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07

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Groupes, Permutations, Anneaux, Arithmétique dans Z, Corps commutatif, Les polynômes formels à une indéterminée à coefficients dans un corps K, Fonctions polynomiales, racines, Espaces vectoriels, K-algèbres, Espaces vectoriels de type fini, Matrices, Déterminants, Fractions rationnelles, Produit scalaire sur un R-ev, Espace vectoriel euclidien, R-ev euclidien orienté de dimension 2, R-ev euclidien orienté de dimension 3, Espaces affines, Géométrie dans un espace affine euclidien

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07

  1. 1. ! " # $%∈ ∈ = & ' () ! * ' ∈ = → +, - - $%∈ ) . +, - ∈ ) ! / # ∈ ! $%0 ∈ ∈λ +, -- +, +, - +, - +, - +, - +, - +, - 00 00 λλ = ×=× +=+ +, - = ," ( + 1 +, 00 00 - -- --- --- = = ×=× +=+ λλ , ++, -- ,+, - 00 +=+∈∀ ( 2 + ) - +,$% → ( ) +$%, ×+ 3 ) ++,, ×+ ! + " # $%∈ ∈λ ! 1 ( λ ,( + ( 4+, - =λ / # $%∈ ∈λ ! λ ( +, λ− ( 3 ! touscours.net
  2. 2. 0 " " 5' +, λ− ! 1 ( ( 3 ( ( +, λ− +−= +, λ & $%∈ $%4∈ ∈= ! " +−= +, λ " -- + - , - +−= λ ()& =+×= +, - 4+, - λλ ! " λ ( 4= +, λ− ( 3 ! 6+ 7 # { }48$%∈ ! # λ ! 1 λ ( ! * ( λ ( ( { }( 3+,9 λ−∈= ! * ( ) 3 ( , + 5 , +( , + 1 3 λ ( 4 λ ) ( ! + * ( / # { }48$%∈ ! # λλλ !!!0 ( ( ( ( ' ααα !!!0 ! ∏= − +, α λ ( 3 ! " * . [ ]+, ∈− λ ( ( $% , ( ( +! ( ( [ ]+, ∈− α λ 9! " ( 3 ! " ( ( 3 ! # { }48$%∈ ( ( ∈ ! ( ( , + ' ! " # ( λλλ !!!0 ( 3 ααα !!!0 ∏= − +, α λ ( ( = α ( 3 ( ≤ = α ! # $%∈ 3 + ( ' 4= ! 1 ! touscours.net
  3. 3. : ) - +,$% → 5 3 # -- = 4 -- =− 4 - =− ; <'<( 4+, - =−∈∀ ! " - − ( ! " 4=− ! " = ! " ( ( ! ( , ) 4 = × ' ) 3 + ) 3 $%∈ =⇔= -- ! 1 ( - ( ∈ = +, ! + " # $%∈ ∈ = & ' () ! . ( 3 ( ( . ∈ − = > ; 1 ( . ( 3 ( =∈∀ = + +;,> +,+, +4, ? ( +, → ;+;, = 6+ • # ≤( +, =( ! @ # 4!!! ++++= − − 0 !!!+,; ++−+= − − − " ( ≥= ;( −= ()& 4( +, = ! @ ∞− = 4 ( 4;= ! +,;;++;,, +,;+;, +,;;+;, !+!,;!+;!, +,;;+;, 4 +,+,+, +,+, +,+,+, ×=• ≥=• =→+=ו =→=• +=+→+=+• − = − λλλλ touscours.net
  4. 4. A " , + 1 +,;+,;+,+;, +=+∈∀ " ;;+;, −−+ ( ( ;;+;, +=+ ! " 2 ,0+ ,:+ ,A+ ( 3 < ! ,B+ ( 3 ) ' ! " ( , ( + ∈ =+, " ∈ − = > ;++;,, ,() , + ,0+ ,A++ # +,;;;++;,, > == ∈ − ! ? ≤≤ − > = − 4 +C, C 4 +, +, = ≠ = C 4 +4+,,+, + ( / . / # . ∈ (≥ ! ∈ +,!!!+,;;+,;+,+, +, CC0 0 ++++=+ " < # 4= 1 3 +4,!!!+4,;+4,+, +, C+++=∈∀ = = 4 ! " ∈ = = 4 +,+, +,+, " C+4+,,+, = ! " C +4+,,+, = ! < 1 +,+, += ! +,+, +,+, +=∈∀ , +;, =+ + " = =∈∀ 4 +, C +4, +, # = =+∈∀ 4 +, C +, +, touscours.net
  5. 5. B / , + # $%∈ ! # ∈ (≥ ! $%∈ = =+ 4 +, C +, +, ! " ∈ = =+ 4 +, C ++,, ++,,++,+,, ,() ( 3 +,= +,= + ")& ) ( . +, + =4 +, C +, ) D ( ! "+ ' / # { }48$%∈ ∈λ >∈ ! λ () ( ( 4+,+!!!,+, +,+,+4, === − λλλ λ ( () ( 9 4+,+!!!,+, +,+,+4, === − λλλ 4+,+, ≠λ ! " # ∈ (≥ ≥ ! +,( 3. +, ((. +, +, C +, !!!+, +C, +, !!!+,;+,+, ++,,+, λ λ λ λ λ λλλ λλ −−≤ − − − ++ − − ++−+= −+= " ( ( 3 ( ( +, λ− ! 1 ( 3 [ ] 4 C +, 4 4 C +, +, 4 ( 3+,(((; +, 4 +, =−∈∀⇔ =−⇔ =⇔ −⇔ − = λ λ λ λλ ( 3 , ) ( 3 ( + # 4 C +, +, 4 +, =− − = λ λ 4 C +, +, 4 +, =−∈∀ − = λ λ ( 4 C +, 4 +, =∈∀ − = λ ! 4 C +, 4 +, = − = λ ()& [ ] 4 C +, 4 +, =−∈∀ λ ! touscours.net
  6. 6. E + " # $%∈ ( ( ≥ ! 1 ( ( ,( + ( 3 ' ( ( 9 ( ' ( ) 9 >∈ ( λλλ !!!0 ( ( ( ααα !!!0 ( > >∈ ∏= −= +, α λ , ( + ' ( ) 9 µµµ !!!0 ( >∈ ∏= −= µ ! @9 : − ( ( $% ( $% 6+ ? . ( + # >∈ ! 1 ( σσσ !!!0 ( ( ∏= << < − = = = =++++++= =++= 0 0: 0:0:000 00 +!!!, +!!!, !!!!!!!!!+!!!, !!!+!!!, λλλλσ λλλλλλσ λλλλλλλλλλλλλλλλλσ λλλλλλλσ [ ] <<< =∈ !!! 0 0 0 !!!+!!!, λλλλλλσ ; <'<( @ ( → ( +!!!,+!!!,+!!!, 0+,+0,+,0 =∈∀∈∀ ! 3 () , + ( σ ! 0+ * # . ( ( ( ≥ ( $% () ( 3 = = 4 () ∏= −= +, λ ! touscours.net
  7. 7. σ +!!!, 0 λλλσ [ ]∈ ! * 3 ( σ − − − =− = =− σ σ σ +!!!, 00 @ λλλλλλ λλλλλλ !!!!!!!!!+,!!! +!!!,+, 0 !!! 0 0 0 0 ++−+ ++++−=− − <<< − < − = ∏ :+ ? # ∈!!!0 ! ( 3 () !!!0 ( = = = +!!!, +!!!, +!!!, +, 0 000 0 σ σ σ 9 ( . +!!!,+,!!! −+−++−= −− " < # !!!0 ( +!!!, 0 ( , + () 0+! < # +!!!, 0 ( , + . ∏= −= +, ) ( 3 +,!!! −++− − ! $% $% + " $% ! / () / . ( ( ≥ ' ( ! . ( $% ( ( ≥ ( ( $% ,( + @ 3 * . ( ( $% 9 . ∈− λλ ! 1 ( ! touscours.net
  8. 8. F 6+ " $% ! # $%∈ ! # ∈λ ( α ! λ ( 3 2 α ! " < # λ ( & = = 4 4!!!+, 4 4 =+++= − − λλλλ " 4+, =λ +, !!! !!!+, 4 4 4 4 λ λλλ λλλλ = +++= +++= − − − − # 4+, =λ ! " λ ( ! < # λ ( ≥α ( 4+,!!!+,;+, +, ==== − λλλ α 4+,+, ≠λα ! " 4+,!!!+,;+, +, ==== − λλλ α , ( 9 +, + 4+,+, ≠λα 4+,+, +,+, == λλ αα ! # $%∈ λ ( ! < ++,, λλ −− ( 3 G ) 3 ∉λ ( ' λλ = ( 3 +, λ− ( 3 0 +, λ− ! / * ( ( $% 9 . ( ( ( ( 0 ! . ' ( ( ≥ ) ( ∏= = & >∈ & { }4+,$%8$% 00 ≥∆∈∈ ! " < " 5' += , ×∈ >+, + ( ! # ++= 0 3 <− ××∈ 4A >+, 0 ( ( $% , ) +;;!+,!, βαβα ++ ( 9 + < # . ( ( 0≥ ! # () ∈λ ! • # ∈λ +, λ−= & ( ≥−= ( ) ( ! • # 8∈λ λ ( λλ ≠ ( ++,, λλ −− ( 3 ! 1 $%+,++,, 0 ∈++−=−− λλλλλλ ! " +, 0 +−= & $%∈ ∈ ∈ ! touscours.net
  9. 9. " 0( −= ! " 0= ( :≥ ) ( ! ) ( () . ( ! # $%∈ ( ( ≥ ( ( ! ( ( λλλ !!!0 ( ααα !!!0 ! µµµµµµ !!!00 9 ( ββββββ !!!00 ! 3 =+ == βα ! ∏∏ ∏∏ == == +−−= −−−= 0 +,+, +,+,+, βα ββα λ µµλ [ ] 0 +? ,01& µµµ µµµ == =+=∈∀ ( H # =4 *) ( 3 () ( 3 () 40 +!!!+++,!!!,,, +++++ −− 1 (( ! touscours.net

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