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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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  1. 1. = !" # $ × ! ! % & ' ⊕ ' ( ) ⋅ * +)+ ⊕ →× ! ⋅ →× λλ ! " , - !⋅⊕ . + + ! - !⊕• % • ∈ ∈µλ =⋅ ⋅⋅=⋅× ⋅⊕⋅=⊕⋅ ⋅⊕⋅=⋅+ !! ! ! µλµλ λλλ µλµλ ( !×+ !! ⋅+ !×+ + " !/0 ×+ + " 1! 2 & !⋅⊕ + " ! 33 =⋅∈∀ ⊕ !⊕ ! ⋅⊕⋅=⋅+=⋅∈∀ 33!333 " 33 =⋅ ! 33 =⋅∈∀ λλ 33!333 ⋅⊕⋅=⊕⋅=⋅ λλλλ 33 =⋅λ " 4! 333 ==⇔=⋅∈∀∈∀ λλλ 5 ⇐ ! !" & - 3=⋅λ - 3≠λ " # - ' 3= " , − λ - 3≠λ !" touscours.net
  2. 2. =⋅=⋅×=⋅⋅ −− !! λλλλ ' 33! =⋅=⋅⋅ −− λλλ ' " 3= 6! !!! −⋅=⋅−=⋅−∈∀∈∀ λλλλ ( ) ( ) 3!!! =⋅−+=⋅−⊕⋅ λλλλ " !! ⋅−=⋅− λλ ( ) 33!!! =⋅=−⊕⋅=−⋅⊕⋅ λλλλ " !! ⋅−=−⋅ λλ " 7! ⋅=∈∀∈∀ " ' !⊕ 8) ( ! 3≥ 6! 3≤ " % $ ⊕ " ⋅ λ− !! ⋅−⋅− λλ !−⋅λ " 9 + !⋅+ " ! ( & :∈ , !"""××× ⊕ ( ⋅ ) % ∈ ∈ ∈ λ !""" !""" !"""!"""!""" +++=⊕ !"""!""" λλλλ =⋅ " !⋅⊕ + " ;) !⊕ % 5 ⊕ !3"""33 - " & ∈ " !"""= !"""= !"""= < ∈""""""""" ⊕⊕== ++++++= ++++++== ⊕⊕=⊕⊕ !""" !!"""!! !!! """!""" !!"""!"""!"""! touscours.net
  3. 3. 4 & ∈ !"""= !"""= " ⊕= +++= +++= ⊕=⊕ !""" !""" !"""!""" ( ' ⊕ " & ∈ !"""= " !"""* −−−= ⊕ " & ∈∈ µλ !"""= !"""= " , ⋅⊕⋅= ⋅⊕⋅= ⊕= +++= +++= ⋅+=⋅+ µλ µλ µµµλλλ µλµλµλ µλµλµλ µλµλ !"""!""" !"""!""" !""" !!"""!! !"""!! ⋅⊕⋅= ⊕= +++= +++= +++⋅=⊕⋅ λλ λλλλλλ λλλλλλ λλλ λλ !"""!""" !""" !!! """! !"""! ! !""" !!! """! !!"""!!! ⋅⋅= ⋅= = =⋅ µλ µµµλ µλµλµλ λµλµλµλµ == =⋅ !""" !""" = & ( + × ' + ∈∈∈ λ** ⋅⋅=⋅ ++=+ !! !**!**! λλλ ××× """ < + " touscours.net
  4. 4. 6 ! 9 > !⋅+ + " % & !""" % % ' " ? % !""" . [ ]∈ @ ⋅++⋅+⋅ λλλ """ * +)+ = ⋅λ < λ " % & ∈ " & 3= ) " & 3≠ ) ∈⋅ λλ " 5 A B ) C '- " %% + ;) % ( D + &@ - & ) & 3= ) ⋅= 3 & 3≠ ' ⋅λ < ∈λ " 3=λ 3= ) & 3≠λ − = λ ) " + E " % - & ∈ " , '- ( ){ } !333F! !3 =⋅+⋅∈∃⇔ ⋅=∈∃=⇔ βαβα λλ ! % G ' % " # - !! " & !" & 3= !3! =βα & 3≠ ( ∈λ - ⋅= λ " !! −= λβα 3=⋅+⋅ βα # - !! " & !" & ( ){ }33F!∈βα - 3=⋅+⋅ βα " & 3≠β ⋅= − β α & 3=β 3=⋅α " , 3≠α ( )33! ≠βα " 3= " !⋅+ ; + " touscours.net
  5. 5. 7 ! % & " , - + ! - • 3 " • $ ∈+∈∀ • ⋅ ∈⋅∈∀∈∀ λλ " & + $ ⋅ ( ) !⋅+ + + ;) !+ % - + !+ ∈3 $ ∈⋅−=−∈∀ ! " + % - - % D ( • + " H + I + { }3 + { }3F∈ { }∈⋅ λλ + " + " > '@ ' + ( ' " • & + - - { }3 ( + " & { }3F∈ { }∈⋅ λλ + " • 5 + 4 ( + { }43 + { }43F4 ∈ { }∈⋅ λλ " + { }43F4 ∈ { }∈⋅+⋅ µλµλ ! + 4 • + ! { }3!! =∈= < % ( " = ' % @ +⋅ ∈ " , B /0 /0 < ∈ ! ! """!3 5' % D 1! > + E E + + " touscours.net
  6. 6. J & ∈! % + " & { }∈∈∀∈== ∈ ∈3 ∈∈∀ 3 $ & ∈ " ∈∈∈∀ ∈+∈∀ " ∈+ ⋅ & ∈∈ λ " ∈∈∀ ∈⋅∈∀ λ ∈⋅λ " ! % - & ⊂ " ∈ ⇔ ∈⋅+∈∀∈∀ ∈ ⇔ ∈⋅+⋅∈∀∈∀ ∈ ⇔ ∈⋅∈∀∈∀ ∈+∈∀ ∈ ⇔ !4 !33 !4 !33 !4 !33 ! ! !33 λλ βαβα λλ 4 ! ∈⋅∈∀∈∀∈∀ = !"""!""": λλλλ !4!! " !4!4 " !4!4 " !4!4 " !!!3!4 & !3!4 - !4 3= ! 4 ! =λ !" '< - " 3! 3 ! A ∅≠ C" ! & + % & ⊂ " 5 + !9 + " touscours.net
  7. 7. K L % 5' ε + ' - ' ' ε∈ + - ε ' ε " { }3!9 =∅• • & { }3F∈ { } { }∈⋅= λλ!9 !9 " • + =!9 " • & + ⊂ ⊂!9 %% ε∈ { }ε∈ =!9 ! • & ⊂ !9!9 ⊂ !9⊂⊂ " !9⊂ " !9!9 ⊂ ' !" & + % & """ " { }!"""9 M !"""9 + % !""" A C !"""9 ' * +)+ { }∈+++ !"""""" λλλλλλ N { }∈+++= !""""""!""" λλλλλλ " !""" 3 $ ⋅ === +=+ !** λλλλ == =⋅ !λλλλ ! !""" + ' + " !"""!"""9 = " 9 • & + % % ! !"""= - % " • & % % - @ % " ( @ % % [ ]!"""33! """3"""33!3"""3 /0 ' @ % " %% - ' % % !""" 8 [ ] !!( ∈ = ≤∈∀ /0 - % (" touscours.net
  8. 8. ∈!""" • [ ]∈ ≠ !"""""""""9!"""""""""9 = • [ ]∈ { }3F∈ !""""""9!""""""9 = • [ ]∈ ∈λ !""""""9!""""""9 =+ λ 4 ! & !""""""9 **** λ+∈ = ≠= = ++== !* µ λλλλ =+ ≠ = λλλ λ µ 5' " , 9 ( ! { } { }∈+−= ∈++= −= −= −−= −= −−−= −−−−= −− !O6 !33!O+33!633 !/33!O33!63309 !/433!O33!63309 !/433!O33!6309 !/433!43!6409 !/JJ33!43!6409 /!J3!63!6409!/O6!3J6!6409 4 4 4 6 4 '- 6 !∈ 4O6!/O6!3J6!6409! +−=⇔∈ & ≤≤ !"""9!"""9 ⊂ " ∈ !"""9!"""9!"""9 =⇔∈ !⋅+ + " touscours.net
  9. 9. O % & ( + " 5 { } { }+=×∈∃∈=∈∈+=+ ! % + + ' B !9 ∪ " %% ;) + + 3 ⋅+ ( % + ! + ' 3+ < ∈3 ! " > ∪ " % + ∪ + $ + ∈ ∈ " ( + != & ' % @ 4≤ ' % 3" !=+ " %% ? ;) " ' & !∈ " 5) ' 3 !! 3 !! ε+++=∈∀ −= != 3" '< ' ' " + 6 { }3!!!339 6 =−=−∈== { } !!33!339! =∈= !!33!33!339=+ " & % & ( + " , - + - + ' - % + ∈ ∈ " - ' P ;) '( % ! ' % 5 !**** !**! ==+=+ ×∈∀×∈∀⇔ ( 5 ( " , '- ! 5 ( % ! ! !333! ===+×∈∀ 4! { }3=∩ ! !****!**! ==+=+×∈∀×∈∀ ! touscours.net
  10. 10. 3 + , ;) - !! ' !33!** = ! + # - !4! " & ! & ∩∈ , 3! =−+ " , ∈ ∈− ∈ ⋅ ! ' ! 3= 3=− ! '< ' " + # - !!4 " & 4!" & ×∈×∈ !**! " & - **+=+ " −=− ** ∈−∈− ** ∩∈−∩∈− ** " 3*=− 3* =− * +)+ *= *= " '< - " N & ⊕ " % & ( + " , - - { }=∩ =+ 3 - ⊕= " ( + ' - + < ∈ ∈ " + " ! % & →ϕ " , - ϕ . + + - !! !*!!** ϕλλϕλ ϕϕϕ ⋅=⋅∈∀∈∀ +=+∈∀ & ϕ ϕ !+ !+ " 9 • 5' ! • ? ' ! M ! " • ? ' % " ! :" 5' % ' " touscours.net
  11. 11. - & →ϕ " !4!*"!!*"!* !!*"!"!*""!*!!! ϕλϕλϕλ ϕβϕαβαϕβαϕ +=+∈∀∈∀⇔ +=+∈∀∈∀⇔∈ %% !4!! " # - !!4 " , - 4! =λ " !*!!*!* ϕϕϕ +=+∈∀ !33!* = 3!3 =ϕ " !!3!!3!3 ϕλϕλϕλϕλϕλ ⋅=⋅+=⋅+=⋅+∈∀∈∀ ( • 5' " • 5' " • 5 ( % ⋅ < ∈ o ;) % ⋅ !*"!!*"!*"!*"* λλλλλ +=+=+=+∈∀∈∀ o > !∈ " ∈ !"!"! == != "! =∈∀ " • 5' * !! → " • 5' ! !! → % !! !! ' ! • 5' ! ! π ψ → !∈" !!!!!!! """" ψψπππψ +=+=+=+ !∈ ∈λ !"!"!!"!" ψλπλπλλψ === " • 5' ! → ' " # ) % ( → % ( ! , - ! → " 1! N @ & !∈ϕ " 5 @ ϕ ' @ { }3!Q =∈= ϕϕ " ϕϕϕ Q*!*!* ∈−⇔=∈∀ " ϕ ; { }3Q =⇔ ϕ " touscours.net
  12. 12. %% + & ϕ ; & ϕQ∈ " !33! ϕϕ == " 3= " '< ' ' " + & - { }3Q =ϕ " & !*! ϕϕ = ϕQ*∈− 3*=− " *= " ϕ ; " ϕQ + " ;) ⊂ϕQ ϕQ3 ∈ " & ∈∈ λ* " , 33"3!*"!!*" =+=+=+ λϕλϕλϕ 5' ϕ { } { }=∈∃∈=∈== !!!> ϕϕϕϕ " ϕ ; =ϕ> " ϕ> + " ;) ⊂ϕ> ϕ>3 ∈ 3!3 =ϕ " ϕ> $ ⋅ & ∈∈ λϕ>* " > ( ∈* - !**! ϕϕ == " !*"!*"!*" λϕϕλϕλ +=+=+ " ϕλ >*" ∈+ " ! > - ' + & !∈ϕ " 5' ϕ ' + + " 5' - ϕ ' + + " !ϕ + ' ϕ> ! { }!3− ϕ + ' ϕQ ! , ! touscours.net
  13. 13. 4 ! & ' ! & & !∈ψϕ ∈λ " , % !! ψϕ ψϕ + →+ !" " ϕλ ϕλ → !" ∈+ ϕλψϕ " , !! ⋅+ !! ⋅+ + B + !! ⋅+ !" ;) % - !! ⋅+ + D ! ! 3 $ ⋅ & !∈ψϕ ∈λ " , ∈* ∈µ !*!"!! !*"!!*"! !*"!*"!*"! ψϕµψϕ ψµψϕµϕ µψµϕµψϕ +++= +++= +++=++ !!*!""!!" !!*""!!" !!*"!" !*""!*"!" ϕλµϕλ ϕµλϕλ ϕµϕλ µϕλµϕλ += += += +=+ !" ∈+ ϕλψϕ ! + !! ⋅+ + " ! 5 - % ( " & !∈ϕ !∈ψ " ϕψ % " ∈* ∈µ !*!"!! !!*"!! !!*"! !!*"!*"! ϕψµϕψ ϕψµϕψ ϕµϕψ µϕψµϕψ += += += +=+ !*∈ϕϕ !*∈ψψ ∈λ !*!* !*!* ϕψϕψϕψψ ϕψϕψϕϕψ +=+ +=+ touscours.net
  14. 14. 6 !6!"!" !4!"!" ϕψλϕψλ ϕψλϕλψ = = ;) % " ∈ !/*0 !!*!!!!*!! !/*!0!/!*0!!/*0 ϕψϕψ ϕψϕψϕψϕψ ϕϕψϕϕψϕϕψ += +=+= +=+=+• '< !" !/*0!!*!! !!*!!!!!*!/!*0 ϕψϕψϕψϕψ ϕψϕψϕψψϕψψ +=+= +=+=+• '< ! ' D ! !!/"0 !!"!!"!/"0!/!"0!!/"0 ϕψλ ϕψλϕψλϕλψϕλψϕλψ = ====• '< 4! !!/"0 !!"!!"!!!"!/!"0 ϕψλ ϕψλϕψλϕψλϕψλ = ===• '< 6! ' ! - % ! !! + !! + % !! ⋅+ + !" ! ! - $ - ! !" % @ ) > " ' ' % " ( !! → !3! " − → " !∈ & ∈∈ λ* !***! == " !*"! !**"!!*"*" !*"*"!!**"!!*" λ λλλ λλλλ += +=++= ++=+=+ !" ∈ & ∈∈ λ* !***! == " !*"! !3**"!3!3!*"*" !*"*"!!**"!!*" "" """ λ λλλ λλλλ += −+−=+−+= ++=+=+ , !! " −− → !33! " → - " touscours.net
  15. 15. 7 4! > ! & !∈ϕ " & ϕ ; !∈− ϕ " , - ϕ " ( - ' ( ' ' " & ∈* ∈λ " , - !*"!!*" −−− +=+ ϕλϕλϕ * +)+ - *"λ+ !*"! −− + ϕλϕ ϕ - *"!!*"!!!!*"! λϕϕλϕϕϕλϕϕ +=+=+ −−−− 9 R ; " R " R ; % " 5' ! " ! !! !" ' ' !! + " ' % " ( & !! −+ → !! " → " ; " " ; >= =− ! = = ≠ +− −+ !3! !3! !! !! " " "" " " 6! & % !" & ∈3 " ' 3!" →φ " %% & ∈∈ λ " !"!!""!"!""!" 333 φλφλλλφ +=+=+=+ ( 5' ! 4 → 4 !****! ∈== ∈λ !*"!*"!!*"*"*"!*" λλλλλλ +=+=+++=+ A ; - 4 C touscours.net
  16. 16. J B ! 4 → ! 4 4 → " + > ! - ∈# #"""! 4 ++ → 4#++= " + 6! - ∈# !3"""! 4 #++ → !3"= !3!" 4 → B !*"*"*"3! 4 #++ → '< !*"*"*""""! 4 ## ++++ → " , - 4 @ " , ) ! ; + " ! S ! % ? @ "α → < ∈α " ∈α ' "αα → α " 3=α ; ' α. 1! ; ! % & ( + " 5 ; ' → < ' - += ∈∈ " % ' + - ∈ ∈ ! , % += →⊕= " 5' @ ' " & ∈∈ λ* " ∈∈∈∈ +=+= *** " ∈∈ +++=+ *"*"*" λλλ !*"!*"!*" λλλ +=+=+ " touscours.net
  17. 17. K N @ & ∈ ∈∈ += " , - ∈⇔=⇔=⇔∈ 33!Q > , ;) - ⊂> " > >⊂ ' ' ( + B " % & → " , - ; - ' ( ( + - ; " 9 ; " ∈ ! ; " E & !∈ " ; =⇔ " & ; < =⊕ = & ∈ " ∈∈ += =! " !!!!! ==== " ' = " & !∈ - = " >= Q= " ;) ( + " # - ' " & ∈ " ∈! !! −+= ∈ 3!!!!! =−=− ∈− ! ;) =+ " # - { }3=∩ & ∩∈ " ∈ " !*= < ∈* " ∈ 3! = 3!!* = " = 3!* = " 3!* == '< ' ' " =⊕ # - ; " & ∈ " ∈∈ −+= !!! " ! % ∈∈ + touscours.net
  18. 18. 2 - & ; { } !>Q ! −= = =∈= %% ∈⇔=⇔=⇔= =− 3! 3!!> % & ; " 5 ; ) ; $ " $ >=+ " !! $+= 5 ; { }3 ' " 5 ; { }3 ' " ! &@ ! % & ( + " 5 @ ) ' −+= →⊕= " touscours.net
  19. 19. O & @ - ) !∈• " %% - - $ >−=−= < ; $ ; ) ! • ; B " >= => ; ! { }3Q = ; ! { } { } !>Q! !>Q! +=−=∈= −==∈=• E & !∈ " @ >=⇔ ⇔ ! ⇔ ' ≤ ! ! ;) " ⇐ - >= " !>Q −= !>Q += " ( + @ ( ' " { }3=∩ ∩∈ =! −=! 3" = 3= 3≠ ! ' ∈∈ += !!!! −++= " , ∈+ ! !!!!!!! +=+=+=+ ∈− ! !!!!!!!! +−=−=−=− % @ ) " %% & ∈∈ += !!+= !!−= " !=− " ! " # ; + " ! % & !"""= % " T % ⇔ - 3 %% " [ ] =∈∀=∈∀⇔ = 33!""" λλλλλ touscours.net
  20. 20. 3 9 !"""• !""" % ⇔ ' " • 5 - !""" @ 3= = λ < λ ' " • - !""" % - " ( + % " + ' % " 5 % ! 3≠⇔ + ' % ! " 1! • & % 3 & 3= 3" 3 = ≠ • & % ( ( & = ≠ ! 3=− • & +% ' % " • & !"""= !""" !!! σσσσ ∈∀ " • & !""" !""" !"""9∈ " %% > ( µλλλ """ - 3"""" =+++ µλλλ " 3≠µ ' λ ≤≤ " = − = λµ • !""" ' " $ " # % & !""" % " !""" !""" % ⇔ % " ⇔ ' - ≤≤ % = " 5 ' !""" " touscours.net
  21. 21. - !""" " & ∈ " !""" ( ∈!""" - = = " & - ' = = * " 3!* =− = " !""" [ ] 3* =−∈∀ [ ] *=∈∀ " '< '( ' ' " ⇐ & - ' - D ;) !""" " 3= = λ [ ] 3=∈∀ λ ( %% 3 ) = λ = "3 " ( !/33!33!330 4 ' - 4 " !/!!0 −−− " 5 ' ! 4 +++ " /!33!6!0 %π 4 & 4 !∈= # " , - ' ( - 4 - """ ++= 5'- - @ =+ =+ =++ # % ! 6 " ! π , − − −= − − = =++ ⇔ 6 6 6 6 " ! π π # # # % ! !! - " touscours.net

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