ment. Montrer que c’est une base de E. duale. Exercice 31 Soit A ∈ Mn (R) telle que A2 +A+I = 0. Montrer que A est inversible.

An1 · · · anm Leur ensemble se note Mnm (K). On dit matrice carr´ee d’ordre n si m = n et on ´ecrit alors Mn (K). En particulier, on consid´ere la matrice   1 0 ··· 0    0 . . . ...    I :=  .  ..  . . 0   .  0 ··· 0 1 Enfin, on dit vecteur colonne   a1  .   .   .  an si m = 1 et vecteur ligne [a1 · · · am ] si n = 1. 2 D´ efinitions Si λ ∈ K, le produit de A par λ est   λa11 · · · λa1m  .   λA :=  ..  . λan1 · · · λanm Si   b11 · · · b1m  .  B :=  .

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