Algèbre linéaire Année 2016

n∈N?σ∈Snσ uσRn
uσ:
x1
...
xn
→
xσ(1)
...
xσ(n)
.
det u=(σ)
a1, ..., an∈Cb1, ..., bn∈CA= (aibj)1≤i,j≤n
a b c
c a b
b c a
,
−1 1 1 1
1−1 1 1
1 1 −1 1
111−1
,
10 0 −5 15
−2 7 3 0
8 14 0 2
0−21 1 −1
.
P, Q, R, S 2
P(−1) P(0) P(1) P(2)
Q(−1) Q(0) Q(1) Q(2)
R(−1) R(0) R(1) R(2)
S(−1) S(0) S(1) S(2)
.
a, b, c ∈Rf:R→Rf(x) =
x a b c
a x c b
b c x a
c b a x
f
f
Mn(R)R
(Mn(R),+,×)
Mn(Q)Mn(Z)Mn(R)
Mn(Q)Mn(Z)
Mn(R)
A∈ Mn(Z)n≥1An= 0
det A= 0 det(In−A) = 1
f(t) = det(In−tA)

ai,j :R→R1≤i≤n1≤j≤n n2
A(t) = (ai,j (t))1≤i,j≤nd(t) : t7→ det A(t)
x∈R
A(x) =
1 + x1· · · 1
1 1 + x
1
1· · · 1 1 + x
.
A= (xA, yA)B= (xB, yB)R2D
A B M = (x, y)R2M∈ D
D
A= (xA, yA, zA), B = (xB, yB, zB), C = (xC, yC, zC)∈R3M= (x, y, z)∈R3
−−→
AB −→ AC PA B C
M= (x, y, z)∈R3M∈ P
P
u, v ∈R3w∈R3x∈R3
det(u, v, x) =< w, x > < ., . > R3
w=u∧v
∧:R3×R3→R3
(u, v)7→ u∧v
∧
u∧v=−v∧u
∧u, v, w ∈R3λ∈R(λu +v)∧w=
λ(u∧w)+(v∧w)u∧(λv +w) = λ(u∧v)+(u∧w)
< u ∧v, u >=< u ∧v, v >= 0
u, v ∈R3\ {0} ku∧vk= sin θkuk kvkθ u
v
u, v (u, v, w)R3
n∈Nz0, ..., zn∈C
V(z0, ..., zn) =
1 1 · · · 1
z0z1· · · zn
zn
0zn 1· · · zn
n
u, v ∈Rn\ {0}θ∈[0, π]
cos θ=<u,v>
kukkvk

V(z1, ..., zn) 0 ≤i<j≤n zi=zj
zi
f:C→Cf(z) = V(z0, ..., zn−1, z)f
n n
fV(z0, ..., zn)V(z0, ..., zn−1)
V(z1, ..., zn)
z0, ..., zn+1 ∈Cw0, ..., wn∈C
P n P (zi) = wi
n∈N?a1, ..., an, b1, ..., bn∈C1≤i < j ≤n ai+bj6= 0
C(a1, b1, ..., an, bn) =
1
a1+b1
1
a1+b2· · · 1 a1+bn
1
a2+b1
1
a2+b2· · · 1 a2+bn
1
an+b1
1
an+b2· · · 1 an+bn
C(a1, b1, ..., an, bn) 0 ≤i < j ≤n ai=aj
bi=bjaibi
f:C\ {−a1, ..., −an} → Cf(z) = C(a1, b1, ..., an, z)
f f(z) = αP(z)
Q(z)α∈C?P Q
n−1n P Q
limz→−an(z+an)f(z)
αC(a1, b1, ..., an−1, bn−1)
C(a1, b1, ..., an, bn)
n∈N Cn[X]Cn
Cn[X]C[X]
Cn[X]n, m ∈N
Cn[X]×Cm[X]
P=Pn
i=0 aiXiQ=Pm i=0 biXin m
an6= 0 bm6= 0
uP,Q :Cm−1[X]×Cn−1[X]→Cn+m−1[X]
(U, V )7→ UP +V Q .
u
B0= ((1,0),(X, 0), ..., (Xm−1,0),(0,1),(0, X), ..., (0, Xn−1))
B1= (1, X, ..., Xn+m−1)

UP,Q uP,Q
P Q Res(P, Q) = det UP,Q P
∆(P)=(−1) n(n−1)
2a−1
nRes(P, P 0)
P=aX2+bX +c a, b, c ∈Ca6= 0 Q=X3+pX +q p, q ∈C
P Q Res(P, Q)=0 P
Q
E0:x2−xy +y2−1,E1: 2x2+y2−y−2.
y0Py0=X2−y0X+y2
0−1
Qy0= 2X2+y2−y−2E0E1
{y=y0}
n
n