Algèbre linéaire II Série 21

V n T :V−→ V
pT(t) = (λ−t)n(T;λ)= 1
vn(T−λid)n−1(vn)6= 0
vn−i:= (T−λid)i(vn) 1 ≤i≤n−1
{v1, . . . , vn}T
A=
5421
0 1 −1−1
−1−1 3 0
1 1 −1 2
.
A
pAMA
A=
2−2 2
222
112
.
S J A =S−1·J·S J
Ann∈N
Vh,i:V×V→RV
k · k :V→R, v 7→ phv, vid:V×V→R, d(v, ˜v) := k˜v−vk
hv, wi=1
2(kv+wk2− kvk2− kwk2)v, w ∈V
kv+wk2+kv−wk2= 2kvk2+ 2kwk2v, w ∈V

d(v, z)≤d(v, w) + d(w, z)v, w, z ∈V
d(v, z)≥ |d(v, w)−d(w, z)|v, w, z ∈V
R3
hx, yi=x1y1+x2y2+x2y3+x3y2+ 2x3y3
hx, yi=x1y1+x2y2−3x2y3+x3y2+x3y3
hx, yi=x1y1+x2y2+x2
3y2
n a1, . . . , an∈R
h−,−i :Rn×Rn−→ R
(x, y)7−→
n
X
i=1
aixiyi
V=C0([a, b],R) [a, b]
Rα > 0
Φ : V×V−→ R
(f, g)7−→ α·Zb
a
f(x)g(x)dx
A An
n≥1
Vh−,−i T:V−→ V
V×V−→ R,(v, w)7−→ hT v, T wi
V f, g ∈V
Zb
a
f(x)g(x)dx ≤Zb
a
f(x)2dx1/2 ·Zb
a
g(x)2dx1/2