Sujet avec corrigé de composition de

N N∗
Z R
Mm(R)mRMm,p(R)
(m,p)R
ERm∈N∗E∗
ImMm(R)M∈ Mm(R)
PM= det (M−XIn)∈R[X]M
Mm,1(R)Rm
A∈ Mm(R)t A A Am(R)
Mm(R)Sm(R)
S++
m(R)
GLn(R)
O(m) = Om(R)
n∈N∗Mn(C) GLn(C)
U(n)
J=Jn0In
−In0
M2n(R)J

Sp(2n)
ω:E×E→R
e= (e1, . . . ,em)E
ω e M = (ω(ei,ej))1≤i,j≤m∈ Mm(R)
(x,y)∈E2, ω(x,y) = t XMY X
x e Y y
e
t
XMY =x1· · · xm
ω(e1,e1)· · · ω(e1,em)
ω(em,e1)· · · ω(em,em)
y1
ym
=x1· · · xm
y1ω(e1,e1) + · · · +ymω(e1,em)
y1ω(em,e1) + · · · +ymω(em,em)
t
XMY =x1· · · xm
ω(e1,y1e1+. . . ymem)
ω(em,y1e1+· · · +ymem)
=x1· · · xm
ω(e1,y)
ω(em,y)
= (x1ω(e1,y) + · · · +xmω(em,y))
t
XMY = (ω(x1e1+· · · +xmem,y))
= (ω(x,y))

ω NE={x∈E, ∀y∈
E, ω(x,y) = 0} {0E}
M ω e
•ω
•M
• ∀f∈E∗,∃!x∈E, ∀y∈E, ω(x,y) = f(y)
(i) ⇔(iii)
ψ:
E→E∗
x7→ E→R
y7→ ω(x,y)
ψR
mker(ψ) = NE
ω⇔NE={0E}
⇔ker(ψ) = {0E}
⇔ψ
⇔ψ
⇔ ∀f∈E∗,∃!x∈E, ∀y∈E, ω(x,y) = f(y)
(i) ⇔(ii)
ω⇔NE={0E}
⇔X∈ Mm,1(R)|∀Y∈ Mm,1(R)nt XMY = 0
⇔∀X∈ Mm,1(R),∀Y∈ Mm,1(R),t XMY = 0⇒X= 0
⇔∀X∈(R),t XM = 0 ⇒X= 0⇔∀X∈ Mm,1(R),t MX = 0 ⇒W= 0⇔ker(t M) = {0}
⇔t M
⇔M
E ω
•ω(x,y)∈E2, ω(x,y) = −ω(y,x)

•ω
R2n
E0=R2n
ω0E0×E0
R2nJ ω0E0
ω0
ω(x,y) = t XJY
=tt XJY 
=t Yt Jt (t X)
J
=t Y(−J)X
=−t Y JX
=−ω(y,x)
X=X1
X2∈ M2n,1
(X1,X2)∈(Mn,1)2Y=Y1
Y2∈ M2n,1(Y1,Y2)∈(Mn,1)2
t
XMY = (X1X2)0In
−In0 Y1 Y2
= (X1X2)Y2
−Y1
=X1Y2−X2Y1
t
Y MX =Y1X2−Y2X1
t
Y MX =t XMY
ω0
Jdet J= det(0) ×det(0) −det(−In) det(In) =
(−1)n+1 det(In)2= (−1)n+1 6= 0 J ω0

ω E
F E F
ω F ◦={x∈E, ∀y∈F, ω(x,y)=0}
F E F ◦
dim F◦= dim E−dim F F E
F⊕F◦=E
ω
F◦E
ψ:
E→F∗
x7→ F→R
y7→ ω(x,y)
ψRF◦
dim(E) = dim(F∗) + dim(ker(ψ))
dim(F∗) = dim(F) ker(ψ) = F◦
dim F◦= dim E−dim F
F E ω F ×F
F⊕F◦=E
˜ω ω F ×F
˜ω
F⊕F◦=E
˜ω F ⊕F◦=E
dim(F) + dim(F◦= dim(E)F⊕F◦=E⇔
F∩F◦={0E}
F∩F◦={0E}
x0∈F∩F◦
x0∈F◦∀y∈F,ω(x,y)=0
x0∈NF={x∈F|∀y∈F, ω(x,y)=0}˜ω
NF={0E}x0= 0E
F∩F◦={0E}F⊕F◦=E