Fiche 3

x1,· · · , xn
a1x1+· · · +anxn=b
a1,· · · , anb(x1,· · · , xn)n
n
x11,· · · , xnn
a1x1+· · · +anxn
b
(α1,· · · , αn)∈na1α1+· · · +anαn=b
b6= 0
b= 0 n
S
(x1,· · · , xn)
S=
−y−2z+3u−4v= 1
2x+3y+z−2t= 0
0x+0y+0z+0t+0u+0v= 0
0x+0y+0z+0t+0u+0v= 5
4 (x, y, z, t, u, v)
z3u5
2y
1x
(α1,· · · , αn)∈n
So`(S)nS
2x+3y+z−2t= 8
−2z+3u−4v= 6
0x+0y+0z+0t+0u+0v= 0
0x+0y+0z+0t+0u+0v= 3
n
n(0,· · · ,0)

p p 6n p
n−p
p=n
n
n
p<n
n
n−p
So`(S)n
S=
x+3y+z−2t+v= 4
−2z−t+3u+4v= 1
t+2u−v= 2
(x, y, z, t, u, v)∈6
x, z, t y, u, v
t= 2 −2u+v
z=−1
2(1 + t−3u−4v) = −1 2(3 −5u−3v) = −3 2+5 2u+3 2v
x= 4 −3y−z+ 2t−v= 4 −3y+3
2−5 2u−3 2v+ 2(2 −2u+v)−v=19 2−3y−13 2u−1 2v
6S
So`(S) = 19
2−3y−13 2u−1 2v , y , −3 2+5 2u+3 2v, 2−2u+v , u , v ∈6|y, u, v ∈
y, u v 6
n
So`(S)
So`(S) = 19
2,0,−3 2,2,0,0+y(−3,1,0,0,0,0) + u−13 2,0,5 2,−2,1,0+v−1 2,0,3 2,1,0,1∈6|y, u, v ∈
619
2,0,−3 2,2,0,0
(−3,1,0,0,0,0) ,−13
2,0,5 2,−2,1,0,−1 2,0,3 2,1,0,1
3
n1
2n
n
n
(0,· · · ,0) n
n
nn1
2n

p n
p
A A =
1 3 1 −2 0 1
0 0 −2−1 3 4
0 0 0 1 2 −1
M(p, n, )p n
A
b b1,· · · , bpp
n
(A|b) (A|b) = 
S T x1,· · · , xn
S⇐⇒ TSo`(S) = So`(T)
λ∈
λ∈

S´ech T´ech
M∈M(p, n, )p n p
(x1,· · · , xn)
M
S Shom =
S0
So`(S)So`(Shom)
n S n
Shom S
spart ∈ So`(S)S
Shom
So`(S) = spart +So`(Shom)
So`(S)nspart
So`(Shom)So`(Shom)spart
n