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Produit scalaire sur un R-ev

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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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Produit scalaire sur un R-ev

  1. 1. ! " # $ % # & ' # ( ) * +,+, ϕ ϕ →× ' • ∈∈ λ) +),+,+), λϕϕλϕ +=+ - ∈∈ λ) +),+,+), λϕϕλϕ +=+ ,. + • ∈ +,+, ϕϕ = , & + • ∈ /+, ≥ϕ ,0 1 + - //+, ==ϕ ,0 # ( 1+ -2 ' ! 3 = →× ++!!!,+!!!,, ϕ ! ϕ ! 4 % ' ! " 5 & ! /+, 6 ≥= = ϕ + 7 &(3 8 9 7 , &(3 8 9+ 3 ϕ ! ∈ ( ) +,+,+, 6 ϕϕϕ ×≤ " 3 ∈ • 3 /= +,ϕ +,ϕ , ϕ # + • 3 /≠ ! ∈λ +,6+,+, +,+,+,+,+, 6 6 / λϕϕϕλ λϕλϕϕϕλλλϕ ϕ ++= +++=++ ≥ touscours.net
  2. 2. 6 4 /≠ ϕ # ( ! " /+, ≠ϕ ! 4 * & : λ ' ; #! 3 # , +! " /+,+,+, 6 ≤−=∆ ϕϕϕ "%< % ! .+ = * " # % $ % → ' +,+,+,+, +,+, +//+,, /+, +≤+∈∀• =∈∀• ==∈∀• ≥∈∀• λλ 3 ϕ ! % +,ϕ , +,+, 6 ϕ=∈ + " " ;* ∈ /+, ≥ϕ ! " +,ϕ > ' /+, =ϕ ' % ' ' /= ! ∈∈ λ +,+,+, 6 ϕλϕλλλϕ == ! 3 ∈ ! 4 ' +,+,+, +,+,+,6 +,+,+,6+, +,+,+,6+,+, +,+,+,+,+,+, ϕϕϕ ϕϕϕ ϕϕϕϕ ϕϕϕϕϕ ϕϕϕ ≤⇔ ++ ≤++⇔ ++≤++⇔ +≤++⇔+≤+ ' +,+,+,+, ϕϕϕϕ ≤≤ , &(3 8 9+ 4 ϕ + " * " # $ % →× ' +,+,+, +,+, +/+,, /+, +≤∈∀• =∈∀• ==∈∀• ≥∈∀• touscours.net
  3. 3. ? 3 % +,+, − * ! " 4 +,+, −= /+,+, ≥−=•• % ' /=− = ! +,+,+,+,+,+,+, +,+,++,,+,+, +≤−+−≤−+−=−=• =−−=−−=−=• 3 * ϕ * % ϕ ! "+ 4 4 % ϕ ! " # 3 ∈ ! 4 ' 2, ϕ + ' /+, =ϕ ! 4 ⊥ ,⊥ & ( ( 0 > * 1+! ' /⊥∈∀ /+/, =∈∀ ϕ ! 7 , & + 3 ϕ ! ∈ %' 666 +,+,+, +=+⇔⊥ " 3 ∈ ! 4 66 6 +,+,6+, +,+,6+,+,+, ++= ++=++=+ ϕ ϕϕϕϕ ⊥⇔=⇔+=+ /+,+,+,+, 666 ϕ ! -+ " 4 % ϕ ! " +,+,+, +,+,6+,6+,+, +,+,6+,+, @ +,-+,+,6+,+, 66 6666 666 666 ϕ ϕ ϕ =−−+ +=−++ +−=− ++=+ touscours.net
  4. 4. A+ A 4 % ϕ ! " # 3 ∈+, # ! • 5 # +, # ⊥≠∈∀⇔ # +, +, #5 δϕ =∈∀⇔ ∈∀ ⊥≠∈∀ ⇔• ,4< =δ = / + = 5 # ∈+, # ⇔ ⊂ # # ∈+, ! ⇔ ⊂ # # ∈+,λ // =∈∀= ∈ λλ 3 ∈+, # , ∈+, # + ∈+, ! " 3 ∈+, # ! 3 ⊂ # ! 3 ∈+,λ # ! 3 ' /= ∈ λ ! 3 ∈ ! = = = ≠ ∈ ≠= ∈ // +,+, /+/, +, ϕλϕλ ϕ λϕ ! " /=λ ! " /=∈∀ λ ! " ∈+, ! " ∈+, ! -2 # 5 ' ! +B6/,C/ π= π6 / +, ! # ∈ # # ! - ## ∈ [ ] /66 6 / /++,, 6 / /++,,++,, 6 ++,,++,, 6 +,+, 6 / 6 / 6 / 6 / 6 / ≠==== ≠=++ + =≠= =− − ++ + =≠ −++= π π π π π ππ touscours.net
  5. 5. , # D∈ π + E+ 3 2 4 % ϕ ! " # 3 ! 4 { }⊥∈∀∈=⊥ , 2* +! ⊥ % ! " ⊥ ∈• / /+/, =∈∀ ϕ • 3 ∈∈ ⊥ λ) ! ⊥ ∈+ )λ ///+),+,+), =+=+=+∈∀ λλϕϕλϕ ! , ' % # ' ' ⊥ > ' ' F + " # ( 3 2 ! 2 ⊥⊥ ⊂⇔⊂⇔⊥∈∀∈∀⇔ # 4 ⊥ ! -2 , ? + ⊥ ⊥ ⊂) ' { } =• ⊥ / /+/, =∈∀ ϕ { }/=• ⊥ ! - ## 3 ⊥ ∈ ! /+, =∈∀ ϕ ! - /+, =ϕ ! " /= ! "%< { }/⊂⊥ % F ⊃• ⊥⊥ +, 0 5 2 * ' 2* 1! 4 3 ∈ ' ⊥⊥ ∈ +, ) * ' ⊥∈∀ ⊥ ' ∈ ⊥∈ ⊥ , # ⊥ + 5% # , # ' # + touscours.net

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