Algèbre - Espaces vectoriels

2 3
n p
R
(S1) : (2x+y= 1
x+ 2y=−1(S2) : (x+y+z= 1 x−y+z= 0 (S3) : 
x+y= 1
x+ 2y= 2
2x+y= 3
(S4) : (x+y+z= 1
x+y+z= 0 (S5) : 
2x+y= 1
x+ 2y=−1
x+y= 0
K R C
K
EK
E
R3
A=R2B=R3
C={(x, y, z)∈R3|x+ 2y−z= 0}D={(x, y, z)∈R3|x+ 2y−z= 1}
F={(x, y, z)∈R3|xy +z= 0}G={(x, y, z)∈R3|x2−2y+y2= 0}

EK
F, G E F ∩G
E F ∪G
A E
A
F+G F, G
E
F G
R3
F E F ={(x, 0,0) |x∈R}R3
R3
F
EK=R C
F= (x1, x2, . . . , xk)
E
F
R3
F
R3
F
R3
(1,1,1)
E
E
dim E
R32

E
dim E
dim E
dim E
dim E
E
((1,2,3),(4,5,6),(3,3,3)) R3
F, G E
F⊂Gdim F6dim G
F⊂Gdim F= dim G F =G
dim(F+G)