CAPES Algèbre linéaire Formes quadratiques

B V
q
u∈V, v ∈V
q(q(u)v−B(u, v)u) = q(u) [q(u)q(v)−B(u, v)B(v, u)] .
q
B(u, v)B(v, u)≤q(u)q(v).
Rn[X]
n n ≥1P, Q ∈Rn[X]
B(P, Q) = Z1
0
tP (t)Q0(t)dt q(P) = B(P, P ).
B
q q
qBn= (1, X, . . . , Xn)
n= 2 q q
R2[X]q

V=½µ a b
c d ¶∈ M2(R); a−d= 0¾J=µ1 1 1−1¶.
B:V×V−→ R
M, N ∈ M2(R)
B(M, N) = Tr(M JN).
B
B=µµ 1 0
0 1 ¶,µ0 1 0 0 ¶,µ0 0 1 0 ¶¶
V
Bq
M∈ M2(R)q(M) = B(M, M).
q q
F⊥q F
F=½µ a0
0d¶∈ M2(R); a−d= 0¾.
q:R3−→ Rq(x, y, z) = 2x2+y2−z2+ 3xy −4xz.
q:R3−→ Rq(x, y, z) = x2+y2−az2+ 3xy −bxz +yz.
a, b ∈R
q:R4−→ R
q(x, y, z, t) = x2+(1+2λ−µ)y2+ (1 + λ)z2+(1+2λ+µ)t2
+2xy + 2xz −2xt + 2(1 −λ)yz −2(1 + λ)yt + 2(λ−1)zt.
λ, µ ∈R
q:R5−→ Rq(x, y, z, t, s) = xy −xt +yz −yt +ys +zt −zs + 2st.
Rn[X]
n n ≥1d1≤d≤n
P∈Rn[X]
q(P) = (P2)(d)(0),
P(d)d P
Tr

qRn[X]
B={1, X, . . . , Xn}
q
q q
Rn[X]
qP P