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# Matrices, Matrix

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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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### Matrices, Matrix

1. 1. ! "# \$ % ! ! ! &# " ' ! ( " ) ! ≤≤ ≤≤" !* ! ! + [ ] [ ]× # \$ " , " " -" . !* ' ≤≤ ≤≤= " ! " ) ! / [ ] [ ]" ×∈ 0 ! 0 # " 1 1111 1 ∈= \$ 0 ! ""###1 ∈ " \$ 0 ! ""###1 ∈ " ' ! ( " * !* ! # 2 0 ! ! !* "###1= # 3 ∈ 1 ! ! touscours.net
2. 2. 1 \$* → 1 "ϕ ! 4 !* = 1 " ! "###1= !* ! ! "∈ / [ ]∈ 0 ! ! ! ! # " # 5+ 11 ,,1 ++=+=−= ! " ! ! 671 "1 " − 8 81 * ! ( )""8"81− ! / ! # 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 "∈ϕ \$ ! ϕ ! ! ) / ! "ϕ ! * """"###" 1 ϕϕϕ "ϕ "∈ ! / "∈ϕ / "ϕ= # * / ! ! ! ! ! # * " "" ϕϕ φ → 4 # 3 = = "ϕ "ϕ ! ϕ ! touscours.net
3. 3. " 2! φ ! 0 ! " ! / φ ! 4 " " 3 5 ! 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 "∈ ! ≤≤ ≤≤= " ! # 3 "∈ ! ≤≤ ≤≤= " ! # [ ]∈ === +=+=+=+ """" \$ ! + ! ! ≤≤ ≤≤= " ! [ ] [ ] +=∈∀∈∀ 3 ≤≤ ≤≤= " ≤≤ ≤≤= " ! + ! " # + ≤≤ ≤≤= " / [ ] [ ]" +=×∈∀ # 9 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 "∈ """ +=+ ! * ! # -" ! \$* ! ) ! "∈ ∈λ 3 ≤≤ ≤≤= " ∈λ # λ ≤≤ ≤≤= ":: / [ ] [ ] :" λ=×∈∀ # touscours.net
4. 4. ; 9 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ "#"# λλ = " \$ 0 " 9 • #"" + 0 • 3 0 ! ! !* 3 0 ! ! !* # " "" ϕϕ φ → ! 0 # • < ! ! ! ! ! 0 "8 ! " • 2! 3 = ! / 5 = ! / * " "" →φ * / ! " "# " 9 " ! ! × ! " ! [ ] [ ]" ×∈ = ! " ! !* ! " / # . / ≤≤ ≤≤= " = # / 3 0 ! ! !* # 3 0 ! ! !* # " ! ! × # " ) "# touscours.net
5. 5. > " 5 ! 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 →ψ # 3 →ϕ # ψϕ ! ! # 3 """ ∈== ϕ 3 """ ∈== ψ 3 """ ∈== ψϕ [ ]∈ [ ] [ ] = = ∈ ∈ = == = == == == === """" ϕϕψϕψϕ [ ] = =∈∀ [ ] [ ] = =×∈∀ " 3 "∈ "∈ # ? × ! " ! [ ] [ ] = =∈∀∈∀ 9 3 0 ! ! !* # 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ψ "∈ϕ # """ ψϕψϕ ×= 5+ −− = − × − 1@11 A ;;1 touscours.net
6. 6. @ -" ! * !* 9 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 "∈ϕ "ϕ= 3 ∈ "∈ ! ! ! # 3 ∈ "∈ ! ! ! # ? * / ×=⇔= "ϕ B ≤≤ ≤≤= " = 1 = 1 ? = = # = − == == === "### !"! 1 "" ϕ ϕϕ [ ] ×=⇔ =∈∀⇔ ⇔= = !C"" ϕϕ 5 =× = = = 11 1 1111 1 5+ 3 "1 ∈ϕ ! = ;1 ! / 1 ! 1 B "11 = , ";,""1," 1 == ϕϕ " ! 1 "::" =ϕ =× : : 3 + + =×= ;1;1: : touscours.net
7. 7. " ! ! ∈∈∈∈ λ"":": " ××=××=×× "" 1" ×+×=×+ :": " :": ×+×=+× ;" "#"#" λλλ ×=×=× >" ! =× ! =× ?= ! ≤≤ ≤≤== " 88 8 8 88 δ =δ = 8 # 5 + 1" 3 0 ! ! !* # 3 0 ! ! !* # 3 0 ! ! !* # 3 ":∈ϕϕ ! * ! # 3 "∈ψ ! -! # ( ) ( ) ( ) ( ) ×+×= += += +=×+ : : : ":": ψϕψϕ ψϕψϕ ψϕϕ ? ! ! C " \$ ! ! # . / ! * !* ! # *( • × C ! × 5+ ! ( " ! ( " ≠ • × × C ! ! C ( 5+ ! ( " ! ( " ≠ • × × C ! ! C ( ! # 5+ ! ( " ! ( " − − = − − × =× − − >> >> 1 1 88 88 1 1 2 *( + 0! " touscours.net
8. 8. A !" # \$ " . • "" = ! !* ! ) ! # • ' 0 ! ! + ! D E !* ) ! / 0 #"+ 0 # 0 E ! D ∈λ ∈ "#"#" λλλ ×=×=× 0 + E + 67" " -" 9 • #"" ×+ 0 • 3 0 ! ! !* * " "" ϕϕ φ → ! 0 # #"" + 0 " • ? / #"" + 0 ! ! 1 "# ! E ! " ! D ! ! F ! ! G# • ? ! 4) / φ ! 0 # "∈ψϕ """""" ψφϕφψϕψϕψϕφ ×=×== !== "2!"2!φ . / : ! * ": "" ϕϕ → 4 ! 0 ! 0 !≠":2! " 5+ 1 [ ]"8"8= [ ]""1:= − − == == 1> ":2! 1 ":2! 8 8 "::2! 8 8 "2! touscours.net
9. 9. H " / ! • . ! * "" ×+ ! 0 #"" + F \$ ! ! G : 0)0! I !* 0 " • B ! "∈ =∈∀ = + !8 • 5 4 ! * " "∈ ! + / = − =+∈∀ 8 " "###"& 1 −−− +++×−=−∈∀ 5+ 3 = 88 18 1 # ! ! # ! ? ! +== 888 188 18 88 18 1 ? 8 888 888 ;88 888 188 18 18 ==== ! ! ! ! " 11 111888 88 1 " "1 """ ! !! −− −−− = − = − − ++= ++=≥ ==+= −+ = − −−− 88 18 "11 1 touscours.net
10. 10. 8 % " 3 "∈ ! ≤≤ ≤≤= " # \$ ! "" ∈ ! " ≤≤ ≤≤= ": = [ ] [ ] : =∈∀∈∀ 5+ == @>; 1 @ >1 ; " -" ∈∈∈ λ"": """ "" """ "" = = +=+ = " "" :": "" λλ * ! # / B ≤≤ ≤≤= " ≤≤ ≤≤= " ≤≤ ≤≤= " " ≤≤ ≤≤= ": " ≤≤ ≤≤= ": "" ≤≤ ≤≤= ": [ ] [ ]∈∈ ::: ==== === """ =" # " ( / ( / 3 "∈ ( / [ ] [ ] " =⇔=∈∀∈∀⇔ ! ( / [ ] [ ] " −=⇔−=∈∀∈∀⇔ ! 5+ 818 11 8 ( / − − 818 18 88 ( / touscours.net
11. 11. \$ "# " ! ( / ( / ! " ! + 0 ! " ! ! 1 "+ 1 "− = − # 888 88 88 88 888 888 88 88 888 88 1 " 888 88 8 88 888 888 88 88 88 88 <" 5 ! C 1 " "! − = C / "# ! J F ! G 0 ! " 1 "!"! # =+ "" #∩∈ ! "8= "# " ! """ # =⊕ " K 3 "∈ !==∈∃⇔ " ! * ! ! * E ! " 3 "∈ + "∈ / !== # * * ! − # ! ! " * ! ! " " # 2 E# ! +" • " E 3 "∈ "∈ " −−− = # • 3 "∈ "∈− =−− " • "! ∈ . / = ! C ( # \$ !* " ! !* * ! " # touscours.net
12. 12. 1 -" 9 9 3 "∈ 3 0 ! ! !* # 3 * 0 ! ! !* : # 3 ":∈ϕ ! ! : # ϕ 4 # 3 * − ! − ϕ ! : # • 3 ! − * →:ψ ! − ! : # 2!=ψϕ :2!=ϕψ 5 !=×= ×= − :::: """ ψϕψϕ !=×= ×= − :: """ ϕψϕψ ϕ 4 ψϕ =− • 3 ϕ 4 # ? ! − ϕ ": − = ϕ # != = = ×=× − − "2! " "" ::: :: :: ϕϕ ϕϕ != = = ×=× − − "2! " "" :: ϕϕ ϕϕ =− 9 3 0 ! ! !* # 3 "∈ϕ "ϕ= ϕ 4 ! "−− = ϕ / 3 0 ! ! !* # " "" ϕϕ φ → ! # touscours.net
13. 13. " 5+ 3 − = 1 # 0 / − L ! + 3 = " \$ # ? / { ;)/A!(⇔== ! 1 ! 3 ϕ * ! ! 1 ! ! / 1 # "1"" 1 +−=∈ ϕ # 3 1 " ∈ # ? / − = + = ⇔ =+ =− ⇔= 11 ""ϕ ϕ 4 − ϕ − MM1 MM ! 1 # − =− 1 + 3 = 1 ;1 3 "1 ∈ϕ ! ! / "1 = ""1<""1;"1<"""<2 === ϕϕϕ # ϕ2 ! ! # ϕ * ! 1 ! ϕ * 4 ! * # " 2 "∈ " ! 3 0 ! ! !* # 3 "∈ϕ ! ! # 4 4 4 ϕ ϕ ϕ ⇔ ⇔ ⇔ touscours.net
14. 14. ; 1" ⇔ ⇔ ⇔ "!3 3 "!3 !! " ! / 3 φ * ! ! " ! ! ! " === 8 8 8 8 1 8 8 ### [ ] "!"" ###" 4 1 φφφ φ ⇔ ⇔ ? [ ] "φ∈ * / 0 ! # N 3 0 ! ! !* "###1= # [ ]∈ ! ! ! ! 0 ! # [ ] !###1⇔ \$ ! C "∈φ ! ! / [ ] "φ=∈∀ " = / ! # \$ : 0)0! 0 " " ( ⇔ "∈= ( # =" = = ! / # 5 #" ! * F φ 4 =φ ! !* 0 ! ! G touscours.net
15. 15. > 5 3 "∈φ ="φ "###1= =∈∃∈∀⇔ =∈∃∈∀⇔ =∈∃∈∀⇔ ⇔ "###O"### ""###O"### "O 4 11 11 φ φ φ ' ( = = ∈ ∈ ∈ !" " " ( ! # 2 ! * / − = ;" 2 ) ! ) 9 ! ! =∈∃⇔ =∈∃⇔ " " 5 ! ) − = ! 4) ! ) ! ! # / 3 / * + "∈ / != 3 0 ! ! !* # 3 "∈ϕ ! ! # 3 "∈ψ ! ! # 2!=ψϕ ϕ 4 ! * ""ψϕ # ϕ 4 # # 5 −−− === ! # 1 / ! C # 2!=ϕψ # ϕ 4 === :"""":"": ϕψϕψϕϕ ϕ 4 # % >" 9 " ⇔ 5 ! "" −− ="" # 3 !== −− touscours.net
16. 16. @ """"""" !! ==== −−−− """","" " !* "−" # . / " """ : 0)0! / # / ###"OO! "!3 3 "!3 ∈⇔ ⇔ ⇔ ⇔ \$ ! ! " " 5" 5+ " \$ ! ? " * ! ! !* ! ) ! # * ! ! O" " ! " C " 3 "∈ = λ λ λ 88 8 8 88 1 # [ ] 8≠∈∀ λ ! = − − − − 1 88 8 8 88 λ λ λ • 3 ! λ ! ! ! * # * # • 3 ! λ ! " !== # =− touscours.net
17. 17. 1" \$ ? "%# * ! !* ! ) ! : 0)0! ! ( ≤≤ ≤≤" = 8=> # * ! ! " "%# ! " " 3 "" %#∈= ≤≤ ≤≤ ! + # • 3 = &88 P PP&8 PPP& & ! " ! "###1 3 8###11 =+++ λλλ ! 8=λ # 8=λ 8≠ " ! % • 3 * ! ###1 !* ! ! − ) * ! ! ( − 8 8 / "###< 1 − = "###1 ! "" "###1 # * # * = * / # . / / * ! # " ? C / C ! / / " %" touscours.net
18. 18. A / ' ( #" : 0)0! ! ( =× = ! " / : 0)0! / " 8 ! ! : 0)0! / * ! " 5+ = =+ =+ λ λ λ 111 0 \$ ( ! / ! + # 0 3 * ! λ ( * # 5 * ! λ # ? [ ]{ }8=∈= λ • 3 = : 0)0! 8=λ 8≠<∀ λ " 0 3 8≠ ( # 0 3 8= / * + / / "###1 ) #" ( #*" ! − / ! !* 1 ### − #*" / / 8 ! "# #" ! ! ! " • 3 #**" ( F G " * / Q / !* ###1++ # o 3 #**" #" * # o 3 #**" 0 3 ! ! #**" 0 #" # 0 3 * "###1++ + ! ! #**" 0 + "######1 + ( #***" 0! " ! 0 / ! / !* ###1 # #" ! ! ! " touscours.net
19. 19. H # =" "∈ = = # ? / "∈ #" * ! " 5 • 3 #" * ! * ! % O • 3 ! 8 # / * !* # * # 3 ϕ * ! / ) # ϕ * 4 # { }8R ≠ϕ # * / "8= ! / 8# &λ+8 =& ! "8= ∈λ ! ! #"# 5 && =+=+×=+× 8" 88 λλ & ! " ! ! !* 0 ! ! # 3 "###1= ! # F G" 3 ":###::: 1= ! # F G" ? / * S ! : ! # !* = !* ! "# / ! ! ! * ! * ! ! ) *# [ ] = =∈∀= = :/!"! :#)!! ": ! * ! " ":2!= ! ! *" / ! ! ) * − ! ! * ) # 5 "2! ∈ # ":2! !* ":2!":2! 5 0 5 = / ! ! * ) # . / ! ! ! ! / ! ! ! ! # touscours.net
20. 20. 18 9 3 ! * ! # 3 ! ! ) *# 3 ∈ ! ! ' * ! ! *# := ! " = ! "2!= : 0)0! := ! ! * 2! O" ! ? ≤≤ ≤≤= " = = : : : # ? = == == === :::: = = [ ] = =∈∀ : # = : : 5+ 1 ! / "= # 3 !* / H1>1" 11 =−+ ! : 0)0! / * ! ! 1 ! " ! "" 3 ! += 1 ( −= ": (!= ! 1 # ? * / ! ! *# ! ! ) * − 1 3 1 ∈ ! ! : : ! *# − = : :1 ? / :: H:H:H H::1:1:;:>::8:>:1::A:A H"::"::11"::>"::11 H1>1 11 11 111111 11 11 =+⇔ =+⇔ =++−+−+++⇔ =−+−−++⇔ =−+⇔∈ touscours.net
21. 21. 1 -" \$ ! ! 9 3 0 ! ! : ! + ! # 3 ! ! ) : # 3 0 ! ! : ! + ! # 3 ! ! ) : # 3 "∈ϕ "ϕ= "::: ϕ= # : − = 2!2! ϕϕ = : ":2!"":2!":: − = ××= ϕϕ ! ! ! " = :::= = = : : ::= # :: − = # ? :::= # : − = 3 "∈ϕ * ! + ! ! ! ) *# 3 "ϕ= ":: ϕ= : − = # & ' ! " . !* 3 "∈ # ! " = ! ! # touscours.net
22. 22. 11 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ϕ ! ! # 3 "###1 ! ! ! ! ! # ""###" 1 ϕ== ? * φ ! " ! / "###1 ! " # : 0)0! = →"φ / "φ = ###1 ! " ? φ * " "###"###" 11 ! == ? "ϕ / """ ###"" 1 ϕϕϕϕ = ""###" 1 ϕ== / 3 "∈ "= ≤ !* ! ! ! ! " ≤ ! " == 8= -" / 3 "∈ # ? ! / / ) / * + "∈ "∈ / − = / 0 * / ! " : − = ( " 3 "∈ 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ϕ ! ! # / ) + : ! : ! / ! ϕ ! : : # touscours.net
23. 23. 1 5 / ) C ! ! ! # 3 : : / "::ϕ= − = = ! ! ) : ! ! ) : # 2 − = ! : ! / ! ! ) : : ! / ! ! ) : # "::ϕ= # \$ F C / ) G " !* / : 0)0! + ( / . + !! − = 3( / 3 − = "" −−−− == 9 3 − = #− = """"" #### −−−−−− ==== " 9 3 "∈ # / "" =⇔ " 3 / 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ϕ ! ! # + : ! : ! / ! ϕ ! : : # : 0)0! "ϕ= "::ϕ= """ == ϕ 1" 3 / == "" \$ 3 "∈ "= ? / / ) ( = 88 88 8 8 88 touscours.net
24. 24. 1; : 0)0! ( ≤≤ ≤≤= "γ = >=≠= ≤= "8γ γ 3 0 ! ! !* # 3 0 ! ! !* # 3 "∈ϕ ! ! # ="ϕ # −="! R ϕ # 3 "###+ ! ϕR # 3 ! ϕR ! # ="! # 3 "### ! # "######: += ! # 3 ### ϕ ! ### # "### # 5 [ ] 8 8### R### 8"###8### 11 11 1111 =∈∀ =+++ ∩∈+++ =+++=+++ α ααα ϕααα αααϕααα ? ! ' "###: = "::( ϕ= / ) ( * = ! ! ! +! ! / ) ( # / # 9 3 "∈ # "" " = B "= # ( ! / ) ( # 2 + ! "∈ "∈ / (− = "− = ( """" # ? "" ∈ "" −− = "" "" ∈ " (( = / ! " " / ) ! # ! " =" "" " = # !* ! ! # . / ! − = 888 88; 88 811 81 # T ! L touscours.net
25. 25. 1> . / / ! "### !* 0 ! ! ( ↔• ≠+←• ≠←• # 8# α λλ ! "###< / # ? / / !* ! ! # > 888 118 88 888 8888 888 118 8888 888 88 1 888 118 8888 888 18 888 888 8888 118 18 1 ; " 1 ; ;1 > >11 > >;; = − −− − ← −← ↔ ↑ = − − −−− −← ↑ = − − −− ↔ ↑ = − − −− −← −← −← −← ↑ = & % " 3 ? * ! ) # ' % !* ) : % → = * ! ! ! * ! & 8≠← λλ & ≠+← α & ↔ 9 3 % !* ) # + "∈ / % ×=∈∀ " 3 "∈ 3 0 ! ! !* "###1= # 3 0 ! ! !* "###1= # 3 "∈ϕ / "ϕ= 3 ": %= touscours.net
26. 26. 1@ • 3 % 8≠← λλ # ? / ":: ϕ= ?= "######: 1 λ= 3 ! ! ! == − : = ! ! ) : : 0)0! ! " 88 8 8 88 −+== ← λλ • 3 % ≠+← α ! C ":: ϕ= "#######: 1 α+= ! == − : = ! # 88 8 8 88 α α +== ← ↑ • 3 % ↔ ":: ϕ= "#########: 1= ! == − : = ! 88 88 88 8 88 ++−−== ← ← ↑↑ * = * + ' 3 !!% ==" # * ! * ! # -" 9 3 * ! ) # ' % !* ) : % → = * ! ! ! * ! & 8≠← λλ)) & ))) ≠+← α & )) ↔ touscours.net
27. 27. 1 9 3 % ! ) # + "∈ / % ×=∈∀ " ? C ! / ! * / " 3 "∈ ": %= # ! / :" ) ! " ! ) % "# + "∈ / "" ×= ": # " ×= ∈ " : . / 3 % ×=∈∀ " "!%= " 2 C ! " ? / * M !* # 5 ! "" ######: ) ∈∈ = ! * / ) ! ! C # 1" ? / / ! ! !* ( 3 # =" F ! ( G F ! 1 ! G" """" ∈=∈=∈= ≤≤ ≤≤ =+++ =+++ =+++ # 11 111111 11 ### ### ### " U * :: =⇔= = * * ! ! ! C % # ! * =⇔= = "∈ # 9 !* ( ! # 5+ =−= = =+− =++ ⇔ =−= = =− =++ \$ # \$ # \$ # \$ # :": >88 18 1 : > 1 1 ": " 88> 81 1 > 1 1 " touscours.net
28. 28. 1A & ( ) " ! 3 "∈ # + ! / ! ) ! 5+ − −−= −← ↑ −− −−= −← −← ↑ = →→ 88 ;8 ; 1H@ ;8 ; 18@ 1>1 ; 1 111 ? / ( ) """ 1 === # ? 11111 1 ;88 88 8; 88 88 8; ;88 ;8 ; ! −← ↑ ←− ←− ↑ − − −← −← ↑ − −−= →→→ 5+ 1 1 1 11 111 88 88 81 ;88 8 ;1 88 8 ;1 8 88 ;1 111 @; ;1 ! ← −← ↑ −← −← ↑ ←− ↑ −↔ ↑ − +← −← ↑ −−− = →→→ →→ • = R# • 3 1≥ # 3 / "−∈ + ! / ! ) −! # 3 "" ∈= ≤≤ ≤≤ # ? " * ! "∈ "# ' ! 8≠ # −← [ ]1∈ ) PPPPP 8 touscours.net
29. 29. 1H 8 1 = = )λ "8###8"###1 ≠λλ =) 0 "0 ! " 8 1 = = λ # \$ ! + C 1≥ " ( ! ) − PPPPP 8 ! 5 −← 1≥ ← ) ! * !* # 5+ = 8@ 1>1 ; ! ! ! ( # ? ) ! ( = = \$ = # ? / \$ −− − − = ←− +← −← +← ↑ ⇔ −− −=−− ←− −← ↑ ⇔ − −= −− −− −← −← ↑ ⇔×=×⇔= − 1 1 @; 88 88 88 ;1 81 88 88 ;8 ; 1 8 81 88 H@8 ;8 ; 188 88 88 8@ >1 1; ; 11 1 ; 111 11 −− − −− =− @ ;H@ A1 touscours.net
30. 30. 8 3 "∈ = 8@ 1>1 ; " ? / −− − −− =⇔⇔=⇔= @ ;H@ A1 88 88 88 ### 88 88 88 8@ 1>1 ; ! ) ! ! ! # -" !* / / 5+ − = ↔ → − ↔ → − ↔ → − ↔ → − +← −← → −−−− − −← +← +← +← → −− −−−− − − = ↑ 888888 88888 88888 @818 118 : 88888 888888 88888 @818 118 88888 888888 88888 @818 118 88888 88888 888888 @818 118 88888 88888 888888 8@18 811 81@18 88888 8@@@18 8@18 811 1 1 88181 11 81; 8181 811 >; >;; @ !,8!/ 1>> 1 ;; 11 ? / ! ; / * / ) *"# ? ! ) ;@>( # touscours.net
31. 31. N 3 "∈ ! # " ' ! !* ! ! ) ! ( = 8888 8888 PPPP& PPPPP8 PPPPPP& ! / ! 8= 8= " 1" ! ) ( ? / / ) ( "" ######( ∈∈ = . / ! ( ! * ! ! N # ' ! ! # / ! ! ) ! # ! 3 "∈ ! 0 3 ! 8= # 0 3 * ↔ ? 8≠ # 3 8= / 8≠ 8≠ " ↔ ? 8≠ # ? −← 1≥ " / ) = : PPPPP 8 & * 4 / *) / * ) 8 PPPP PPPP PPPP 88 8 & P8 PP& touscours.net
32. 32. 1 & *+ " + " " ' ( ! / ) ! =+++ =+++ =+++ # 11 111111 11 ### ### ### " ?= "" ∈= ≤≤ ≤≤ ! ( = ! = ! ! # . ! #" * ! * 3 ! : 0)0! * ! 0 ∈"### / ! #" # -" 2 #" ! ! ( *= = = ! * ! ! !* ! ! + " " ? / • #" ! "###< 1*∈ • #" ! "###1 # 5 ! " • 3 "###1 0 *( ! 8 0 ( ! "###1 # 3 ":###:: 1 # / ":###::"### 11 = # ? *== == : = =− 8": # [ ] :=∈∀ • 3 "###1 + "8###88"###1 ≠λλλ / 8= = λ # "###1 ! #" "###11 λλλ +++ # touscours.net
33. 33. . / "###1 ! = " #" ! / / / * #" ! ! ( *="ϕ =ϕ !* ! ! ! ! ! ! ! = * ! ! ! = " ! ! / "###1= "ϕ " • #" ! ϕ2∈* # • #" ! ϕ 4 C ! " • 3 ϕ 4 = " #" / / ! # • 3 ϕ * 4 ! #" ! ! ! ! 0 3 ϕ2∉* #" * ! 0 3 ϕ2∈* #" # o 3 ϕ 4 # o 3 " { }ϕR8 ∈+ = 8 + ! #"# 5 8 ϕϕϕ R"" 88 ∈−⇔=⇔ # = ϕ 4 ⇔ ϕ 4 "# 3 ϕ * 4 0 3 ϕ2∉* #" * ! 0 3 ϕ2∈* #" ! # #" ! = = = " " " 11 ϕ ϕ ϕ ?= ϕ ! ! ' ! = → +ϕ !* ( : 0)0! / ! ! " ( ! , = ∈ =, * ( ϕR # touscours.net
34. 34. ; . / ! * ! ! # =" * ! ,#&& ∈+8 = 8 ! "# ,# * ! ! 8" =, ) #"# " . ! ! !* " ) = − − # 1 1 8888 8888 PPPP& PPPPP8 PPPPPP& " ? ! 4) / #" [ ] 8=+∈∀ # • 3 [ ] 8≠+∈∃ #" • 3 [ ] 8=+∈∀ ( #" / ( #*" = − # PPPP&8 PPPPP PPPPPP& " ? / !* ###1 ( + ( ( 8 ! − − = += += &8 P PP& \$ ( ! F + − G# 3 #" ) / ! # • 2 ( − ! ! • \$ / * ( ! ! − ! ! # • 3 = ( 4 8 ! ! " • 3 = / ( ( / # touscours.net
35. 35. > • 3 == ( # • 3 #" 4 " * ! 0 ! ! − # " " ( V ! 3 "∈ # 67∈ ! = = 8 = = 8 " !=8 "# * " "67 →φ ! 0 : 0)0! / 67∈ ∈λ != ×= +=+ "" """" """" 67 λλ < = " 9 ! ( V ! ! ! 1 ≤ ( V ( V / 8" = "# 5 1 ###8 1 + ! " # [ ]1 8 " ∈ # 2 + ! "8###88"### 18 ≠λλλ / 8 1 8 = = λ # \$ ( V = = 1 8 λ ! 8" = # ? C / φ * 4 / { }8R ≠φ "# 2 + 67∈ / { } { }678"67 ∈==∈ # / ) # 5 ? ( V ! ! " ! # 3 - ( V # \$ ! ! ! - ! - += !! < """" 88 - +×= == # 8" = # 8= * ! ! # ! -# touscours.net
36. 36. @ 1" 3 "∈ # ? ! / / ) " / * + "∈ / − = ? / F C ) G !* / # 5 / F C / ) G " : 0)0! / F C ) G F C / ) G# / = ;1 = 8 8 ! / ! C " / !− = !== − ) C ! ! ! # 5 ! 0 ! ! !* "∈ϕ / "ϕ= ) ⇔ + * ! / ":ϕ= # \$ ! C / * / " ' ) ! ! " 5+ = 8 # / * ! # 3 0 ! ! 1 !* "1= 3 "∈ϕ / "ϕ= 0 * / ":ϕ ! L 3 / * + ! "::: 1= + ∈1λλ / :": λϕ = 111 :": λϕ = # 3 ! ! : ! # = λ ! = =+ λ λ 3 ≠λ 8== / : !* # =λ C 1 =λ 2!=ϕ / ! 1!≠ # * ! # touscours.net
37. 37. " 9 3 "" ∈= ≤≤ ≤≤ = = ! "9 \$* "9 " → ! " "∈ "9"9 = 3 ≤≤ ≤≤= " ≤≤ ≤≤= " ≤≤ ≤≤== " ./ ≤≤ ≤≤== " ? "9"9 "9"9 /. ==== ==== == = = == == / 3 C / " "9"9""9"9"9 ==== −−− 0 + / 1 8 8 9 8 9 == 8 * ) 8 8 / ! !* ! "9 ϕ ! * / / "ϕ= # touscours.net