Feuille 2 Fichier

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R3
F1={(x, y, z)∈E|x−y= 0 x+z= 1}
F2={(x, y, z)∈E|y−2z= 0 x+y= 0}
F3={(x, y, z)∈E|y−2z= 0 x+y= 0}
F4={(x, y, z)∈E|x2+|y|+z4= 0}
F5={(x, y, z)∈E| |z+ 1|2− |z−1|2= 0}
F6={(x, y, z)∈E|x2+y2+z2=−1}
F7={(x, y, z)∈E|x−5y+z= 0,2x+y−7z= 0,−x+3y−z= 0 3x+2y+z= 0}
E=R[X]
F1={P∈E|P(1) = 0}
F2={P∈E|P(1) = 2}
R R2R3
R3
E F G E
F∪G E ⇐⇒ F⊂G G ⊂F.
E=F(R,R)
E1
E2
E3
E4
E5
E6
D1D2K2
D1=D2
K2=D1⊕D2
E=R4
F={(x, y, z, t)|x−y= 0 z−t= 0}H=V ect((1,−1,0,0),(0,0,(1,−1)) .
F H E
E=F⊕H

CE
ER3v1= (2,3,−1)
v2= (1,−1,−2) F w1= (3,7,0) w2= (5,0,−7) E F
v1= (1,0,0,1) v2= (0,0,1,0) v3= (0,1,0,0) v4=
(0,0,0,1) v5= (0,1,0,1) R4
{v1, v2} {v3}R4
{v1, v2} {v4, v5}R4
{v1, v3, v4} {v2, v5}R4
{v1, v4} {v3, v5}R4
FR R
F
P
I
P I F F =P ⊕ I
ker fi
fifi
f1:R2→R2f1(x, y) = (2x+y, x −y)
f2:R3→R3f2(x, y, z) = (2x+y+z, y −z, x +y)
f3:R3→R2f3(x, y, z) = (y2, x +z+ 2)
f4:R2→R4f4(x, y) = (y, 0, x −7y, x +y)
f5:R3[X]→R3f5(P) = P(−1), P (0), P (1)