Feuille 3 Fichier

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v1= (0,1,1) v2= (1,0,1) v3= (1,1,0)
R3w= (1,1,1) (v1, v2, v3)
v1= (1,1,1) v2= (−1,1,0) v3= (1,0,−1) R3
e1= (1,0,0) e2= (0,1,0) e3= (0,0,1) w= (1,2,−3)
(v1, v2, v3)
R3
R3
t∈R(1,0, t),(1,1, t),(t, 0,1)
R3
v1= (1,−1, i)v2= (−1, i, 1) v3= (i, 1,−1)
C3
v= (1 + i, 1−i, i)
EK
f E K
H E u ∈E\H D =K.u
H
H E dim H= dim E−1
(x, y, z, t)∈R42x−3y+ 2z−4t= 0
(x, y)∈R2−x+ 3y= 0
(x, y, z)∈R3x+y= 0
(x, y, z, t)∈R4x−z+t= 0
R4
v1= (1,2,3,4), v2= (1,1,1,3), v3= (2,1,1,1), v4= (−1,0,−1,2), v5= (2,3,0,1).
F{v1, v2, v3}G{v4, v5}
F G F ∩G F +G
E=R2[X] 2
(1, X + 1, X2+X+ 1) E P =X2
E=Rn[X]{P0, P1, . . . , Pn}deg Pi=i
i= 0,1, . . . , n E
F G E
E=F+G
F∩G={OE}

dim(E) = dim(F) + dim(G)
f f :R2−→ R2f(x, y)=(y, 0)
Ker(f) = Im(f)
E=Im(f)⊕Ker(f)
F G F ⊂ FG ⊂ G
BF×Gdim(F×G) = dim(F) + dim(G)
F G E
Ψ : F×G−→ F+GΨ(x, y) = x+y
Ψ
Ker(Ψ) F∩G
fRn[X]R[X]P(X)∈Rn[X]f(P(X)) =
P(X+ 1) + P(X−1) −2P(X).
fRn[X]
n= 3 fR3[X]
f1, X, X2, X3
f
e1, e2, e3R3φ
φ(e1) = e3φ(e2) = −e1+e2+e3φ(e3) = e3
A φ (e1, e2, e3)
f1=e1−e3f2=e1−e2f3=−e1+e2+e3e1, e2, e3f1, f2, f3
f1, f2, f3R3
φ(f1), φ(f2), φ(f3)f1, f2, f3B φ (f1, f2, f3)
φ
P=
1 1 −1
0−1 1
−1 0 1
P P −1A B P
P−1
fR2A=22
3
−5
2−2 3
e1=−2
3e2=−2 5
B0= (e1, e2)R2B0(f)
Ann∈N
∀n∈N
xn+1 = 2xn+2
3yn
yn+1 =−5
2xn−2 3yn
e+, e−:R−→ Re+(x) = exe−(x) = e−x
e= (e+, e−)E=V ect(e)
Sh Ch e
h= (Ch, Sh)E
C C
C R C R
C R

EC= (c1, c2, ..., cn)
EdimC(E)
E
C R E
EdimR(E)