ALGÈBRE IV - CORRECTIONS DES QUELQUES EXERCICES DES

q
B
q B
MB(q) = 
211
110
101
.
X= (x1, x2, x3)tu
q(u) = XtMB(q)X.
α E
MB(α)B
α(u) = MB(α)X.
Y v E
hv, ui=YtX.
hα(u), ui= (MB(α)X)tX=XtMB(α)tX,
hα(u), ui=q(u)
MB(α)t= MB(q),
MB(α) = MB(q),
MB(q)α
α
MB(q) 0,1,3
B0
MB(q)
V0= vect (1,−1,−1)t,
V1= vect (0,1,−1)t,
V3= vect (2,1,1)t.
P
B0
P=
√3
30√6 3
−√3
3
√2
2
√6
6
−√3
3−√2 2
√6
6
.

PtMB(q)P= diag(0,1,3).
P P −1=PtB0
α
MB0(α) = P−1MB(q)P= diag(0,1,3).
ϕ A1, A2, A, B1, B2, B E
λ1, λ2∈C
ϕ(λ1A1+λ2A2, B) = tr((λ1A1+λ2A2)tB) =
=λ1tr(At
1B) + λ2tr(At 2B) =
=λ1ϕ(A1, B) + λ2ϕ(A2, B),
ϕ
ϕ(A, µ1B1+µ2B2) = tr(Atµ1B1+µ2B2) =
=µ1tr(AtB1) + µ2tr(AtB2) =
=µ1ϕ(A, B1) + µ2ϕ(A, B2),
ϕ
ϕ(A, B) = tr(AtB) = tr(BtA) = tr(BtA) = ϕ(A, B),
ϕ
ϕ ϕ
Btr(At
iAj)
tr(At
iAj)Aj16
MB(ϕ) = I4.
A=a1a2
a3a4,
ϕ(A, A) = (a1, a2, a3, a4)MB(ϕ)(a1, a2, a3, a4)t=
=a1a1+a2a2+a3a3+a4a4=
=ka1k2+ka2k2+ka3k2+ka4k2,
ϕ
MB(f)
JA1J=1i
−i1, JA2J=i1 1−i,
JA3J=−i1
1i, JA4J=1−i i1.
f
f(Aj)f(A1) = A1+iA2−iA3+A4
MB(f) = 
1i−i1
i1 1 −i
−i1 1 i
1−i i 1
.

Bf∗
MB(f∗) = MB(f)t=
1−i i 1
−i1 1 i
i1 1 −i
1i−i1
.
f∗
A, B ∈E
hf∗(A), Bi=hA, f(B)i= tr(AtJBJ) = tr(AtJBJ).
hJAJ, Bi= tr(JAJ)tB) = tr(JtAtJtB) = tr(JAtJB),
J
J
tr(JAtJB) = tr((J−1JAtJBJ) = tr(AtJBJ) = hJAJ, Bi.
f∗(A) = JAJ.
f
MB(f)MB(f∗) MB(f∗)MB(f)
f(f∗(A)) = JJAJJ = 2I2A2I2= 4A,
f∗(f(A)) = JJAJJ = 2I2A2I2= 4A,
f◦f∗=f∗◦f
fMB(f)
diag(λ1, λ2, λ3, λ4)λj
f
det(MB(f)−λI4)
2 2 2i, −2i
g= 1/2f g4= idEg
1±1±i g 2
{1,1,1,−1},{1,1, i, −i}.
g2= idE