Algèbre Linéaire 1 L1 MI - DS 2

E
E
E
E
{u1, u2, u3, u4}E{u1, u2, u3}
F=
x
y
z
t
∈R4:x+y+z= 0 x+ 2z−t= 0
.
FR4
F
R4
u1=
1
2
−1
1
, u2=
−1
4
1
5
, u3=
2
2
−2
0
, u4=
−1
0
1
1
.
Vect(u1, u2, u3, u4)
F⊕Vect(u1, u2, u3, u4) = R4
R2[X]
P0=X2−2, P1= (X−1)(X+ 1), P2= (X−2)(X+ 1), P3= (X−1)(X+ 2).
R2[X]
P0P2P3
(P1, P2, P3)R2[X]
M=
−1 1 −2
2−1 1
−1 0 2 
M M−1
E x ∈E(x)B
B E

B1= (e1, e2, e3)E
f1=−e1+ 2e2−e3, f2=e1−e2, f3=−2e1+e2+ 2e3.
B2= (f1, f2, f3)E
u, v ∈E B1B2
(u)B1=
1
1
−1
(v)B2=
2
−1
2
.
u B2v B1
w∈E w B2(w)B2= a
b
c!
B1(w)B1=M a
b
c!M
B1B2